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Phytoplankton-zooplankton interactions : data analysis and modelling (with particular reference to Ocean.. Parslow, John Stanley 1981-12-31

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PHYTOPLANKTON-ZOOPLANKTON AND P  MODELLING  I N T E R A C T I O N S : DATA  (WITH P A R T I C U L A R  (5Q°N,145°S)  AND  R E F E R E N C E TO  CONTR3LLED  ANALYSIS  OCEAN  ECOSYSTEM EXPERIMENTS  By  JOHN B.Sc.,  A  THESIS THE  STANLEY  University  SUBMITTED  DOCTOR  PARSLOK  of Queensland,  IN P A R T I A L  REQUIREMENTS  FOR  OF  THE  1974  F U L F I L MENT OF DEGREE  OF  PHILOSOPHY  in THE  F A C U L T Y OF  (DEPARTMENT  We  accept to  THE  ©  this  GRADUATE OF  the rs^uired  John  STUDIES  MATHEMATICS)  thesis  U N I V E R S I T Y OF  STATION  as conforming standard  BRITISH  COLUMBIA  January  1981  Stanley  Parslov,  1981  ).  In  presenting  requirements of  British  it  freely  agree for  this for  an  available  that  I  by  understood  that  his  that  or  be  her or  shall  Date  dint,  flfrrtl  , t<?f>.  University  and  study.  I  copying  granted  by  the  of  publication be  allowed  Columbia  of  this  It  this  without  make  further  head  representatives.  not  The U n i v e r s i t y of B r i t i s h 2075 W e s b r o o k P l a c e Vancouver, Canada V6T 1W5  the  the  shall  W#7H£ MflncS  of  at  of  Library  permission.  Department  fulfilment  the  extensive  may  copying  f i n a n c i a l gain  degree  reference  for  purposes  or  partial  agree  for  permission  scholarly  in  advanced  Columbia,  department  for  thesis  thesis  of  my  is  thesis my  written  i i  ABSTRACT The Pacific  anomalous h a s been  classical grazing of  attributed  models  a basis  present  a t O.S.P.  C.mg and  have  Chia" .ly" ),  production time  surface  model  series  compared  levels  are sensitive  could  previous  i s low a n d m i g h t  Phytoplankton  or  (about  radiation  by p h y s i c a l  low a n n u a l Predicted  production production  rate,  'Marra'  P r e d i c t e d year  be h i g h e r  are  A- n u m e r i c a l  of r e s p i r a t i o n  response  to  (<0.5 mg  in solar  and d r i v e n  yields  type  are analysed  efficiency  changes  results  of net p r o d u c t i o n .  this  low c o n c e n t r a t i o n s  estimates.  light  interaction,  i n the s p r i n g .  to the choice  simple  and  effect to  i f variation  results  year  i n Secchi  depth  be a c c o u n t e d f o r . In  a phytoplankton-zooplankton  production  model,  strategies  a r e both  identification parameters is  on t h e s e  of a r a p i d  a doubling  variation  with  model.  to seasonal  the weatherships  levels  in  In  to obtain  a t O.S.P.  uniformly  inhibition  based  from  introduction  i n the Subarctic  control.  low p h o t o s y n t h e t i c  adapt  1  most  realistic  i n almost  3  show  cycle  t o be n e c e s s a r y  observations  f o r a more  Chla.m" ), 1  to grazing  a r e found  Weathership  provide  mg  seasonal  of the phytoplankton-zooplankton  thresholds  control.  0.4  phytoplankton  spring  techniques  into  estimates. to late  problems  thresholds  necessary  f o r cohorts  introduced  these  grazing  (biomass) and  for grazing  are adapted  model  control.  grazers.  the phytoplankton-zooplankton As a  summer  result, and f a l l  are associated with  Systems population  Cohort model  attention i s shifted where  on t h e  over-wintering  to estimate  of the dominant  based  sensitivity  the over-wintering  and  structure using  from the stability  departure  of the  dominant An  grazers. approximate  mathematical  analysis  nutrient-phytoplankton-zooplankton and  e l a b o r a t i o n of  cyclic  solutions  constant  shown  i s focused  on  nutrient-limited  behavioural)  also  used  parameters compared  from  to  model,  explanation densities A found  on of  as  a  the the  of  observed  stable  case  of  an  diffusivity idealised  and  growth  in  a  terms  and  food  techniques  growth  estimates  are  and  discussed with  provides  of  a  a consistent  p e r s i s t e n c e at  Bessel  undergoing growth  low  functions is diffusion  profile.  complete  biological  Subsurface  rate,  profile.  phytoplankton-nutrient  series  relatively  rates.  sinking  growth  The  to  and  declining  statistical  estimates,  profile,  parameters  maxima  are  Solutions to  equations  for  also  coupled  corresponding  and Non-  d i s c u s s i o n of  previously reported  s t e p - f u n c t i o n growth  exponential  approach  nutrient-phytoplankton-zooplankton  exponential  physical  population  to  and  for  thresholds.  bloom)  feeding  series.  time  populations  effects and  limitation  grazing  response  the  Stable  (physiological  phytoplankton  solution  allows  varying  A  results.  nutrient-limited equilibrium.  dimensionalization of  and  the  parameter  for phytoplankton i n the  time  (1974)  explanation  identification  zooplankton  experiments.  mathematical  sinking  Systems  values  of  broader  the  nutrient  (spring  zooplankton  copepod  limitations  future based  CEPEX  allows  under  absence  a  Steele's  numerical  transient and  for  estimate  exist  the  the  literature  deterministic mind  in  i s proposed. to  with  authors' to  cycle  framework  concentrations are  are  mortality rates  Attention the  previous  model  of  to  on  the  profiles  constant  for  an  obtained  for  non-linear subsurface  the  maxima  of the n u t r i e n t - t r a p type  boundary-layer and  magnitude  approximate from  techniques. of these  theory  a complex  simulation  obtained  The dependence  maxima  agrees  are also  well  on p a r a m e t e r s with  model.  using  of the depth, i s explored.  previously published  shape The results  V  CONTENTS  ABSTRACT  i i  L I S T OF TABLES  ix  L I S T OF FIGURES  xi  ACKNOWLEDGEMENTS  XX i  PREFACE  xx i i  CHAPTER 1.  INTRODUCTION AND ANALYSIS OF SIMPLE GRAZING MODELS.  1.1 G e n e r a l  Introduction  .•  1.2 P h y s i c a l O c e a n o g r a p h y o f t h e S u b a r c t i c 1.3 B i o l o g y  of the S u b a r c t i c  Pacific  Pacific  1 2 8  1.4 S i m p l e P h y t o p l a n k t o n - Z o o p l a n k t o n M o d e l s f o r t h e S u b a r c t i c Pacific  14  1.5 P r e v i e w  of Chapters  CHAPTER 2.  QUALITATIVE ANALYSIS OF A COMPLEX SIMULATION MODEL.  2.1 I n t r o d u c t i o n  2-4  33  ...38  2.2 M o d e l a n d A n a l y s i s  40  2.3 S i m u l a t i o n  47  Results  2.4 C o n c l u s i o n s  CHAPTER 3.  59  PHYTOPLANKTON AT O.S.P.: DATA ANALYSIS AND MODELLING.  3.1 I n t r o d u c t i o n  68  3.2 D a t a A n a l y s i s  68  3.2.1  D e s c r i p t i o n of t h e Data Set  68  3.2.2  C h l o r o p h y l l Data  69  3.2.3  14  C Data.  75  3.2.4 N i t r a t e D a t a 3.2.5  101  N i t r a t e Concentration  and P r o d u c t i o n  3.3 A P h y t o p l a n k t o n G r o w t h M o d e l 3.3.1  Introduction  3.3.2  Physical Structure  3.3.3  Biological  3.3.4 S i m u l a t i o n 3.3.5  CHAPTER 4.  104 I l l I l l  Basis  and D r i v i n g V a r i a b l e s  f o r t h e Model  117  Results  Primary Production  113  124 and N i t r a t e D e p l e t i o n  148  HERBIVOROUS ZOOPLANKTON AT O.S.P.: DATA ANALYSIS AND  MODELLING. 4.1 P a r a m e t e r E s t i m a t i o n  150  4.1.1  D e s c r i p t i o n of Data.  4.1.2  Review of Parameter E s t i m a t i o n  4.1.3  A p p l i c a t i o n t o O.S.P.  4.1.4 S t a t i s t i c a l  150 Techniques  Data  151 156  Considerations  160  4.1.5  Results  f o r Calanus plumchrus  -..161  4.1.6  Results  f o r Calanus c r i s t a t u s  168  4.1.7 O t h e r S p e c i e s 4.1.8  Secondary Production  174 Estimates  175  4.2 B i o m a s s M o d e l f o r Z o o p l a n k t o n 4.2.1  Introduction  4.2.2  Formulation  4.2.3  Choice of Zooplankton Parameters  180 180  of a Biomass G r a z i n g  Model  181 183  4.2.4 S i m u l a t i o n R e s u l t s a n d D i s c u s s i o n  187  4.3 A C o h o r t M o d e l f o r Z o o p l a n k t o n 4.3.1  198  Introduction  198  4.3.2 M o d e l F o r m u l a t i o n  200  4.3.3 P a r a m e t e r s  201  4.3.4 S i m u l a t i o n R e s u l t s a n d D i s c u s s i o n  202  4.4 C o n c l u s i o n s  CHAPTER 5.  224  PARAMETER ESTIMATION AND S T A B I L I T Y FOR A CEPEX  ENCLOSURE. 5.1 I n t r o d u c t i o n  250  5.2 E s t i m a t i o n  of Parameters i n a Z o o p l a n k t o n Growth Model.253  5.3 E s t i m a t i o n  Results  261  5.4 S t a b i l i t y o f t h e P h y t o p l a n k t o n - z o o p l a n k t o n 5.5 C o n c l u s i o n s  CHAPTER 6.  Interaction.298 ;  D I F F U S I O N , SINKING AND GROWTH OF  303  PHYTOPLANKTON.  6.1 I n t r o d u c t i o n  310  6.2 R e v i e w o f a S i m p l e M o d e l  312  6.3 A More R e a l i s t i c  321  6.4 E f f e c t  Model  of a Mixed Layer  6.5 A G e n e r a l  327  Necessary C o n d i t i o n F o r Growth  6.6 D i s c u s s i o n  CHAPTER 7.  337  MATHEMATICAL ANALYSIS OF DEEP CHLOROPHYLL  7.1 I n t r o d u c t i o n 7.2 A P h y t o p l a n k t o n - N u t r i e n t  336  MAXIMA. 342  Model  344  vi i i  7.3  Effect  of  a Mixed Layer  354  7.4  Effect  of  Nutrient  356  Dependent S i n k i n g R a t e s  7.5 D i s c u s s i o n  358  CHAPTER 8'.  373  BIBLIOGRAPHY  CONCLUDING REMARKS  377  L I S T OF TABLES.  Table I .  Parameters  Table I I . 1964  used  i n S t e e l e ' s Model  (2.1)  P r e d i c t e d a n n u a l p r i m a r y p r o d u c t i o n a t O.S.P.,  t o 1976  128  Table I I I . Monthly  means o f p r e d i c t e d d a i l y  p r o d u c t i o n a t O.S.P. u s i n g t h r e e l i g h t  net primary  a d a p t a t i o n time  scales  134  Table IV.  T a b l e V.  Parameter  Parameter  estimates f o r Calanus  plumchrus  estimates f o r Calanus c r i s t a t u s  Table V I .  Secondary  Table V I I .  Average  production estimates  relative  Table V I I I .  Final  parameter  Calanus  179  e s t i m a t e s a n d SSQ e r r o r s f o r 269  E x p o n e n t i a l growth  Final  169  233  Pseudocalanus  T a b l e X.  163  s e a s o n a l abundance of  microzooplankton  Table IX.  42  parameter  rates  f o rCalanus  275  e s t i m a t e s a n d SSQ e r r o r s f o r 278  Table  XI.  Paracalanus  Final  parameter  estimates  and  SSQ  errors  for .  xi  L I S T OF FIGURES.  Figure  1.  Seasonal c y c l e  in vertical  s t r u c t u r e a t O.S.P.  Figure  2.  Schematic diagram of s u r f a c e c u r r e n t s and  domains i n t h e S u b a r c t i c P a c i f i c  Figure  3.  Seasonal c y c l e  D e p a r t u r e Bay, S t r a i t  Figure and  4.  6  i n c h l o r o p h y l l a a t O.S.P. a n d  of Georgia.  . . .'  16  S e a s o n a l c y c l e a t O.S.P. i n p r i m a r y  zooplankton  standing  5  production  stock  17  Figure  5.  Phase p l a n e  Figure  6.  Type I I I f u n c t i o n a l  Figure  7.  Phase p l a n e  portraits  f o r t h e s y s t e m 1.6  29  Figure  8.  Phase p l a n e  portraits  f o r t h e s y s t e m 2.3  45  Figure  9.  Comparison of g e n e r a t i o n  and  those  Figure  obtained  10.  i n numerical  Stable c y c l i c  GX=0.05 a n d ( a ) F=0.4. ( c ) F = 0.6  portraits  f o r t h e s y s t e m 1.2  21  responses  times  28  from e q u a t i o n  2.4  s o l u t i o n s o f 2.2  48  s o l u t i o n s o f t h e s y s t e m 2.2 f o r  ( b ) F=0.2  49 51  Figure  11.  B=10. w i t h :  B e h a v i o u r o f t h e s y s t e m 2.1 f o r GX=0.05 a n d ( a ) F=0.4, D=100.  (b) F=0.2, D=100. D=175.  Figure  (stable cycle)  (unstable o s c i l l a t i o n s ) .  ...  52  ( c ) F=0.2,  (stable cycle)  12.  Simulation  s y s t e m 2.1 w i t h :  53  o f s p r i n g a n d summer u s i n g t h e  ( a ) B=10., F=0.4, D=100. a n d GX=0.05 .....58  (b) B=10., F=0.4, D=175. a n d GX=0.05  60  ( c ) B=10., F=0.4, D=175. a n d GX=0.04  61  (d) B=10., F = 0.2, D=250., GX=0.05 a n d E=0.3  62  Figure  13.  weatherships  Figure  14.  Surface  observations  of c h l o r o p h y l l a from t h e  a t O.S.P  Surface  smooth, s e a s o n a l  70  c h l o r o p h y l l a ( c r u i s e medians,  f i t plus  residuals)  annual  for:  (a) a n d ( b ) 1964-68  72  ( c ) and ( d ) 1969-76  73  Figure  15.  Observations  from t h e w e a t h e r s h i p s  Figure  16.  Surface  smooth, s e a s o n a l  Figure  17.  a  a t O.S.P  76  C h i b/Chl a ( c r u i s e medians,  annual  f i t p l u s r e s i d u a l s ) 1969-76  Surface  smooth, s e a s o n a l  of C h i b/Chl a and C h i c / C h l  C h i c/Chl  f i t plus  a ( c r u i s e medians,  r e s i d u a l s ) 1969-76  77  annual 78  Figure  18.  Surface observations  of p r o d u c t i v i t y  per unit  C h i a f r o m t h e w e a t h e r s h i p s a t O.S.P  80  Figure  19.  Frequency  81  Figure  20.  P ( 0 ) ( c r u i s e medians, a n n u a l smooth,  fit  plus  histograms f o rP(0)  seasonal  r e s i d u a l s ) f o r : ( a ) a n d ( b ) 1964-68  82  ( c ) a n d ( d ) 1969-76  83  Figure  21.  Scatter plot  of P(0) vs I  Figure  22.  Scatter plot  of A vs I  Figure  23.  Depth p r o f i l e s  Figure  24.  M o n t h l y a v e r a g e s o f ex.  94  Figure  25.  S c a t t e r p l o t o f B v s I. f o r 1964-68  96  Figure  26.  Scatter plot  98  Figure  27.  Monthly averages of e s t i m a t e s of  l i g h t adaptation (b) B  f o r 1964-68  f o r 1964-68  of P  of P  parameters:  M A X  28.  vs I  0  f o r 1964-68  (a) B  Surface observations  f r o m t h e w e a t h e r s h i p s a t O.S.P.  91  92  99 100  s  Figure  85  of n i t r a t e  concentration 102  xiv  Figure  29.  Surface n i t r a t e concentrations  a n n u a l smooth, s e a s o n a l  Figure  30.  ( c r u i s e medians,  f i t plus residuals)  Nitrate concentrations  103  from depth  ( l a y e r a v e r a g e s , a n n u a l smooths, s e a s o n a l  fits  profiles plus  residuals)  105  (a) 0 - 20m  106  (b) 20  - 40m  107  ( c ) 40  - 80m  108  (d) 80  - 130m  109  (e) 130  Figure  - 200m  31.  110  P(0)  v s low  s u r f a c e n i t r a t e v a l u e s , May  to  O c t o b e r , 1964-68.  Figure  32.  112  Time s e r i e s of t o t a l  t e m p e r a t u r e and  solar radiation,  surface  mixed l a y e r depth used t o d r i v e s i m u l a t i o n  model  Figure  125  33.  Predicted daily  net  production  using  standard  parameter set  Figure  34.  127  P r e d i c t e d net  mixed l a y e r C:Chl a r a t i o s e t and  Figure  seasonal  35.  adaptation  As  light  production, f o r 1976  mixed l a y e r Chi a  using  standard  and  parameter  adaptation  f o r F i g 34,  but  with  130  'instantaneous'  light 131  XV  Figure  36.  As  Figure  37.  P r e d i c t e d C:Chl a r a t i o  1976  using  Figure y  38.  f o r F i g 34,  oc = 0.5  and  but  w i t h 3-day a d a p t a t i o n  133  i n the mixed l a y e r f o r  B =2.0  Predicted daily  136  net  production  on  increasing  t o 0.1  Figure  139  39.  Predicted daily  C:Chl a r a t i o  Figure  40.  for  net  production  0 c = l . O , B =2.0  production  parameter set w i t h Marra e f f e c t  introduced  41.  Observed Secchi  mixed l a y e r  w i t h c o n s t r a i n t V^IOO.  net  Figure  Predicted daily  and  for  ..140  standard 142  d e p t h s a t O.S.P. v s  time  of  year  143  Figure  42.  Predicted daily  p a r a m e t e r s e t and (b) l o w e r  Figure ratio  production  using  (a) u p p e r e n v e l o p e t o S e c c h i  envelope to Secchi  43.  net  standard  depths.  depths  Predicted seasonal  145  c y c l e i n mixed l a y e r C:Chl a  using optimality c r i t e r i o n  Figure  44.  regions  ...144  P r o j e c t i o n o f a p p r o x i m a t e 95%  147  confidence  f o r Calanus plumchrus parameter estimates  on  (a)  (9,R ) plane  164  (b)  U,6)  165  T  plane  xvi  Figure  45.  regions on  (a)  (b)  confidence  f o r Calanus c r i s t a t u s parameter estimates  171  (6,R ) plane  172  T  (y,B)  Figure  plane  46.  lengths  Figure  P r o j e c t i o n o f a p p r o x i m a t e 95%  1;  47.  173  Regression on  c o e f f i c i e n t s W,-  vs  corresponding  log-log scale  177  P r e d i c t e d m i x e d l a y e r C h i a and  b i o m a s s f o r 1976  using  standard  Figure  48.  E f f e c t of d e c r e a s i n g  Figure  49.  E f f e c t of  herbivore  parameter set  188  D on F i g 47  191  introducing over-wintering  strategy  i n F i g 47 Figure 25 mg  Figure  194  50. Cm"  3  51.  E f f e c t of  reducing  spring recruitment  on F i g 49  195  P r e d i c t e d C h i a and  grazer  standing  stock  b i o m a s s m o d e l w i t h Type I f u n c t i o n a l r e s p o n s e and recruitment  Figure 1976,  52.  to  for  spring  ;  P r e d i c t e d C h i a and  using weight t h r e s h o l d s  zooplankton  for departure  199  biomass f o r i n cohort  model 203  Figure  53.  As  f o r F i g 52,  but  with fixed residence  times  xvi i  and  lower  Figure Fig  g r o w t h a n d m o r t a l i t y r a t e s f o r C.  54.  E f f e c t of q u a d r u p l i n g  Cristatus.  ..209  spring recruitment i n  53  Figure  211  55.  E f f e c t of h a l v i n g s p r i n g r e c r u i t m e n t  i n F i g 53. 213  Figure  56.  E f f e c t of i n c r e a s i n g m o r t a l i t y r a t e s f o r  C a l a n u s i n F i g 53  Figure  57.  214  E f f e c t of d e c r e a s i n g  g r a z i n g p a r a m e t e r s CO and  D i n F i g 53.  Figure  58.  216  E f f e c t of i n c r e a s i n g m e t a b o l i c  r a t e s i n F i g 57. 217  Figure  59.  zooplankton fixed  Figure  c a r b o n f o r 1964-76, u s i n g  recruitment  60.  a t O.S.P.  Figure  P r e d i c t e d m i x e d l a y e r C h i a and  61.  62.  t h e p a r a m e t e r s and  l e v e l s o f F i g 53  Observed zooplankton  wet w e i g h t s  219  (10 day  means)  1956-78  221  P r e d i c t e d mixed l a y e r C h i a and  c a r b o n f o r 1964-76 u s i n g c o u p l e d  Figure  total  Average seasonal  zooplankton  recruitment  c y c l e s i n (a) o b s e r v e d  223  xvi ii  z o o p l a n k t o n wet w e i g h t s  (10 day means, 1 9 5 6 - 7 8 )  228  (b)  ingestion  v a r i a b l e V I (10 day means, 1 9 6 9 - 7 8 )  229  (c)  ingestion  v a r i a b l e V2  230  Figure  63.  (10 day means, 1 9 6 9 - 7 8 )  A n n u a l v a r i a t i o n (10 d a y means) i n  (a)  ingestion  v a r i a b l e VI  237  (b)  ingestion  v a r i a b l e V2  238  Figure  64.  Phytoplankton carbon  (0-20m a v e r a g e ) i n CEE5  f o r W -=0.4 jug C, W j ^ =2.0 pq C ...262  F i g u r e -65.  J2fj(W),  Figure  66.  Functional  Figure  67.  Observed d e n s i t i e s of Pseudocalanus  Figure  68.  Comparison of p r e d i c t i o n s and o b s e r v a t i o n s  Pseudocalanus (b)  trial  ^(W)  (a) t r i a l  ..257-  J  response data  f o r Pseusocalanus  ....264  i n CEE5 ...266  for  1  270  6  271  Figure  69.  O b s e r v e d d e n s i t i e s o f C a l a n u s i n CEE5  Figure  70.  Comparison of p r e d i c t i o n s and o b s e r v a t i o n s  for  Calanus.  (a) t r i a l  (b)  trial  4  281  (c) t r i a l  6  283  (d)  7  284  trial  1  276  279  xix  Figure  71.  O b s e r v e d d e n s i t i e s o f P a r a c a l a n u s i n CEE5.  Figure  72.  Observed d e n s i t i e s of t o t a l  Figure  73.  Comparison  Paracalanus  (a) t r i a l  nauplii  ...288  i n CEE5 ...289  of p r e d i c t i o n s and o b s e r v a t i o n s f o r 1  292  (b) t r i a l  2  293  (c) t r i a l  3  295  (d) t r i a l  4  297  Figure after  74.  Predicted phytoplankton concentrations  i n CEE5  d a y 30  302  Figure  75.  An i l l u s t r a t i o n  of the e i g e n c o n d i t i o n  Figure  76.  C o n t o u r p l o t o f t h e f u n c t i o n JVioo, Sg )  317  Figure  77.  C h a r a c t e r i s t i c p r o f i l e s p{S)  319  Figure  78.  An i l l u s t r a t i o n  Figure  79.  C o n t o u r p l o t o f t h e f u n c t i o n A. (UJ,/3)  Figure  80.  C h a r a c t e r i s t i c phytoplankton p r o f i l e  of the e i g e n c o n d i t i o n  6.6.  6.9.  ...324  .325  1  f o r c o = 0.7,  13 = 20  328  Figure (a)  ...316  81.  j3=0  Contour p l o t of the f u n c t i o n  _A (/l,ip f o r 3  332  XX  (b)  (3=1  333  (c)  /?=3  334  (d)  (3 = 10  335  F i g u r e 82.  Contour  plot  of  S,/(3 v s co,(3 f o r  F i g u r e 83.  Characteristic  F i g u r e 84.  Phytoplankton p r o f i l e s  profiles  P(S)  P(<p)  S=0.1  359  v s S//5  361  v s <p f o r 6 = 1.  and  9 = 0.1  365  F i g u r e 85. 0=1.  and  Contour  p l o t s of f T  <p v s c  5=0.1  (3,UJ* f o r  and  0.1  F i g u r e 86.  366  Contour  p l o t s of  ^  vs  j3,co  for  =w  and  w =5.w  368  4  F i g u r e 87.  C o m p a r i s o n of p h y t o p l a n k t o n p r o f i l e s  by c o m p l e x s i m u l a t i o n model and  predicted  by a s y m p t o t i c a n a l y s i s .  ..372  xx i  ACKNOWLEDGEMENTS I  would  like  advice  o f my  thesis  other have  members to thank  Biological data study  to acknowledge  o f my Mr  thesis  station, in this  of  parameter  stimulating N.C.Sonntag.  and  studies  studentship  from  T.R.Parsons,  Mr  O.D.Kennedy  assisting  with  providing  t o an  also  were  supported  Australia.  and  the  discussions. the much  and by  I  Pacific of  the  initial,  J.B.L.Matthews  i n Canada CSIRO,  of  support  in i t s interpretation.  owes much Dr  and  for helpful  for kindly  estimation  collaboration My  and  Nanaimo,  study  Dr  committee  J.D.Fulton  used  graduate  supervisor,  the encouragement,  Mr a  post-  raw The  xxi i  PREFACE E c o l o g y has been v a r i o u s l y c a l l e d a s o f t  science,  an  i m m a t u r e s c i e n c e , o r e v e n a p r e - s c i e n c e (Kuhn,1970) by with the c l a s s i c a l  fields  of p h y s i c s and c h e m i s t r y .  d i s c i p l i n e s have a c q u i r e d a t h e o r e t i c a l c o r e w h i c h (and u s u a l l y m a t h e m a t i c a l l y ) f o r m u l a t e d . in  these f i e l d s  and  without nervous  The  latter  i s rigorously  A mathematician  i s e s s e n t i a l l y concerned with deducing  i m p l i c a t i o n s of t h i s mathematics  comparison  the  theory a c c o r d i n g to the p r i n c i p l e s  logic.  He c a n , f o r t h e most p a r t ,  b a c k w a r d g l a n c e s t o see  working  of  face t h i s  i f t h e t h e o r y has  task  just  changed. T h i s i s not t o say t h a t the upon h i m .  The  q u e s t i o n about observation.  ' r e a l w o r l d ' does not  mathematical problem the r e a l Moreover,  system  intrude  usually corresponds to a  i n v o l v i n g experiment  and/or  i n many i n t e r e s t i n g c a s e s , an  exact  general s o l u t i o n to the a p p r o p r i a t e mathematical problem  cannot  be f o u n d and  the d e r i v a t i o n  problem  is necessary  ( a p p l i e d mathematics  art  o f an a p p r o x i m a t e , t r a c t a b l e has been c a l l e d  in part  of j u d i c i o u s a p p r o x i m a t i o n ' ( G r e e n s p a n , 1 9 6 9 ) ) .  'the  It is this  p r o c e d u r e w h i c h demands o f s u c c e s s f u l a p p l i e d m a t h e m a t i c i a n s f a m i l i a r i t y w i t h the s c i e n c e , or t e c h n i c a l mathematical A soft of  'physical  intuition',  a  as w e l l  as  ability.  s c i e n c e s u c h as e c o l o g y i s d i s t i n g u i s h e d by  an a c c e p t e d t h e o r e t i c a l c o r e .  F o r any p a r t i c u l a r  i t s lack  real  system,  t h e r e i s no u n i v e r s a l l y a c c e p t e d , m a t h e m a t i c a l l y f o r m u l a t e d model.  An a p p l i e d m a t h e m a t i c i a n  two c h o i c e s . analyze  v e n t u r i n g i n t o such a f i e l d  He c a n t a k e a p r e v i o u s l y p u b l i s h e d m o d e l  i t s mathematical p r o p e r t i e s , paying l i t t l e  has  and  attention  to  xxi i i  the  (sensitive)  develop large  which  body  issue  of  is increasingly  of  ecological  mathematical  elegance,  which  correspond  do  theory (it  not  can  be  i s used  reality  The  who  study  this  thesis),  but  modelling, of  his  with  the  wonder  familiarity  theory  reality.  There  idealised  to  any  system.  of  a  the  real  particular,  gap  can is a  considerable  traditional  between  biologist  own  the  alternative  models This  real  system  theory  lacking  and  become  a  and  formal  computer, and  then with  very  version  science  is a  i f biologists, science,  who  i s , someone  reality.  i s not.  While  With  the  s i m u l a t i o n models  advanced  presumably  need  the  prerequisite for  ability  little  into  that  of  l a r g e , complex  studied with  their  plunge  modeller;  mathematical  biological  i s to  s o p h i s t i c a t e d mathematical  constructed  might  field  of  A  training.  formulates  advent  the  controversy  familiarity  be  i n the  accuracy. from  often  in detail  mathematician's  biological  theory,  concerning  overwhelm  mathematical  divorced  useful  in  may  its biological  can  mathematics. have  One  more  applied mathematicians  at  all. The  answer  computers analysts  have in  to not  the  clumsy  tool  set  parameter  of  simulation the  usual  question  i s yes,  for  the  same  eliminated a l l mathematicians  physical sciences.  which  produces values.  models,  involving  a n a l y s i s ) can  properties.  a  It  A  a  many  i s the  a  in  the  parameters  small  give  outcome  number very  task  of  of  for a  case and  of  that  numerical is a specific complicated  interactions,  simulations  misleading the  reason  except  s i m u l a t i o n model  particular  Particularly  c o n s i d e r a t i o n of  (sensitivity model's  this  applied  p i c t u r e of  the  mathematician  xxi v  to  uncover  parameter  the general values  mathematical such  structure.  physical  are  not usually  qualitative Some  idealised  models models  formulation biological  input  both  In  an  human  construct thing  rigorously  observes  simulation  a n d may  between  As i n equations and  2 ) , or of  involve with  of  simple  the  considerable  a clear  picture  t h e m o d e l Vs.  new has  mathematical  and parameter  model  the  system  being  copy,  t h e human  values),  mind  make  perfectly  which  of the  allowed  us t o  n o t b e a t a l l t h e same  I  share  sufficiently point  about  t o agree  event)  with  complex  to construct a  p i e c e s and even  (an u n l i k e l y  The  one t o d e f i n e  one.  : an a b i l i t y  component  intelligence,  or the computer.  personal  t h e same  of  a blueprint  system,  is a difficult  i n ecology from  given  would  of understanding  and would  models  of the n o t i o n  computer  be a h i g h l y  notion  complicated natural  field  the analysis  (Chapter  case,  (structure  that  working  'understanding'  statement  high.  t o model  involve  models  In each  discussion  or of a complex  Weizenbaum's  i t s  biologically.  an e x a c t ,  of  4).  and input  as understanding  concept  in a  both  parameter  i s very  Others  of o r i g i n a l  (Chapter  interesting  mind,  in  of approximation  thesis  1,6,7).  of the r e l a t i o n s h i p  Weizenbaum(1976)  his  (Chapters  interpreted  both  on  important.  c o n s t r u c t e d by o t h e r s  (behaviour)  important  solutions  and techniques  are  and a n a l y s i s  sought  output  available  structure  general  behaviour  are implicit  uncertainty regarding  sciences, exact  analysis  which  i s especially  of the s e c t i o n s of t h i s  simulation  been  where  This  and t h e a p p r o p r i a t e model  the  of a model's  and the assumptions  as ecology,  values  dependence  t o have  large, i t mimic  i s n o t t h e same  XXV  thing  as an  understanding  qualitative, in and  this  thesis,  number These  All  at a  simple,  of understanding In  addition of types  range  techniques  statistical  of data  analysis  relatively  ambitious of data  model.  analysis  I n many  underlying  model  facilitate  the computation  analysis  of t h e i r  computers now  deliberately  complicated  natural  in this of t h i s  models  from  or experimental  thesis  represents  approach.  The  techniques 5.  underlying  usually  the  linear, to  and the  development  algorithms  of  (Benson,1978)  b i o l o g i c a l l y meaningful observations  systems. an  a  thesis.  of Chapter  estimates  properties.  lacks.  techniques,  simple,  of parameter  model-fitting  complex  on a n  of the c l a s s i c a l  statistical  i n more  descriptive  a r e based  that  of models,  in this  procedures  the p o s s i b i l i t y . o f estimating  parameters  potential  kept  and numerical  offers  analysis  was  provide  model  and a n a l y s i s  model-fitting  assumptions  level,  are presented  the  and c a r r i e d out  simulation  conventional  that  between  graphical  a complex  find  above  relationships often  which  I  described  to the construction  from  the rather  analyses  by u n c o v e r i n g  predictions  sense  to  approximate  of the ecosystem.  Much  exploration  on of the data  of the  1  CHAPTER 1 INTRODUCTION AND ANALYSIS OF SIMPLE GRAZING MODELS.  1.1 G e n e r a l The an  problems d i s c u s s e d  initial  Subarctic section their  Introduction.  interest Pacific.  i n t h e marine p l a n k t o n i c ecosystem of the The g e n e r a l  i s intended  as a b r i e f  interrelationships.  p h y s i c a l and b i o l o g i c a l given  i n t h i s t h e s i s have a l l a r i s e n o u t o f  i n t r o d u c t i o n given survey  A detailed  of t h e problems t r e a t e d and i n t r o d u c t i o n to the  oceanography of the S u b a r c t i c P a c i f i c i s  i n S e c t i o n s 1.2 a n d 1.3 r e s p e c t i v e l y .  Comprehensive  i n t r o d u c t i o n s t o the d e r i v e d problems a r e l e f t corresponding  this  study  commenced, i n 1 9 7 6 , m o d e l l i n g o f  c y c l e s i n p l a n k t o n i c n u t r i e n t - p l a n t - h e r b i v o r e systems,  particularly  t h e phenomena o f s p r i n g a n d f a l l  b l o o m s , was w e l l e s t a b l i s h e d . empirical seasonal  standing  anomaly.  A number o f s i m p l e  seemed r i p e  l o n g time  11  observed  showing l i t t l e  o r no  s t o c k , s t o o d o u t a s an  et al,1960;  Heinrich,1962)  f o r a more r i g o r o u s e x a m i n a t i o n  s e r i e s of b i o l o g i c a l  Ocean S t a t i o n  the  v e r b a l h y p o t h e s e s h a d been p r o p o s e d  problem v i a modelling, e s p e c i a l l y  at  understanding,  account f o r t h i s c y c l e ( M c A l l i s t e r the time  phytoplankton  t h i s background of  c y c l e i n the Subarctic P a c i f i c , i n phytoplankton  and  Against  knowledge and t h e o r e t i c a l  variation  to  to the  chapters.  At t h e time seasonal  in this  of the  i n view of t h e e x i s t e n c e of a  observations  from t h e  weatherships  P" ( h e n c e f o r t h a b b r e v i a t e d O.S.P. ) a t 50°N,  145°W i n t h e S u b a r c t i c  Pacific.  P r e l i m i n a r y c o n s i d e r a t i o n s , u s i n g some c l a s s i c a l  population-  2  i n t e r a c t i o n models  ( C h a p t e r 1) s u g g e s t e d  c o u l d be i m p o r t a n t a t O.S.P.  that grazing thresholds  T h i s r e s u l t p r o m p t e d an  q u a l i t a t i v e a n a l y s i s o f a complex n u m e r i c a l model L a n d r y , 1 9 7 6 ) whose b e h a v i o u r  investigation  useful  (Steele,1974;  i n computer s i m u l a t i o n had provoked  some d i s c u s s i o n o f t h r e s h o l d s . a n a l y s i s proved  approximate  The i n s i g h t s o b t a i n e d f r o m  i n a more q u a n t i t a t i v e , d e t a i l e d m o d e l l i n g  of ecosystem  d y n a m i c s a t O.S.P. ( C h a p t e r s 3 , 4 ) .  A n a l y s i s of the time s e r i e s of b i o l o g i c a l weatherships, conducted  as p a r t of t h i s  o b s e r v a t i o n s from t h e  investigation, involved  some n o v e l s t a t i s t i c a l  problems,  systems i d e n t i f i c a t i o n  technique f o r estimating population  parameters (Chapter series  from z o o p l a n k t o n  including  time s e r i e s  4) t o o v e r c o m e i n c o n s i s t e n c i e s  f r o m O.S.P.  t h e a d a p t a t i o n of a  ( P a r s l o w e t a l ,1979) i n t h e .zooplankton  A v a r i a n t of the parameter  from a c o n t r o l l e d ecosystem  CEPEX programme response in  (Chapter  parameters  time  enclosure studied during the  5 ) , i n an a t t e m p t  to estimate  functional  r e l e v a n t t o b i o l o g i c a l q u e s t i o n s which  arose  t h e t h e o r e t i c a l a n a l y s i s o f c h a p t e r two. Chapters  6 and 7 i n v o l v e a t h e o r e t i c a l a n a l y s i s of  phytoplankton p o p u l a t i o n s undergoing and  time  estimation  t e c h n i q u e was a p p l i e d t o p h y t o p l a n k t o n a n d z o o p l a n k t o n series  this  nutrient  limitation.  While  c o n s i d e r a t i o n of l i g h t - l i m i t e d proved  t o be o f more i n t e r e s t  environments  growth,  diffusion,  sinking  o r i g i n a l l y m o t i v a t e d by growth  a t O.S.P. , t h e r e s u l t s  i n cases of l o w - t u r b u l e n c e  and s u b - s u r f a c e c h l o r o p h y l l  maxima.  1.2 P h y s i c a l O c e a n o g r a p h y o f t h e S u b a r c t i c P a c i f i c . This introduction  t o t h e p h y s i c a l oceanography of t h e  3  Subarctic  Pacific  models c o n s i d e r e d  i s i n t e n d e d as background t o t h e e c o l o g i c a l later.  Two a s p e c t s a r e c o n s i d e r e d h e r e .  The  s e a s o n a l c y c l e i n w a t e r c o l u m n s t r u c t u r e a t O.S.P. i s i m p o r t a n t i n the m o d e l l i n g of primary O.S.P. r e l a t i v e  production there.  t o the broad  Subarctic Pacific  c i r c u l a t i o n p a t t e r n s of the  i s d i s c u s s e d i n an a s s e s s m e n t o f h o r i z o n t a l  a d v e c t i v e e f f e c t s on b i o l o g i c a l The  weatherships'  profiles, basis  The l o c a t i o n o f  processes.  o b s e r v a t i o n s of temperature  and s a l i n i t y  a s w e l l as m e t e o r o l o g i c a l v a r i a b l e s , have p r o v i d e d a  f o r a number o f t h e o r e t i c a l a n d e m p i r i c a l s t u d i e s o f  seasonal  c y c l e s a t O.S.P. , p a r t i c u l a r l y  t h e heat  budget  a s s o c i a t e d w i t h f o r m a t i o n and breakdown of t h e s e a s o n a l thermocline presented O.S.P.  ( T a b a t a , 1 9 6 5 ; Denman,1972).  quantitatively  i n Chapter  w i t h a .model of' p r i m a r y  3; a b r i e f  description  On t h e b a s i s o f s a l i n i t y , t h r e e permanent zones  a lower  1000m.  be  production at  i s given  here.  Dodimead e t a l ( 1 9 6 3 )  distinguish  : an u p p e r zone f r o m 0 t o 100m, a h a l o c l i n e  f r o m 100 t o 200m i n w h i c h s a l i n i t y and  This cycle w i l l  i n c r e a s e s f r o m 32.8%<>to 33.8%o,  zone, i n which s a l i n i t y  i n c r e a s e s s l o w l y t o 34.4%,at  The t o p o f t h e h a l o c l i n e c o r r e s p o n d s  t o t h e maximum  depth  of t h e s u r f a c e m i x e d l a y e r , a t t a i n e d i n M a r c h a t the. e n d o f t h e p e r i o d of n e t heat thermocline April  l o s s through  i s e s t a b l i s h e d over  the surface.  A  seasonal  t h e p e r i o d of net heat  t o September, w i t h t h e mixed l a y e r being  typically  30m d e e p a t t h e e n d o f t h i s p e r i o d .  Surface  increase  this period.  f r o m a b o u t 5°C t o 13°C o v e r  gain,  from  about  temperatures The s e a s o n a l  t h e r m o c l i n e a n d an a s s o c i a t e d s e a s o n a l h a l o c l i n e a r e e r o d e d t h e c o o l i n g p e r i o d , f r o m O c t o b e r t o M a r c h , by c o n v e c t i v e  over  overturn  4  and  storm  activity.  characteristic some  of  This  the  geographical  seasonal  oceanic  variation  cycle  Subarctic  in  the  ( F i g 1) Pacific  magnitude  is  qualitatively  although  and  timing  there of  is  the  cycle. Two  reviews  Pacific ocean  have  In  domains  were  Subarctic surface body  the  of  identified  which  Bering  about  and  miles/day not  in  by  42°N  off  the of  to The  the  coast  Alaska,  Alaskan  eventually  Gyre, join  circulation  of  at  2  that  while East the  part  north  nautical  miles/day  to  4  To  the  north  again,  the  east  at  about  part  of  the  Oyashio  eastward  flowing  part  the  the  the  and  part  northward  intense  Alaskan moves  Kamchatka entire  2  travelling  flowing  relatively  of  to  of  and  Kuroshio  America,  and  a  for  the  These  North  the  time  Immediately  moving  Part  systems  isohaline  current  Steam).  current  al  vertical  domain.  forming  et  almost  mixing  of  Pacific  limit  Kuroshio.  California  Alaskan  form  Sea  the  North  the  southern  c o n s i s t i n g of the  into  The  latitude.  as  Subarctic  Favorite  principal  moving, . e a s t w a r d  with  the  investigations  ,1963;  2.  an  by  of  International al  Fig  formed  and  mix  Gulf  (the  4-6  as  of  the  described  the  Oyashio.  review,  defined  water  et  was  form  to  (Dodimead  current  south  south  the  transition  divide  current  by  salmon  the  currents  around  of  as  does  to  part  about  0  currents  nautical  as  was  34% at  oceanography  published  summarized  warmer  Subarctic  physical  earlier  region  of  Oyashio  the  Commission  ,1976).  was  been  environment  Fisheries  a  of  boundary  Stream north  into  current  system  turns  or  the the  i s given  as  years.  O.S.P.  is  south-easterly  located  north  edge  the  of  of  the  Alaskan  transition Gyre.  The  zone,  on  following  the quote  is  net h e a t i n g  0  wind mixing  p\ convective mixing  seasonal  50  thermocline  CL CD  a 100  —  opimarcnt  J  F M A  M  J  a'ocline  J A S O N  D  Month Figure  1.  Seasonal cycle i n v e r t i c a l structure (Adapted from Tabata, 1965).  a t O.S.P.  7  particularly  relevant f o rmodelling  O.S.P. a l t h o u g h salinity '.  i t was p r o m p t e d by c o n s i d e r a t i o n s o f  the geostrophic  flow  i n the v i c i n i t y  o f Ocean  "P" i s g e n e r a l l y z o n a l , b u t , more i m p o r t a n t l y , s l o w (2  miles/day). (seasonal)  Hence, w i t h i n t h e time p e r i o d c o n s i d e r e d ' the waters i n the area  conditions similar be  seasonal  and temperature c y c l e s t h e r e : . .  Station  the p l a n k t o n i c ecosystem a t  regarded  t o those  as having  are subjected  here,  to climatic  a t Ocean S t a t i o n "P"; t h u s ,  resided there.  '  they can  . . ' (Dodimead e t a l  ,1963) . The pattern  second review  ( F a v o r i t e e_t a l ,1976) p r e s e n t s  f o r the general  circulation,  t h e c u r r e n t a n d domain s t r u c t u r e s . feature presented interpretation  w h i c h may a f f e c t  f o r O.S.P.  a small clockwise Charlotte  w i t h f u r t h e r e l a b o r a t i o n of T h e r e i s , h o w e v e r , a new  b i o l o g i c a l modelling  Subarctic Current.  and d a t a -  The d y n a m i c t o p o g r a p h y f o r J u l y shows  gyre i n the s u r f a c e c i r c u l a t i o n  Islands, east  a similar  of t h e n o r t h - s o u t h  Associated  o f f t h e Queen  branching  of the  w i t h t h i s gyre i s a ' D i l u t e  Domain' o f w a t e r s h o w i n g t h e e f f e c t s o f c o a s t a l r u n o f f . review  i s ambiguous as t o t h e westward e x t e n t  Circulation patterns total  wind-stress  o f t h i s domain .  s u g g e s t e d by d y n a m i c t o p o g r a p h y ,  t r a n s p o r t and n u m e r i c a l  t h a t t h e f l o w a t O.S.P.  from t h e e a r l i e r the  review.  o f O.S.P.  i s e s s e n t i a l l y as described  above  H o w e v e r , t h e D i l u t e Domain d e f i n e d on  b a s i s o f t h e 33%„ i s o h a l i n e c o n t o u r  t o a l m o s t 160°W a n d i n c l u d e s O.S.P. (1976),  integrated  model r e s u l t s a l l  i n d i c a t e that the c o a s t a l gyre l i e s w e l l t o the east and  The  the Subarctic Current  a t 100m e x t e n d s w e s t w a r d  I n F i g 41 o f F a v o r i t e et. a l  i s portrayed  as d i v i d i n g  to the  8  west o f t h e D i l u t e Domain a n d O.S.P. The salinity  D i l u t e Domain i s d i l u t e  w a t e r s o f t h e R i d g e Domain t o t h e n o r t h w e s t  upwelling the  i n the Alaskan  southeast  clear  by c o m p a r i s o n w i t h t h e h i g h e r  Gyre),  t h e c o a s t a l u p w e l l i n g domains t o  and t h e t r a n s i t i o n  t o what e x t e n t  t h e lower  domain t o t h e s o u t h .  salinity  the annual excess of p r e c i p i t a t i o n throughout the eastern  Subarctic  s u p e r i m p o s e d on a z o n a l predominantly  flow.  (due t o  a t O.S.P.  I t i s not  may be due t o  over e v a p o r a t i o n  which  occurs  (Dodimead e t a l , 1 9 6 3 ) ,  I f t h e f l o w t h r o u g h O.S.P. i s  z o n a l and p a r t of t h e e a s t w a r d - f l o w i n g  Subarctic  c u r r e n t , t h e n a s i n d i c a t e d by t h e q u o t e a b o v e , a m o d e l o f seasonal  changes e i t h e r i n the p h y s i c a l s t r u c t u r e of the water  c o l u m n o r i n t h e p l a n k t o n i c c o m m u n i t y may r e a s o n a b l y c o n s t r u c t e d -without account. on  I f O.S.P.  coastal circulation, and  taking large-scale horizontal advection  The m o d e l s p r e s e n t e d  this basis.  others  a more e x p l i c i t  really  of S u b a r c t i c oceanic treatment  A very considerable Pacific  exhaustive  approach-.  the c u r r e n t  above  c y c l e a t O.S.P.  r a t h e r than c o a s t a l  is  waters),  e f f e c t s may be r e q u i r e d .  Pacific.  literature  e x i s t s on t h e b i o l o g y o f t h e  survey  makes no p r e t e n c e  The f o l l o w i n g i n f o r m a t i o n  as an i n t r o d u c t i o n t o t h e q u e s t i o n s modelling  created  f o r reasons c i t e d  seasonal  of advective  and t h i s b r i e f  review.  into  lies within a small-scale  w h i c h seems u n l i k e l y  1.3 B i o l o g y o f t h e S u b a r c t i c  Subarctic  i n t h i s t h e s i s h a v e been  ( f o r example, t h e n i t r a t e  characteristic  be  i s presented  partly  addressed here using a  I t i s also intended  s t a t e of b i o l o g i c a l  a t an  p a r t l y a s an o v e r v i e w o f  knowledge, so t h a t a  reader  9  unfamiliar with this here, in  with their  l i t e r a t u r e can p l a c e  necessary  broader p e r s p e c t i v e .  surrounding  Probably in  will  Relevant  be m e n t i o n e d where  o v e r 20 y e a r s  ago ( S e m i n a , 1 9 5 8 ) , a n d c o n f i r m e d a t O.S.P.  August of 1959, m a c r o n u t r i e n t s  c a r b o n .were m e a s u r e d  ugat.l"  1  normally  observed exponential some 40 t i m e s  zooplankton phytoplankton in  grazing  weathership  a cruise in July  silicate,  e t a_l , 1 9 6 0 ) . 1  Nutrient  NO", >16 u g a t . l "  1  levels  SiO , P  >1.2  when n u t r i e n t d e p l e t i o n m i g h t  This observation,  together  increase of phytoplankton  with the  i n batch  once g r a z e r s  t h e a r g u m e n t , a d v a n c e d by S e m i n a  culture were  (1960),  that  i s r e s p o n s i b l e f o r the constancy of  concentration  the Subarctic P a c i f i c  i n the Subarctic.  The s e a s o n a l  cycle  was c o n t r a s t e d w i t h t h e ' s p r i n g b l o o m '  c y c l e s of c o a s t a l a r e a s and t h e N o r t h (1962).  by t h e  During  the i n i t i a l concentration  removed, s u p p o r t e d  f o r t h e B e r i n g Sea  (nitrate,  (>6 u g a t . l "  PO^), a t a time of year be e x p e c t e d .  ecosystem  ( c h l o r o p h y l l a) a n d p a r t i c u l a t e o r g a n i c  (McAllister  were c o n s i s t e n t l y h i g h  to  T h i s was r e p o r t e d  (Parsons,1965).  phosphate), phytoplankton  from  i s t h e absence of a s p r i n g i n c r e a s e i n  abundance.  and  information  appropriate.  phytoplankton  observations  intensively  t h e best-known f e a t u r e of t h e p l a n k t o n i c  the sub-Arctic P a c i f i c  focus,  be c o n c e r n e d  w h i c h i s by f a r t h e most i n the region.  areas w i l l  considered  l i m i t a t i o n s and r a t h e r narrow  Much o f t h e r e v i e w  w i t h s t u d i e s a t O.S.P. sampled ocean s t a t i o n  t h e models  Atlantic  by H e i n r i c h  He a t t r i b u t e d t h e d i f f e r e n c e t o t h e l i f e h i s t o r y  s t r a t e g i e s of t h e dominant h e r b i v o r o u s plumchrus and Calanus c r i s t a t u s ,  copepods,  i n the Subarctic  A p r e l i m i n a r y p i c t u r e of zooplankton  Calanus Pacific.  abundance,  composition  10  and  v e r t i c a l d i s t r i b u t i o n a t O.S.P.  was g i v e n by M c A l l i s t e r  ( 1 9 6 1 ) , b a s e d on s u r f a c e t r a w l s a n d v e r t i c a l weatherships zooplankton from A p r i l April and  f r o m 1956 t o 1 9 5 8 .  h a u l s made f r o m t h e  He d e s c r i b e d a w i n t e r minimum i n  b i o m a s s f r o m December t o M a r c h a n d a summer maximum to July.  S u r f a c e t r a w l s were d o m i n a t e d by c o p e p o d s i n  a n d May a n d by a m p h i p o d s i n J u n e a n d J u l y a n d i n November  December.  predominantly times.  Vertical copepods  hauls  (150m t o s u r f a c e ) were  ( c a 75%) a n d c h a e t o g n a t h s  Two l a y e r s o f maximum a b u n d a n c e o f z o o p l a n k t o n  found,  above and below t h e permanent h a l o c l i n e .  weight  i n t h e t o p 150m r a n g e d  mg.m  i n May, 1 9 5 7 .  -3  (15%) a t a l l  f r o m c a 10 m g . n r  3  were  Zooplankton  wet  i n w i n t e r t o 80  P e r h a p s t h e s i n g l e most c o m p r e h e n s i v e s t u d y o f t h e b i o l o g y of.the Subarctic Pacific  t o date  T h i s study of p r e d a t o r - p r e y  i s t h a t of L e B r a s s e u r  relationships  i n the Gulf of Alaska  included a d e t a i l e d a n a l y s i s of nine years from O.S.P.  The l i f e  (1956-64) of  zooplankton  data  zooplankton  s p e c i e s were d i s c u s s e d , b a s e d on a v e r a g e  h i s t o r i e s of t h e dominant  p a t t e r n s o f a b u n d a n c e by s t a g e a s r e f l e c t e d s u r f a c e tows.  t o biomass, as noted  Subarctic Pacific  important  in vertical  h a u l s and  b u n g i , makes a s i g n i f i c a n t  f o r other l o c a t i o n s  (Heinrich,1962; Sekiguchi,1975).  copepods ( Pseudocalanus, are  seasonal  N e x t t o t h e two s p e c i e s m e n t i o n e d a b o v e , a t h i r d  l a r g e h e r b i v o r o u s copepod, Eucalanus contribution  (1969).  in fall  Calanus  and w i n t e r .  Smaller  pacif icus, Metridia pacif ica) A mixed c o l l e c t i o n of  p l a n k t o n i c primary c a r n i v o r e s , i n c l u d i n g chaetognaths elegans, Eukrohnia  i n the  (Sagitta  hamata) and t h e t r a c h y m e d u s a A g l a n t h a  were a l s o d i s c u s s e d .  An a t t e m p t  was made t o e s t i m a t e t h e  digitale  11  standing  s t o c k s of forage organisms  annual carbon the t e r t i a r y Due  (myctophid  f l u x through a s i m p l i f i e d consumers  (salmon,  and s q u i d ) and t h e  f o o d web l e a d i n g up t o  baleen whales and p o m f r e t ) .  t o t h e n a t u r e o f t h e s a m p l i n g p r o g r a m a t O.S.P. , much  more i s known o f t h e h e r b i v o r o u s a n d c a r n i v o r o u s z o o p l a n k t o n of t h e t r o p h i c  l e v e l s above o r below.  F o r example, t h e  p o p u l a t i o n d y n a m i c s o f s q u i d , an i m p o r t a n t carnivore, are virtually  unknown.  primary/secondary  The s e a s o n a l p a t t e r n i n  s p e c i e s c o m p o s i t i o n o f p h y t o p l a n k t o n a t O.S.P. c o m p a r a t i v e l y p o o r l y known.  than  Weathership  i s also  phytoplankton  samples  have n o t y e t been a n a l y s e d q u a n t i t a t i v e l y .  I n f o r m a t i o n on  abundance o f n e t p h y t o p l a n k t o n  from t h e s h i p s of  o p p o r t u n i t y program zooplankton data J u l y , August,  (Venrick,1971)  (discussed l a t e r ) .  1959 ( M c A l l i s t e r  c r u i s e o f 1969 ( P a r s o n s , 1 9 7 2 ) stations dominated  i s available  (R. W a t e r s ,  and from w e a t h e r s h i p  micro-  O b s e r v a t i o n s a t O.S.P.  in  e t a l , 1 9 6 0 ) , d u r i n g t h e Transpac and p r e l i m i n a r y a n a l y s i s of L i n e P  p e r s . comm. ) s u g g e s t  i n b i o m a s s by s m a l l f l a g e l l a t e s ,  that phytoplankton are l e s s t h a n 10 pm i n  diameter. T h e r e h a v e been a number o f t h e o r e t i c a l a n a l y s e s o f p r i m a r y and  secondary  p r o d u c t i o n a t O.S.P.  or i n i t s v i c i n i t y .  The  s e a s o n a l c y c l e o f c h l o r o p h y l l a a n d p r i m a r y p r o d u c t i o n a t O.S.P. ( b a s e d on (1965).  1 4  C  uptake  m e a s u r e m e n t s ) was d e s c r i b e d by  In t h a t paper,  (Sverdrup,1953)  was u s e d  Sverdrup's  critical  depth  model  t o explore the interaction  depth, mixed l a y e r depth and s u r f a c e i r r a d i a n c e . n u t r i e n t s were d e s c r i b e d a s n o n - l i m i t i n g  Parsons  of s e c c h i  A l l macro-  throughout  the Gulf of  A l a s k a a n d a n a n n u a l p r i m a r y p r o d u c t i o n o f c a 60 g C m "  2  was  12  estimated,  b a s e d on  1 4  C  measurements and t h e s e a s o n a l d e c r e a s e i n  n i t r a t e and phosphate. expanded t o c o v e r LeBrasseur spring this  The c r i t i c a l  the f u l l  (1968).  G u l f of A l a s k a  later  by P a r s o n s a n d  The l a r g e - s c a l e p a t t e r n i n t h e t i m i n g o f t h e  increase i n zooplankton  standing  s t o c k was p r e d i c t e d i n  study. R e s u l t s of primary  and  d e p t h a p p r o a c h was  production  s t u d i e s on t h e T r a n s p a c c r u i s e  a s h i p s of o p p o r t u n i t y program conducted  from American M a i l  L i n e c r u i s e s b e t w e e n S e a t t l e a n d Yokohama were r e p o r t e d by Parsons and Anderson and on  Menzel's  (1970).  (1962) e q u a t i o n  the Transpac c r u i s e .  ranging  to overestimate  (ug C.ug C h i a ^ . l y " ) 1  production efficiency  in this  p r o v i d e d a b e t t e r f i t f o r the Transpac data, but values f r o m 0.07 t o 3.1 (jug C.ug C h i a ^ . l y " ) 1  o b t a i n agreement w i t h d a t a The  conversion  a t O.S.P.  estimated  of primary  h a s been s t u d i e d by M c A l l i s t e r  (1969,1972).  production  dark  (estimated  respiration, Zooplankton  of zooplankton  uncertainty  i n zooplankton  1 4  C  production  measurements), minus  was c a l c u l a t e d a s a  stock.  Assimilated ration  as secondary p r o d u c t i o n . respiration  of secondary p r o d u c t i o n  t o a maximum o f 23g C . m ~ . y r . was c h o s e n a s most  likely.  _ 1  minus  Because of  r a t e s (eg S t e e l e , 1 9 7 4 ) ,  represented  the d i f f e r e n c e of  l a r g e , u n c e r t a i n q u a n t i t i e s and ranged from n e g a t i v e 2  Daily  was assumed t o be i n g e s t e d by  standing  was t a k e n  estimates  from  respiration  respiration  to  cruises. t o secondary  zooplankton. fraction  f r o m t h e AML  were n e c e s s a r y  production  phytoplankton  two  was f o u n d  form of S t e e l e  Reducing the photosynthetic  p a r a m e t e r f r o m 0.24 t o 0.17 equation  A depth-integrated  An e s t i m a t e  values  o f 13 g C . i r r . y r ~ 2  l  13  For the be  t h e p u r p o s e s o f t h i s t h e s i s , an i n t e r e s t i n g summary o f  current  state  of b i o l o g i c a l knowledge f o r t h i s l o c a t i o n can  o b t a i n e d by a s s e s s i n g  for a d e t a i l e d , the  r a t i o n a l m e c h a n i s t i c model  ecosystem there.  goal,  i t s s t r e n g t h s and weaknesses as a  Whether a model of t h i s k i n d  i s debatable  provide a useful  1 4  C  way t o s t r u c t u r e  productivity  isa  desirable  quantitative  ( P i a t t e t a_l , 1 9 7 5 ) , b u t i t d o e s  There i s a long time s e r i e s a,  ( P i a t t e_t a l ,1975) o f  p a r t i c u l a r l y i f we w i s h t o make s u c c e s s f u l  predictions,  basis  an a p p r o a c h t o e x i s t i n g  data.  of o b s e r v a t i o n s of c h l o r o p h y l l  a n d m a c r o n u t r i e n t s a t O.S.P.  However, f o r a  model o f p h y t o p l a n k t o n d y n a m i c s , a knowledge o f v a r i a t i o n s i n p h y t o p l a n k t o n carbon and c a r b o n r c h l o r o p h y l l photosynthesis vs l i g h t scales),  relationship  attempt w i l l available  The  time  and i n l i m i t i n g  i f a n y , w o u l d a l l be d e s i r a b l e .  An  be made t o i n f e r some o f t h e s e i n d i r e c t l y f r o m t h e  o b s e r v a t i o n s i n C h a p t e r 3.  and/or s p e c i e s dependent e f f e c t s require  (on s h o r t a n d l o n g  i n phytoplankton r e s p i r a t i o n rates  e f f e c t s of m i c r o n u t r i e n t s ,  r a t i o s , i n the  Investigation  of s i z e  i n biological interactions  would  o b s e r v a t i o n s a t a comparable l e v e l of d e t a i l . l o n g t i m e s e r i e s o f 150m v e r t i c a l h a u l s  zooplankton data source, providing  information  i s the p r i n c i p a l  on t o t a l wet  w e i g h t , s p e c i e s c o m p o s i t i o n a n d some s t a g e a n d / o r s i z e T h e s e d a t a a r e s u p p o r t e d by s t u d i e s p a t t e r n s of v e r t i c a l m i g r a t i o n  on s e a s o n a l a n d d i u r n a l  (Vinogradov,1968; F r o s t and  McCrone,1974; S e k i g u c h i , 1 9 7 5 ; Marlowe and M i l l e r , 1 9 7 5 ) . m e a s u r e m e n t s h a v e been made o f g r a z i n g  rates,  locations  f a r f r o m O.S.P.  Some  chemical  c o m p o s i t i o n and r e s p i r a t i o n of t h e dominant h e r b i v o r e s in coastal  structure.  primarily  ( P a r s o n s e t a l ,1969;  14  Ikeda,1972;  Taguchi and I s h i i , 1 9 7 2 ; F u l t o n , 1 9 7 3 ; Ikeda,1977).  H o w e v e r , i n f o r m a t i o n on t h e f u n c t i o n a l a n d n u m e r i c a l r e s p o n s e s o f t h e s e dominant coastal  copepods i s poor compared w i t h b e t t e r  s p e c i e s such as Calanus p a c i f i c u s ( P a f f e n h o f f e r , 1 9 7 0 ;  Frost,1972,1975) The  average  or Pseudocalanus  Kennedy,1972), very poorly  ( P a f f e n h o f f e r and H a r r i s , 1 9 7 6 ) .  s e a s o n a l p a t t e r n o f m i c r o z o o p l a n k t o n ( r e t a i n e d by 44  jam mesh) a t O.S.P.  of  h a s been r e p o r t e d  ( L e B r a s s e u r and  b u t t h e a u t e c o l o g y o f most o f t h e s e o r g a n i s m s i s  known.  A similar  s t a t e of ignorance e x i s t s  the primary c a r n i v o r e s mentioned  abundance of f o r a g e organisms study of forage organisms  above,  i s poorly  i s c o m p l i c a t e d by t h e i r  known.  very  The  large  (multi-year) generation  F o r such l o n g - l i v e d organisms, and of c o u r s e f o r wide-  r a n g i n g p r e d a t o r s such as w h a l e s , salmon c o n d i t i o n s at O.S.P. variability to  f o r most  w h i l e even t h e  such as squid  d i u r n a l and s e a s o n a l m i g r a t i o n s and l o n g times.  studied  and p o m f r e t , a model of  becomes m e a n i n g l e s s a n d t h e l a r g e - s c a l e  and c i r c u l a t i o n  of the S u b a r c t i c  Pacific  would  have  be m o d e l l e d .  1.4 S i m p l e P h y t o p l a n k t o n - z o o p l a n k t o n M o d e l s  f o r the Subarctic  Pac i f i c . As t h e summary o f S e c t i o n 1.3 d e m o n s t a t e s ,  an a t t e m p t t o  c o n s t r u c t a d e t a i l e d , c o m p l e t e e c o s y s t e m m o d e l f o r O.S.P. , o r rather  f o r t h e G u l f of A l a s k a o r t h e whole  w o u l d be p r e m a t u r e ,  t o say t h e l e a s t .  l e v e l s and l a r g e h o r i z o n t a l arguments  scales,  For multiple  the simpler  of L e B r a s s e u r (1969) and Sanger  appropriate at present.  Subarctic  A much n a r r o w e r  Pacific, trophic  tropho-dynamic  (1972) a r e more range of q u e s t i o n s i s  15  addressed  h e r e , c e n t e r i n g on t h e o b s e r v e d  c h l o r o p h y l l a c o n c e n t r a t i o n s a t O.S.P. addressed  using s i m p l i f i e d  T h e s e q u e s t i o n s c a n be  s i m u l a t i o n models which  commensurate w i t h t h e p r e s e n t  s t a t e of b i o l o g i c a l  The s e a s o n a l c y c l e o f C h i a a t O.S.P. 3.  The c y c l e  contrast.  i n the S t r a i t  of Georgia  While the decrease r a d i a t i o n a t O.S.P.  increase  i n primary p r o d u c t i v i t y  (Parsons,1965)  standing stock (as (wet w e i g h t ) o f  grazing results  r a t e which  ina  balances phytoplankton  questions.  Why s h o u l d i t o c c u r  i n turn raises  even more t r o u b l e s o m e ,  i n view  i s the tight  balance  variability  Atlantic?  of t h e r a p i d growth  of p h y t o p l a n k t o n , s e a s o n a l and d a i l y v a r i a t i o n s r a t e , and t h e observed  other  i n the oceanic S u b a r c t i c P a c i f i c  not i n c o a s t a l areas, nor i n the oceanic North  growth  growth  the year.  T h i s h y p o t h e s i s of g r a z i n g c o n t r o l  Perhaps  i s not  ( F i g 4 ) . T h i s seems t o be c o n s i s t e n t w i t h  phytoplankton mortality  and  this  i n an  s e a s o n a l l y i n a s i m i l a r manner t o  the hypothesis that zooplankton  throughout  in Fig  and i n c r e a s e  i n t h e s p r i n g does r e s u l t  m e a s u r e d by C h i a ) . I n s t e a d , t h e b i o m a s s  primary p r o d u c t i v i t y  i s presented  i s also presented f o r  i n an i n c r e a s e i n p h y t o p l a n k t o n  z o o p l a n k t o n a b o v e 150m v a r i e s  a r e more knowledge.  i n mixed l a y e r depth  in solar  reflected  constancy of  i n this  i n zooplankton  rate  growth  standing stock,  r e q u i r e d between g r a z i n g and p h y t o p l a n k t o n  by t h i s h y p o t h e s i s .  The f i r s t follows.  q u e s t i o n was a n s w e r e d by H e i n r i c h  (1962) a s  The d o m i n a n t g r a z e r s i n t h e N o r t h A t l a n t i c a n d i n many  c o a s t a l a r e a s , copepods such as Calanus  f i n m a r c h i c u s and C a l a n u s  p a c i f i c u s , o v e r - w i n t e r as l a t e copepodite  s t a g e s or a d u l t s and  30  CO  JL 2 0 D  i "i  Q_ 10 O  s  /1  11  v  _o  U  0  M  M  O  N  Month  Figure 3.  Seasonal cycle i n chlorophyll a at O.S.P. ( s o l i d l i n e ) and i n Departure Bay,  Strait  of Georgia (dashed l i n e ) . (Adapted from Parsons, 1965).  D  J  F  M  A  M  J  J  A  S  O  N  MONTH Figure 4.  Seasonal cycle at O.S.P. i n (a) primary production and (b) zooplankton standing stock, (from M c A l l i s t e r , 1 9 6 9 ) .  D  18  cannot  reproduce  i n the spring u n t i l  t h e y have a c c u m u l a t e d  egg  t i s s u e by f e e d i n g on a d e q u a t e p h y t o p l a n k t o n c o n c e n t r a t i o n s . f u r t h e r p e r i o d i n which so t h a t a s i z e a b l e zooplankton  nauplii  l a g occurs  t o the spring  A  do n o t f e e d f o l l o w s r e p r o d u c t i o n ,  i n t h e n u m e r i c a l response of  i n c r e a s e i n phytoplankton growth.  The  d o m i n a n t c o p e p o d s i n t h e S u b a r c t i c , C. p l u m c h r u s a n d C. c r i s t a t u s ,  reproduce  a t depth  l a i d down t h e p r e v i o u s summer. thereby r e c r u i t e d response  i n the spring, using f a t stores An a c t i v e l y - g r o w i n g p o p u l a t i o n i s  t o t h e s u r f a c e i n t h e s p r i n g w i t h o u t any l a g i n  t o i n c r e a s i n g p h y t o p l a n k t o n growth  some c o a s t a l a r e a s s u c h a s t h e S t r a i t and  t h e Sea o f J a p a n  of G e o r g i a  Heinrich attributed  p h y t o p l a n k t o n growth consequent  failure  There a r e (Parsons,1965)  ( H e i n r i c h , 1 9 6 2 ) where C. p l u m c h r u s a n d / o r  C. c r i s t a t u s d o m i n a t e b u t a p h y t o p l a n k t o n occur.  rates.  rate  this  s p r i n g bloom does  t o an e a r l i e r  in stratified  increase i n  c o a s t a l waters  and a  i n t i m i n g of the s p r i n g r e c r u i t m e n t of these  copepods. While question will  t h i s a r g u m e n t seems t o o f f e r a s o l u t i o n  (a s u f f i c i e n t  clearly  address  ensure  t h e second  l a g i n zooplankton  a s p r i n g phytoplankton bloom), question.  t h i s q u e s t i o n of balance stability  i n the spring i t does not  of t h i s  section,  by c o n s i d e r i n g t h e  p r o p e r t i e s o f some s i m p l e b i o m a s s m o d e l s o f  regarded as r e a l i s t i c  interactions.  T h e s e s h o u l d n o t be  o r p r e d i c t i v e models but r a t h e r as  of paradigm,  consideration w i l l grazing  In the remainder  i s addressed  phytoplankton-zooplankton  statements  response  to the f i r s t  i n t h e sense  of Kuhn(1970).  Their  a l l o w us t o r e l a t e  the question of balancing  l o s s and p h y t o p l a n k t o n growth  t o some i m p o r t a n t c u r r e n t  19  issues  in marine  ecology.  A  classical  starting point  prey  interactions  will  be  regarded  differential  (as for  the the  equation  for  simple  models  of  zooplankton-phytoplankton rest  model  of  this  section)  is  predatorinteraction  the  :  x = r . x - a . x . y  1.1a  y  1.1b  =  e.a.x.y  -  m.y  (Lotka,1925).  In  t h i s model,  intrinsic  of  growth  rate  successful  encounter  density  that  a.x.y  e  the  so  predators;  is  rate  that  increase  in  predator,  rate  predators  their  of  robustness  e.a.x.y  1.1  small  the  single  total  while  the  stability  under  a  m  r  parameter  of  the  is a of  per  constant food.  per  The  originally  attracted  properties  and  an  is  the  unit  prey of  o f .consumed  consequent  in  rate  prey prey  by to  of  capita  loss  oscillatory  some  consequent  s t r u c t u r a l changes  a  consumption  conversion  absence  represents  predator  rate  represents  y,  in  system  neutral  by  x;  e f f i c i e n c y of  so  solutions  parameter  prey,  i s the  predator  for  of  the  the  interest, lack model  but  of led  to  its  replacement. The  model  linearities saturation such  a  can  to of  model  improved  account predators  for  by  introducing  further  resource-limitation  (Holling,1959).  One  of  prey  possible  nonand form  for  is  x  =  r.x.(l-x/K)  y  =  e.i .x.y/(D+x) M  be  -  i .x.y/(D+x) M  -  m.y  1.2a 1.2b  20  Here,  K  i s a carrying  predators, predator reaches the  i  M  Two  i s the prey  i t s maximum  parameter  this  i s t h e maximum  and D half  a  of phase  isocline  always  isocline  a vertical  point  forms  right  o f t h e 'hump',  and  trajectories  To must  spiral  be f o u n d  i / D i s comparable  a r e shown 'hump'  behaviour  The prey  and the predator  geometric  When  trajectories  of  May,1974).  i n F i g 5.  intersection  i t .  to  M  ( f o r a review,see  the predator  into  these  results  (x,y) i s a  rule  critical  determines i t s  isocline  (x = x ) l i e s  the predator spiral  model.  The o b v i o u s  t,  the seasonal  reflecting resulting  t o t h e O.S.P.  to incorporate seasonal  growth  system  to treat  parameters  to a  to  stable  isocline  outward  lies  stable  homogeneous  system.  equilibrium  i s sufficiently  time  over  scales solution  which  fast  presently  productivity  homogeneous Suppose  (x(t),y(t))  I f the time  this  time  r and K change,  t o t h e non-homogeneous  t a r e such  (Fig  4).  that  that the a  stable  f o r the corresponding  f o r approach  compared  time,  and consequently  f o r t h e moment  exists  scale  a way  i s t o allow the  i n primary  i n 1.2 a t a n y p a r t i c u l a r equilibrium  into  cycle,  r a n d K t o be f u n c t i o n s o f  cycle  analytically.  seasonal  effects  approach  i s no l o n g e r  non-trivial  the  consumption  ( F i g 5a), (x,y) i s asymptotically  phytoplankton  parameters  prey  solution.  relate  difficult  a t which  of  of prey per  The q u a l i t a t i v e  simple  When  o f t h e hump,  homogeneous  The  1.1.  Their  and a  the  cycle  of consumption  portraits  line.  i n the absence  The r a t i o  a quadratic  properties.  limit  value.  plane  of t h e system  the l e f t  rate  much d i s c u s s i o n  stability  to  f o r prey  density  i n equation  model has seen types  capacity  with  i t seems  system,  to  this  the (seasonal) reasonable  i f i t starts  that  near  21  predator isocline  C Q)  a  ^ ^ isocline  D "O  \  o  prey  prey  Figure 5.  density  density  K  x  x  Phase plane portraits, f o r system (a) equilibrium s t a b l e . unstable.  1.2.  (b) equilibrium  22  (x(t),y(t)), solution. ,1978),  will  remain  By t h i s  close to this  quasi-equilibrium  s e p a r a t i o n of time s c a l e s (Ludwig  a s e a s o n a l c y c l e c a n be e n v i s a g e d w h i c h  approximated  by t h e q u a s i - e q u i l i b r i u m s o l u t i o n .  are determined  eta l  i s closely Now x ( t ) , y ( t )  by  e.i .x/(D+x) = m  1.3a  M  y(t) = r(t).(l-x/K(t)).(D+x)/i  T h e r e a r e two i m m e d i a t e l y expressions.  1.3b  M  encouraging  The q u a s i - e q u i l i b r i u m p h y t o p l a n k t o n c o n c e n t r a t i o n  i s c o n s t a n t over time, depending The  a s p e c t s of these  o n l y on z o o p l a n k t o n  parameters.  quasi-equilibrium zooplankton concentration i s proportional  to the p h y t o p l a n k t o n growth s t a n d i n g s t o c k s which as o b s e r v e d  rate.  .Phytoplankton and z o o p l a n k t o n  f o l l o w e d ( x ( t ) , y ( t ) ) c l o s e l y would behave  a t O.S.P. , w i t h p h y t o p l a n k t o n c o n c e n t r a t i o n  a p p r o x i m a t e l y c o n s t a n t and z o o p l a n k t o n c o n c e n t r a t i o n v a r y i n g w i t h primary  productivity.  T h e r e i s however a s e r i o u s p r o b l e m The  stability  criterion  with this 'explanation'.  g i v e n a b o v e c a n be w r i t t e n a s  D > K - 2.x  1.4  K r e p r e s e n t s the s t a n d i n g stock of phytoplankton a t which resource l i m i t a t i o n  causes  t h e growth  T h e r e a r e two o b v i o u s p o t e n t i a l l y p h y t o p l a n k t o n , namely n u t r i e n t section  rate t o drop t o zero.  limiting  resources f o r  s u p p l y and a v a i l a b l e  1.3, n u t r i e n t c o n c e n t r a t i o n s a t O.S.P.  light.  In  were d e s c r i b e d a s  23  non-limiting.  The  of M c A l l i s t e r  value  of K r e a c h e d i n t h e c u l t u r e e x p e r i m e n t  e t a l ( 1 9 6 0 ) was  a b o u t 40  usual non-linear Michaelis-Menten nutrient concentration r a t e of value  small  o f x/K  of 1/40  Increases  logistic  in phytoplankton  phytoplankton  negative  i n x above x w i l l  i n the  x.  In f a c t ,  f e e d b a c k on  model w o u l d  d e n s i t y can  be q u a n t i f i e d f o r a homogeneous m i x e d - l a y e r  + k  (adapted  decrease This effect  population  by and  from Parsons et a l , 1977), combined w i t h a . (P) v s  light  Steele,1962).  ( I ) r e l a t i o n s h i p (eg P=  #.I.exp(-  I n t e g r a t i n g over depth g i v e s  an  f o r g r o w t h i n t h e m i x e d l a y e r as a f u n c t i o n of  layer depth, z , M  surface  and  I  x.  Differentiating  light  , the parameters k  intensity  and  0  k  4  this expression  and  for x = x y i e l d s a value  1/K  For  i n 1.2.  z  M  =' 30m  (late  ( S t e e l e , 1962), t h i s y i e l d s a value  T a k a h a s h i and  x.  This value  0  the Chi a  mixed  I parameters a concentration, to x  and  suitable for insertion  summer), k  (Lorenzen,1980;Megard et a l ,1980), k  times  I , P vs  with respect  substituting  a b o u t 20  k,  .x  t  expression  M A X  can  x,  photosynthesis I/I MAX)'  growth  suggest.  e m p l o y i n g a r e l a t i o n s h i p between e x t i n c t i o n c o e f f i c i e n t ,  k = k„  and  be much s m a l l e r t h a n a  growth r a t e s through s e l f - s h a d i n g .  chlorophyll,  the  r e l a t i o n s h i p between growth  means t h a t any  increases  times  4  = .02  = .1 n r , 1  0  of K,  jght  agrees w i t h the  as  m .mg C h i  a" ,  2  and  = 8 mg  I /I 0  M A  x  Chi a.nr  results  1  = 3  2. or  of  Parsons(1972).  B o t h c o n s i d e r a t i o n s of n u t r i e n t l i m i t a t i o n  and  self-shading  24  suggest  that  maintained  , a t O.S.P. ', t h e p h y t o p l a n k t o n  population i s being  a t c o n c e n t r a t i o n s about o n e - t w e n t i e t h of i t s c a r r y i n g  capacity or l e s s .  A c c o r d i n g t o t h e c o n d i t i o n 1.4, f o r t h i s  e q u i l i b r i u m t o be s t a b l e , D must be a b o u t 20 t i m e s x w h i c h imply that zooplankton phytoplankton  a t O.S.P.  a r e growing  concentrations sufficient  maximum r a t i o n .  While  would  and r e p r o d u c i n g a t  t o s u p p l y 1/40 o f t h e i r  l a r g e v a l u e s of t h e g r a z i n g h a l f -  s a t u r a t i o n c o n s t a n t o f t h i s o r d e r have been r e p o r t e d , s u c h a l o w relative  ingestion  metabolic  requirements  particular specific  r a t e seems u n l i k e l y  of the zooplankton.  assumptions  of t h e g r a z e r s ' f u n c t i o n a l  Of c o u r s e , t h e  response,  u n d e r l y i n g t h e e s t i m a t e o f K.  i s more f u n d a m e n t a l  than  the s t a b i l i t y  response  g r a z e r s has r e c e i v e d support functional  response  equilibrium  of t h i s  i s always  may b e .  The l o c a l  as w e l l as the  However, t h e problem  c r i t e r i o n alone  F o r example, a p i e c e w i s e l i n e a r ,  (Holling,1965) functional  x/K  even t h e b a s a l  v a l u e 1/40 c a n be q u e s t i o n e d a s i t d e p e n d s on t h e  form  suggest.  t o cover  would  o r Type I  on t h e p a r t o f z o o p l a n k t o n  ( M u l l i n and Brooks,1975). type  i s substituted  i n t o 1.2, t h e  a s y m p t o t i c a l l y s t a b l e , no m a t t e r r a t e of approach  If a  how s m a l l  t o e q u i l i b r i u m , on t h e  o t h e r h a n d , i s t h e n g i v e n by r . x / K , a n d t h e q u a s i - e q u i l i b r i u m e x p l a n a t i o n d e p e n d s n o t o n l y on ( x , y ) b e i n g s t a b l e , b u t on t h e f u r t h e r c o n d i t i o n t h a t approach time  t o e q u i l i b r i u m o c c u r s on a f a s t  s c a l e compared w i t h changes i n ( x ( t ) , y ( t ) ) .  Values  of o r d e r l / 2 0 t h a n d a v a l u e o f r o f 0.2 d a y : , e s t i m a t e d 1  McAllister  et a l (1960), g i v e a c h a r a c t e r i s t i c  t o e q u i l i b r i u m o f o r d e r 50 d a y s . the seasonal time  time of  o f x/K from approach  T h i s i s n o t s h o r t compared w i t h  s c a l e of changes i n p h y t o p l a n k t o n  growth r a t e  and  y(t).  Large  abundance  would  subjected  to  homogeneous when in  x/K  terms  oscillations occur  i s very of  i f such  seasonal system  in phytoplankton  cycles  with  a  small,  a  weakly-stable  type  I  functional  to  track  a  zooplankton  system  in productivity.  i t is little  its ability  and  was  Although  response  better  than  seasonally  the  is  stable,  neutrally  stable  varying  equilibrium. The  system  observations seems  at  promising  system  so  as  retaining  the  solution One  to  a  However,  and  might  t r y to  the  stability  one  overcome  the  consistent explanation  O.S.P.  approach-which  is  to  behaviour  doesn't  Introducing  b r i n g about  models  proceed time  by  modifying  scale  the  problem,  while  the q u a s i - e q u i l i b r i u m  with  (Bazykin,1974)  type  I  a  quadratic  and  loss  been  in  for  suggested  can  be  for predators  simple  (Landry,1976).  response,  here  term  stability  has  models  functional  i s mentioned  in A  predator-  other  model  written  of  =  for  e.a.x.y  x  loss  at rate  -  1.5a  u.y  intraspecific  1.5b  2  sub-maximal (u.y)  with  ration  levels.  predator  competition  switching  or  aggregative  model  be  rejected,  can  this  as  x = r . x - a . x . y y  of  q u a s i - e q u i l i b r i u m approach  of  work  asymptotic  phytoplankton-zooplankton kind,  provide  (x,y(t)).  sake.  prey  cannot  qualitative  interest known  1.2  for  increase  density could resources  responses  without  (An  result  other  in higher  considering  than  i n per from food,  carnivores.)  its  capita  stability  or  from  This  26  p r o p e r t i e s , as the q u a s i - e q u i l i b r i u m s o l u t i o n f o r s e a s o n a l l y varying  r(t) i s  y(t)  = r(t)/a  x(t)  =  The  u.y(t)/(e.a)  q u a s i - e q u i l i b r i u m phytoplankton  constant not  concentration  but v a r i e s w i t h y ( t ) i n t h i s model.  consistent with observations  x ( t ) i s not  This  i s certainly  a t O.S.P.  Two p o s s i b l e f u n c t i o n a l r e s p o n s e s have a l r e a d y considered  i n 1.2 : a h y p e r b o l i c  (Type I ) r e s p o n s e  been  (Type I I ) a n d p i e c e w i s e  (Holling,1965).  Within  the context  of  linear simple  b i o m a s s m o d e l s s u c h a s 1.2, t h e s e c a n be c h a r a c t e r i z e d a s destabilizing functional  and n e u t r a l l y s t a b l e r e s p e c t i v e l y .  response discussed  A third  by H o l l i n g (1965) i s t h e s i g m o i d o r  Type I I I f u n c t i o n a l r e s p o n s e , i n c o r p o r a t i n g a r e d u c e d r a t e by c o p e p o d s a t l o w f o o d d e n s i t i e s . response has a s t a b i l i s i n g  This  e f f e c t i n simple  grazing  responses of t h i s type,  feeding  b e l o w some t h r e s h o l d  densities  or a reduction  (Frost,1975).  a copepod t r y i n g  biomass models.  filtering  activity  S t e e l e and The  suggested  i n v o l v i n g e i t h e r a c e s s a t i o n of  food  density  in filtering  ( P a r s o n s et_ a_l ,1969; r a t e a t low food  Theoretical considerations  to optimize  clearance  type of f u n c t i o n a l  A number o f s t u d i e s o f p e l a g i c c o p e p o d s h a v e  Frost,1972),  form of  energy  intake should  suggest  that  reduce i t s  a t l o w f o o d d e n s i t i e s (Lam a n d F r o s t , 1 9 7 6 ;  Frost,1977).  e f f e c t of a l l o w i n g a type I I I g r a z i n g  seen i n a phase p l a n e a n a l y s i s of t h e system  r e s p o n s e c a n be  27  k = r.x -  1.6a  f(x).y  y = e.f(x).y  - m.y  where t h e g r a z i n g or  sigmoid form  1.6b  function  ( F i g6b).  f ( x ) has t h e t h r e s h o l d  t h e case of a g r a z i n g  asymptotes t o the v e r t i c a l at  under t h e c o n d i t i o n s  a t O.S.P.).  Phase p l a n e p o r t r a i t s For  ( F i g 6a)  (The p h y t o p l a n k t o n s e l f - l i m i t i n g t e r m  (-x/K) h a s b e e n d r o p p e d a s i t i s n e g l i g i b l e prevailing  form  some p o i n t  approaches  threshold l i n e x=x  a t x=x„ , t h e p r e y c  at the l e f t ,  (The p a r a m e t e r  i  M  represents  r a t i o n . ) The z o o p l a n k t o n  as. f o r s y s t e m 1.2.  i n F i g 7. isocline  h a s a minimum  x*, a n d a s y m p t o t e s ' t o t h e l i n e y = r . x / i  oo.  zooplankton  f o r t h e s y s t e m 1.6 a r e g i v e n  M  as x  t h e maximum  isocline  is a vertical  line  These i n t e r s e c t a t t h e e q u i l i b r i u m g i v e n  by  f (x) = m/(e.a)  1.7a  y = r.x.e/m  1.7b  By  linearizing  shown t h a t  this equilibrium i s asymptotically  if  x < x*; t h a t  of  t h e minimum  condition distinct  i n t h e n e i g h b o u r h o o d o f ( x , y ) , i t c a n be  i s , the zooplankton i n the phytoplankton  i s s a t i s f i e d , there phase p o r t r a i t s .  stable  isocline  must l i e t o t h e l e f t  isocline.  are s t i l l  two  When x i s v e r y  i f and o n l y  Given that  this  qualitatively  l a r g e , the system  1.6  becomes a p p r o x i m a t e l y  x = r.x - i  . y  1.8a  y=(e.i -m).y  1.8b  M  M  28  a  x  o  Figure 6.  prey density  x  prey  x  density  Type I I I functional (a) threshold.  responses:  (b) sigmoid.  29  0  F i g u r e 7.  prey  density  Phase plane p o r t r a i t s (a) e . l ~ m > r . M  x f o r system  (b) e . i ^ - n K r .  1.6.  30  If  e.i^-m > r , o r , e q u i v a l e n t l y ,  zooplankton exceeds that from l a r g e x and s m a l l  of phytoplankton, t r a j e c t o r i e s s t a r t i n g  y will  phytoplankton concentrations impossible  i f t h e maximum g r o w t h r a t e o f  a l w a y s c y c l e back t o low (Fig 7a).  In t h i s sense, i t i s  f o r p h y t o p l a n k t o n t o escape zooplankton  permanently.  I f e . i - m < r , ( a s seems more l i k e l y ) , M  b e h a v i o u r o f t r a j e c t o r i e s f o r l a r g e x d e p e n d s on conditions,  (x, ,y,- ) .  c y c l e , but o t h e r w i s e , trajectory  will  x will  approach  increase  (+00,+ 0 0 ) .  t r a j e c t o r i e s which c y c l e  similar,  the  initial  I f y i i s l a r g e enough, t h e t r a j e c t o r y  p l a n e p o r t r a i t ( F i g 7b)- t h e r e  portraits  control  indefinitely  and t h e  i n the corresponding  i s a s e p a r a t r i x which  from those which d o n ' t .  will  phase  divides  The p h a s e p l a n e  f o r sigmoid f u n c t i o n a l responses are q u a l i t a t i v e l y the p r i n c i p a l d i f f e r e n c e  being that  the phytoplankton  i s o c l i n e a s y m p t o t e s t o x=0. The can  local  r a t e of approach of t r a j e c t o r i e s t o e q u i l i b r i u m ,  be f o u n d by l i n e a r i z i n g  clearance  about  r a t e of z o o p l a n k t o n  of a p p r o a c h t o e q u i l i b r i u m  (x,y).  (given  i s given  In f a c t , i f h(x) i s the  by f ( x ) / x ) ,  the l o c a l  rate  by  r.x.h'(x)/h(x)  The  stability  criterion  x < x* i s simply  r a t e of approach, x* b e i n g given equilibrium increases  (x,y) i s l o c a l l y  with  by h ' ( x * ) = 0 .  stable provided  That  to a positive i s , the  the clearance  rate  x a t x=x.  In t h e case of a t h r e s h o l d , as  equivalent  u s e d by S t e e l e  (1974),  hyperbolic  functional  response  31  f(x)  and  = i  M  . (x-x  explicit  ) /(D+(x-x ) ) , 4  0  +  e  formulae  c a n be w r i t t e n  f o r x* and f o r t h e r a t e o f  approach:  x*-x  =  0  /D7X  0  rate  of approach  That  i s , the clearance rate  greater  than  saturation  x does  occur  ((x-x ).D+(x-x„) ) 2  e  a t a d e n s i t y x* w h i c h i s  by t h e g e o m e t r i c  and t h e t h r e s h o l d  i s o f t h e same o r d e r  growth  then  on r e l a t i v e l y  assumption  growth,  x  mean e  .  of the h a l f -  Provided the  t o x*, t h e r a t e  of  approach  as the phytoplankton  rate,  r.  Note  predicts,  and  a zooplankton  production. approximate capable  f o r the approach  fast  scales,  f o r seasonal that  as b e f o r e ,  The s i m p l e  stock  biomass  based  of q u a l i t a t i v e l y  to equilibrium  growth  as  f o r the quasi-equilibrium  variation  i n phytoplankton cycle  phytoplankton  which model  i n 1.6  o f t h e same o r d e r  the quasi-equilibrium  a constant  standing  argument  time  and consequently  t o be v a l i d  1.7  cycle  i s a maximum  not l i e too close  i s possible  phytoplankton  be  /  2  r. It  to  the threshold  equilibrium  rate,  2  0  c o n s t a n t , D,  equilibrium to  = r.((x*-x ) -(x-x„) )  varies  given  standing  with  reproducing  stock  primary  1.6, a c c o r d i n g t o  on s e p a r a t i o n o f t i m e  by  scales,  the observed  this should  seasonal  a t O.S.P. Reflection  perspective  on t i m e  scales  on t h e d e v e l o p m e n t  traditionally  associates  with  allows  an  so f a r .  interesting, The time  phytoplankton  intuitive  scales  growth  one  a r e short and  32  one  would expect  tightly  which  regulates phytoplankton density  t o o p e r a t e on a s i m i l a r  traditionally and  an a g e n t  time s c a l e .  The t i m e  a s s o c i a t e s w i t h c o p e p o d d y n a m i c s a r e much l o n g e r  t h i s appears  t o be a p r o b l e m  h y p o t h e s i s a t O.S.P.  with a grazing control  Where n u t r i e n t s a r e l i m i t i n g ,  e f f e c t s on p h y t o p l a n k t o n g r o w t h  a p p a r e n t l y n o t t h e c a s e a t O.S.P. response  feedback  can t i g h t l y r e g u l a t e  phytoplankton d e n s i t y , as d i s c u s s e d i n Chapter  functional  s c a l e s one  2, b u t t h i s i s  The s i g m o i d o r t h r e s h o l d  i n 1.6 e s s e n t i a l l y  i n t r o d u c e s a new f a s t  s c a l e , a zooplankton b e h a v i o u r a l time s c a l e ,  time  i n t o the problem.  T h i s development augments r a t h e r n e a t l y H e i n r i c h ' s ( 1 9 6 2 ) e x p l a n a t i o n o f t h e d i f f e r e n c e s between s e a s o n a l c y c l e s S u b a r c t i c and e l s e w h e r e , the dominant g r a z e r s . North A t l a n t i c  based  He a t t r i b u t e d  abundance.  I f t h e maximum g r o w t h  7b i s a p p r o p r i a t e .  growth  rate  The s p r i n g  isocline.  easily  i n the system  result  leaving the l o c a l region  i n which  A delay  stability  growth  r a t e of p h y t o p l a n k t o n a phase p l a n e p o r t r a i t  as i n  increase i n phytoplankton  i s equivalent to a vertical  phytoplankton  of t h e  increase i n phytoplankton  exceeds t h a t of zooplankton biomass, Fig  s t r a t e g i e s of  t h e s p r i n g bloom i n t h e  t o a delay i n the numerical response  dominant g r a z e r t o t h e s p r i n g and  on t h e l i f e - h i s t o r y  i n the  shift  i n the  i n zooplankton  response  can then  b e i n g o v e r t a k e n by t h e s e p a r a t r i x , domain about  (x,y) and e n t e r i n g a  p h y t o p l a n k t o n have escaped  A c c o r d i n g t o 1.6, t h e t r a j e c t o r y  will  zooplankton  approach  control.  ( + oo, + o o ) .  p r a c t i c e , of c o u r s e , n u t r i e n t s a r e d e p l e t e d and t h e s p r i n g  In bloom  terminates. Heinrich(1962) attributed  t h e o c c u r r e n c e o f s p r i n g blooms i n  33  coastal  r e g i o n s o f t h e S u b a r c t i c where C. p l u m c h r u s  a failure  i n t i m i n g of r e c r u i t m e n t .  does o c c u r e a r l i e r locations  i n the S t r a i t  inappropriate.  stratification  of Georgia  and a consequent  failing  equilibrium cycle separatrix, The perhaps  to track  that the timing i s i n F i g 7b s u g g e s t s subject to run-off,  increase i n phytoplankton  T h i s may r e s u l t the rapidly  growth  i n t h e system  shifting  state  quasi-  ( x , y ( t ) ) a n d a g a i n b e i n g o v e r t a k e n by t h e  resulting  i n a spring  bloom.  o b s e r v a t i o n s of F u l t o n (1973) suggest a s i m p l e r and more c o n v i n c i n g e x p l a n a t i o n : n a m e l y , t h a t  C. p l u m c h r u s  i n the s t r a i t  fails  to coincide  than t i m e , w i t h t h e s p r i n g bloom. C. p l u m c h r u s  o c c u r s o n l y i n water  r e c r u i t m e n t of  i n space,  C. p l u m c h r u s  rather  S u c c e s s f u l o v e r - w i n t e r i n g of deeper  t h a n 300m, w h i c h  o c c u p i e s o n l y o n e - f o u r t h of t h e area of t h e s t r a i t .  subject  reach the  i n February and March  In c o a s t a l waters  rate can occur very r a p i d l y . (x(t),y(t))  o f C. p l u m c h r u s  The p h a s e p l a n e p o r t r a i t  explanation.  bloom  of Georgia than i n oceanic  ( F u l t o n , 1 9 7 3 ) and i t i s n o t c l e a r  another  Although the spring  (Parsons,1965), the n a u p l i i  s u r f a c e waters of t h e S t r a i t  i s present to  i n the remaining areas  i s presumably  A r r i v a l of  delayed,  to horizontal advection.  1.5 P r e v i e w o f C h a p t e r s 2-4. E a c h o f t h e s i m p l e m o d e l s c o n s i d e r e d a b o v e c a n be r e g a r d e d as a c o m p o s i t e O.S.P.  There  hypothesis concerning trophic  interactions at  i s c l e a r l y a need f o r independent  t e s t s of a s p e c t s of these hypotheses.  experimental  F o r example, t h e  functional  r e s p o n s e s o f t h e d o m i n a n t c o p e p o d s a t O.S.P.  a r enot  well-known  and t h e e x i s t e n c e of z e r o or reduced c l e a r a n c e r a t e s  34  at low p h y t o p l a n k t o n d e n s i t i e s  i s an e s s e n t i a l p a r t o f t h e model  1.6. The t h e o r e t i c a l p o s s i b i l i t i e s the above a n a l y s i s .  Given  have h a r d l y been e x h a u s t e d by  the comparative  wealth of i n f o r m a t i o n  i n t h e l o n g t i m e s e r i e s o f o b s e r v a t i o n s a t O.S.P. , more t h a n rough q u a l i t a t i v e agreement of a model w i t h a v e r a g e c y c l e s c a n be demanded. and  A q u a n t i t a t i v e comparison  of p r e d i c t i o n  o b s e r v a t i o n r e q u i r e s a more c a r e f u l l y c o n s t r u c t e d , more  d e t a i l e d model. h a v e been u s e d zooplankton  F o r e x a m p l e , t h e s t a t e v a r i a b l e s x a n d y above rather loosely  t o r e p r e s e n t p h y t o p l a n k t o n and  s t a n d i n g s t o c k , a l t h o u g h t h e y h a v e been  i d e n t i f i e d w i t h C h i a i n mg.irr mg.nr  2  varies  respectively.  dominated  distribution  needs c a r e f u l  examination.  Zooplankton  summary v a r i a b l e , r e p r e s e n t i n g  from c a r n i v o r e s as w e l l as h e r b i v o r e s , a l t h o u g h  by h e r b i v o r o u s c o p e p o d s a t most  (LeBrasseur,1969). available  and ( h y p o t h e t i c a l l y ) 3  i s a rough  contributions  a n d z o o p l a n k t o n wet w e i g h t i n  ( M c A l l i s t e r , 1 9 6 9 ) , t h e u s e o f C h i a i n mg.m" a s  a state variable clearly wet w e i g h t  3  implicitly  As t h e p h y t o p l a n k t o n p o p u l a t i o n a t O.S.P.  seasonally i n vertical  in C:Chl a r a t i o  view  seasonal  More d e t a i l e d  f o r O.S.P.  times  s i z e and s p e c i e s i n f o r m a t i o n i s  and t h i s c e r t a i n l y deserves a t t e n t i o n i n  of recent t h e o r e t i c a l  results  ( S t e e l e , 1 9 7 4 ; S t e e l e and  Frost,1977). These problems a r e a d d r e s s e d Chapter  3, an a t t e m p t  i n Chapter  i s made a t a more r e a l i s t i c  s i m u l a t i o n model o f p h y t o p l a n k t o n growth partly  based  particularly  on an o r i g i n a l 1 4  C  uptake  3 and Chapter  a t O.S.P.  4.  In  quantitative The m o d e l i s  a n a l y s i s of t h e phytoplankton  data,  r a t e s , o b t a i n e d from t h e w e a t h e r s h i p s .  35  In Chapter  4, a p a r a m e t e r  copepod time s e r i e s  e s t i m a t i o n technique developed f o r  i s applied to estimate population  f o r t h e d o m i n a n t h e r b i v o r e s a t O.S.P. of t h e p h y t o p l a n k t o n - z o o p l a n k t o n  Two more e l a b o r a t e m o d e l s  interaction are constructed,  u s i n g these r e s u l t s and those of Chapter  3.  Some e f f e c t o f  h e r b i v o r e s i z e - s t r u c t u r e and s p e c i e s c o m p o s i t i o n the second  model.  parameters  i s considered i n  The m o d e l s a r e s t u d i e d u s i n g s i m u l a t i o n a n d  the q u a l i t a t i v e t e c h n i q u e s and r e s u l t s of t h i s c h a p t e r and Chapter  2.  Three hypotheses i n t e r a c t i o n a t O.S.P. explain  concerning the phytoplankton-zooplankton h a v e been c o n s i d e r e d i n S e c t i o n 1.4. A l l  the seasonal cycles  i n p h y t o p l a n k t o n and z o o p l a n k t o n  s t a n d i n g s t o c k as the r e s u l t of t h e system's a b i l i t y track a quasi-equilibrium seasonal cycle.  The h y p o t h e s e s  i n t h e mechanisms r e s p o n s i b l e f o r t h e s h o r t term feedback  necessary  forthis  to closely differ  stabilising  t r a c k i n g of a s e a s o n a l l y s h i f t i n g  equilibrium.  The model 1.2, i n v o l v i n g  phytoplankton  growth,  r e s o u r c e - l i m i t a t i o n of  i s c e r t a i n l y capable of the a p p r o p r i a t e  q u a l i t a t i v e b e h a v i o u r , b u t h a s been r e j e c t e d on t h e b a s i s o f independent quadratic  experimental evidence.  l o s s r a t e f o r g r a z e r s , cannot  seasonal c y c l e . grazing  The model 1.6, w h i c h  not  reproduce  the actual  the observed  assumes a r e d u c t i o n i n  r a t e a t low p h y t o p l a n k t o n d e n s i t i e s ,  reproducing the observed earlier,  The m o d e l 1.5, i n v o l v i n g a  i s capable of  seasonal cycle q u a l i t a t i v e l y .  functional  responses  As n o t e d  o f g r a z e r s a t O.S.P. a r e  known. The  based  s i m u l a t i o n s c o n s i d e r e d i n Chapter  upon t h e s y s t e m  4 will  1.6 a n d t h e a s s u m p t i o n  be p r i m a r i l y  of g r a z i n g  36  thresholds. will  However,  become  modified  particular, strategies an  an  show  period  interaction  and  .problem  of  against  the  hypothesis  will  and  be  variation  a  new  2  one.  In  for on  a  in  necessary  loss  by  term  for  for  models, 2.  the  be  even  in  model  the  in will  the  Sea,  into  A  of  and  the fall  during  by  this  predictions  and  the  grazing  The  possibility  of  late  and  growth  in  threshold  summer  importance be  concerned  with  events  of  spatial  discussed.  The  in plankton  at  idea  O.S.P. that  ecosystems  is  not  phytoplankton-zooplankton  Steele(1974)  found  behaviour.  found  that  Steele's based  An  that  inserting a  model  their  of  time  insights  theoretical  into  the  need  conclusions  analysis i s give  the  was  quadratic  eliminated  scales,  thresholds  alternative  q u a l i t a t i v e mathematical  provides  on  of  departure  will  separation  analysis  summer  model  potential  realistic who  focus  robustness  between  of  over  absence  control  model  In  nauplii  the  over-wintering  important  authors  results.  The  and  of  1.6  phytoplankton-zooplankton  theoretical interest.  herbivores  on  recruitment  alternatives.  directly  obtain  based  of  grazing  North  Both  life-history  attention  phytoplankton  Landry(1976),  simulation  Chapter  may  copepod  bloom,  of  in  elaboration.  re-evaluation  numerical  to  thresholds.  these  of  general  thresholds  proposed  lack  the  implicit  model  control  effect  a  search  i s not  more  interactions were  a  of  in d e t a i l  in maintaining  i s of  feeding  result,  reconsidered  Chapter but  a  A  force  limitation  will  spring  discrepancies  observations  fall  a  grazing  copepods.  of  the,spring  destabilising  dominant  nutrient  treatment  As  hypothesis  process  stabilise  maintaining  and  the  prevent  thresholds.  composite  that  can  grazing  period  in  explicit  will  extended  the  the  of  in  results  of  37  Steele for here  and  marine  Landry  which  ecosystem  f o r O.S.P.  in  appear  models  t o have  in general,  particular.  interesting and  implications  the models  considered  38  CHAPTER 2 QUALITATIVE ANAYLSIS OF  2.1  A COMPLEX SIMULATION MODEL  Introduct ion. In  North  a theoretical  Sea,  Steele  t r e a t i s e on  the p l a n k t o n i c e c o s y s t e m of  (1974) e x a m i n e d t h e  i n t e r a c t i o n s between d i f f e r e n t  t o be  of t h e o r e t i c a l  simplified  step i n complexity  predator-prey  on  nutrient-phytoplankton-copepod  a b a s i s f o r more c o m p l e x r e a l i s t i c as a f i r s t  of  H i s c o n c l u s i o n s were b a s e d  a s i m u l a t i o n model of a s i m p l i f i e d  and  importance  t r o p h i c l e v e l s as ' c o n t r o l  mechanisms f o r the whole s y s t e m ' .  food c h a i n which c o n t i n u e s  relative  the  interest,  m o d e l s ( S t e e l e and and  both  as  Frost,1977)  r e a l i s m above the h i g h l y  m o d e l s of t h e L o t k a - V o l t e r r a  type  (Lotka,1925;May,1974). S t e e l e ' s c o n c l u s i o n s were b a s e d p r i m a r i l y  on c o m p a r i s o n s of  a number of c o m p u t e r s i m u l a t i o n s of t h e model i n v o l v i n g assumptions concerning interactions.  A key  the  f o r m and  f i n d i n g was  magnitude of t r o p h i c  t h a t the model c o u l d not p r e d i c t  timestreams which agreed q u a l i t a t i v e l y North  Sea  response  copepods.  Landry  ( H o l l i n g , 1 9 5 9 ) , was  (1976) o b t a i n e d  S t e e l e ' s model w i t h o u t  invoked  realistic  t h r e s h o l d s by  of  for  the  herbivorous  behaviour  from  i n t r o d u c i n g a per c a p i t a  r a t e on h e r b i v o r e s w h i c h i n c r e a s e d i n p r o p o r t i o n  h e r b i v o r e numbers a t low  densities.  by L a n d r y a s a s i z e - d e p e n d e n t abundant) but quadratic  with observations  u n l e s s a t h r e s h o l d - f e e d i n g mechanism, or type I I I  functional  predation  various  i t s significance  term  T h i s was  partly  l o s s term f o r h e r b i v o r e s  introduced  ( s m a l l e r copepods being  seemed t o l i e i n t h e (Steele,1976);  to  more  resulting  the  39  i n t r o d u c t i o n of equilibria  such a l o s s term i s well-known t o produce  in otherwise unstable  simple  Lotka-Volterra  stable  models  (Bazykin,1974). B o t h S t e e l e and  Landry presented t h e i r  suggestive  evidence f o r the  thresholds  and  respectively particular  existence  and  r e s u l t s as a t  importance  density-dependent per-capita  i n r e a l ecosystems.  b i o l o g i c a l questions  This  i s recognised  p o t e n t i a l c o n t r i b u t i o n of m o d e l l i n g  studies  as  a  in general,  a r o u s e d c o n c e r n i n g the  of  ( M u l l i n e t a l ,1975; F r o s t , 1 9 7 5 ) .  argument t h a t r e s u l t s are One  observation  reason  t o be  obtained  for this,  conceptual  elaboration  particular  c o n c e p t can  n e v e r be  the  Given the not be  and  Landry's which are  large uncertainty  criteria using  on  the  incapable  b a s e d on  of  the  b a s i s of a s m a l l  necessity  of  in  the  any  computer  such  as  simulation.  f u n c t i o n a l form  number of" n u m e r i c a l  t h e o r e t i c i a n , at  about the  (as  should  s a t i s f y i n g a s e t of q u a l i t a t i v e  This  b e h a v i o u r of  solutions  c r i t i c i s m can  least in principle,  q u a l i t a t i v e m a t h e m a t i c a l a n a l y s i s of information  caution.  i n most e c o l o g i c a l p a r a m e t e r s , i t i s  p a r t i c u l a r parameter v a l u e s .  a d d r e s s e d by  realistic  studies, i s that  to studies  c l e a r t h a t a model h a v i n g a p r e s c r i b e d j u d g e d t o be  if  any  established.  A second reason a p p l i e s p a r t i c u l a r l y Steele's  feeding  of a model i s a l w a y s p o s s i b l e  c a s e of L a n d r y v e r s u s S t e e l e ) , so t h a t  it  discussion  must, a l w a y s be v i e w e d w i t h  common t o a l l m o d e l l i n g  and  However,  some a s p e c t of a m o d e l i s n e c e s s a r y  on  valuable  the  thresholds  rates  of a t t e n t i o n  a p p e a r s t o h a v e been s u c c e s s f u l h e r e i n v i e w of experimental  of  predation  focusing  least  t h e model c a n  be as  provide  s o l u t i o n s over regions  in  40  parameter  space,  this  can also  type  behaviour thereby  allow  useful  in  1 which  and  with  discrepancy simple there  models  of simple  that  grazing  ecosystem.  of the S u b a r c t i c P a c i f i c  phytoplankton  summer,  and  stability. through  this  to invoke only  why  analysis  thresholds  much  of the  be n e e d e d f o r  i s sought  here  model.  and A n a l y s i s model  developed  by S t e e l e  (1974)  and used,  R  + U. ( E . ( P - P l ) / ( D + P ) + F ) . Z . W -  = V.(RO-R)  0  of change  zooplankton  of n u t r i e n t  equals  mixing  excretion - phytoplankton  = A.R.P/(B+R)  (rate  phytoplankton  Sea s i m u l a t i o n  discrepancy  by L a n d r y ( 1 9 7 6 ) i s :  P  puzzling  thresholds in  should  of S t e e l e ' s  important  Sea, a  throughout  alterations,  (rate  of S t e e l e ' s  is superficially  because  In the North  explanation of t h i s  the q u a l i t a t i v e  Model The  An  analysis  thresholds are  are nutrient-limited  i t i s not c l e a r  and  L o t k a - V o l t e r r a models i n  While  necessary  are not n u t r i e n t - l i m i t e d .  model,  2.2  for a qualitative  I t was  of the  on p a r a m e t e r s  Steele's conclusion f o r the North  exists.  A n a l y s i s of  insights.  by t h e s t u d y  suggested  them.  understanding  i t sdependence  biological  the Subarctic P a c i f i c  consistent  at points within  to a better  motivation  i s provided  Chapter  than  lead  of the model  Additional model  rather  of change  - V.P  - C.Z.W  0 7  7  with  - A.R.P/(B+R)  through  2.1a  thermocline  +  uptake)  .(P-P1)/(D+P)  of phytoplankton  certain  = growth  - mixing  2.1b -  grazing)  41  W =  ( (0.7.C-E) . (P-P1)/(D+P)  (rate  of i n d i v i d u a l growth  - F).W 0  2.1c  7  = net assimilation - active  and  basal  metabolism)  Z = -GW.(W-Wl).(Z-Z1)/(H+Z.W) (zooplankton 1inear  When  mortality  rate  - GX.Z  2.Id  = nonlinear,  copepods  naupli  reach  adult  store,  S,  are released,  weight,  W2,  for a period  ZO b e i n g  X  notation  here  biomass.  The  +  given  2.1  G  that  t o be  i s referred  represents  ZO  2.1e  naupliar  weight.  of Steele(1974), reserved  to Steele  for non-trivial  corresponding  The  except  that  f o r zooplankton  for a detailed  set of four involving  of n a u p l i i solutions  of the behaviour  regions  i n parameter  checked  in this  simultaneous, thresholds  and there  in closed  and approximations  a r e then step  a  equations  recruitment  understanding  The.basic  which  by  the i n i t i a l  I) f o l l o w s  differential  simplifications  results  after  to  of the model.  discontinuous  some  a n d WO  i n 2.Id to allow  system  nonlinear,  looking  (Table  The r e a d e r  derivation  i s diverted  •  i s efficiency  i s used  growth  of J days,  ZO = X.S/WO  GW  term  term).  reproductive  where  weight-dependent  is little  form.  i s employed  and extended  These  by c o m p u t e r  qualitative analysis  A  here  of t h e model space.  and the point  in  s e r i e s of to  obtain  over approximate simulation.  i s the r e c o g n i t i o n of  42  Table I. Parameters used i n Steele's model (2.1).  R...nutrient  concentration  (carbon equivalent)  RO...nutrient concentration  i n mixed l a y e r .  (carbon equivalent) below mixed l a y e r .  V...mixing rate through thermocline. U . . . f r a c t i o n of excreted  nutrient recycled.  P...phytoplankton carbon concentration  i n mixed l a y e r .  A. ..maximum phytoplankton growth rate (day ^ ) . B. ..half-saturation constant (carbon equivalent)  for nutrient-  dependent growth. Z...zooplankton density (#/l). W...zooplankton weight (ug C/ind). C. . . f i x e s maximum zooplankton ingestion rate. Pi..threshold for zooplankton grazing on phytoplankton carbon. D. . . f i x e s zooplankton grazing rate above PI. E. . . f i x e s component of metabolic rate proportional to ingestion. F. . . f i x e s basal metabolic rate. GW. .maximum of weight and density-dependent mortality r a t e . Wl..weight threshold for m o r t a l i t y . Zl..number threshold for m o r t a l i t y . H...'half-saturation'  constant f o r mortality.  GX..constant mortality rate. Z O . . i n i t i a l number of n a u p l i i i n cohort. WO..initial naupliar weight. W2..adult weight. S...reproductive store. J...period over which reproductive  store accumulates.  43  two  aspects of the seasonal behaviour  Landry;  namely, the t r a n s i e n t  concentrations  response  (the s p r i n g bloom),  s t u d i e d by S t e e l e a n d to high i n i t i a l  and t h e approach  cyclic  p a t t e r n i n the n u t r i e n t - l i m i t e d p e r i o d which  latter  i s more l i k e l y  is  t r e a t e d here The  distinct (For  to a  stable  follows.  The  t o be a m e n a b l e t o q u a l i t a t i v e a n a l y s i s a n d  first.  a n a l y s i s proceeds time  nutrient  through  t h e r e c o g n i t i o n of t h r e e  s c a l e s i n t h e model under n u t r i e n t  limitation.  an i n s t r u c t i v e e x a m p l e o f t h e u s e o f m u l t i p l e t i m e  scales i n  the a n a l y s i s of a c o m p l i c a t e d e c o l o g i c a l model, see Ludwig,Jones and  H o l l i n g ( 1 9 7 8 ) . ) A time  o b t a i n e d by d i v i d i n g saturation constant of  scale for nutrient  the source  t e r m , V.RO, by t h e h a l f -  f o r n u t r i e n t uptake,  V ( 0 . 0 1 , d a y ) a n d RO 1  t u r n o v e r c a n be  B.  For Steele's values  (760 p g C ( e q ) . ! - ) , V.RO 1  equals  S t e e l e u s e d a r a t h e r h i g h v a l u e o f B (96 p g C ( e q ) . l to  Landry).  of  v e r y low n u t r i e n t c o n c e n t r a t i o n s i n t h e oceans,  _ 1  7.6.  according  Recent chemostat r e s u l t s , ' c o m b i n e d w i t h o b s e r v a t i o n s suggest  that  h a l f - s a t u r a t i o n c o n s t a n t s f o r g r o w t h s h o u l d be s m a l l e r t h a n of  o r d e r 0.1 )jg a t N . l "  (McCarthy nutrient  1  or approximately  and Goldman,1978).  This results  1  scale for that of  (maximum g r o w t h r a t e 0.2 d a y , y i e l d i n g a t i m e 1  s c a l e of 5 days) or zooplankton. t r e a t i n g R as a f a s t v a r i a b l e ; concentration adjusts rapidly  equation  i n a time  t u r n o v e r o f o r d e r 1 d a y , much s h o r t e r t h a n  phytoplankton  so t h a t R  10 jug C ( e q ) . ! -  this,  We p r o c e e d  t h e r e f o r e by  t h a t i s , by a s s u m i n g t h a t t o changes i n o t h e r s t a t e  nutrient variables  R(P,Z,W), where R makes t h e r i g h t - h a n d s i d e o f 2.1a z e r o .  S u b s t i t u t i n g R = R i n e q u a t i o n 2.1b g i v e s  44  P = V.RO - V.P - Z.W-. . ( (C-U.E) . f (P) - U.F) , 0  2.2a  7  where f ( P ) s t a n d s f o r ( P - P 1 ) / ( D + P ) , a n d t h e t e r m V.R h a s been neglected  s i n c e R i s assumed t o be o f o r d e r  o n e - h u n d r e d t h RO. equation been  When c o m b i n e d w i t h  2.2a f o r m s a s y s t e m  (2.2),  B or approximately  e q u a t i o n s 2.1 c , d , e ,  f r o m w h i c h n u t r i e n t s have  eliminated. The  provided  second and t h i r d  t i m e s c a l e s c a n be d i s t i n g u i s h e d  t h e copepod m o r t a l i t y r a t e  i s low enough t h a t  Z changes  s l o w l y c o m p a r e d w i t h p o t e n t i a l g r o w t h r a t e s o f P a n d W. can  Then Z  be t r e a t e d a s a s l o w v a r i a b l e a n d t h e b e h a v i o u r o f P a n d W  considered  with  Z fixed.  The s y s t e m  P = V.RO - V.P - Z.W - .( (C-U.E) . f ( P ) - U.F)  2.3a  W = ( (0.7.C-E) . f (P) - F).W°-  2.3b  0  has  7  7  t h e phase plane p o r t r a i t  equilibrium  shown i n F i g 8.  The n o n - t r i v i a l  s o l u t i o n (P,W(Z)) o f 2.3 i s s t a b l e p r o v i d e d  p o s i t i v e , a c o n d i t i o n which i s always s a t i s f i e d value as  of PI.  Then, a c c o r d i n g  regardless  to the slow-variable  quasi-equilibrium  and  Z(t).W  0 , 7  reach adult density  solution (P,W(Z(t))).  are both constant.  h a s d r o p p e d by a f a c t o r  c a p i t a m o r t a l i t y i s constant  ZO.exp(-GX.t) and, a l l o w i n g generation  t i m e i s g i v e n by  According  track  t o 2.3, P  I t follows that a cohort  w e i g h t , W2, f r o m an i n i t i a l  of t h e  approximation,  Z ( t ) decreases through m o r t a l i t y , P ( t ) and W(t) should  the  per  f'(P) i s  will  w e i g h t , WO, when t h e  (WO/W2) . 07  F o r example, i f t h e  (GW = 0, GX f 0 ) , Z ( t ) =  f o r the incubation  period J , the  Figure  8.  Phase plane p o r t r a i t  f o r t h e s y s t e m 2.3-  46  T = J + 0.7.1n(W2/WO)/GX. So  2.4  f a r , o n l y t h e g r o w t h o f a s i n g l e c o h o r t h a s been d e a l t  w i t h b u t an a p p r o x i m a t e possible.  During  treatment  t h e p e r i o d of J days over  stores a r e accumulated, equation  of reproduction i s also which r e p r o d u c t i v e  W i s f i x e d a t a d u l t weight  2.3b a n d t h e e q u i l i b r i u m v a l u e P a r e n o t r e l e v a n t . I t  is consistent with the slow-variable approximation P  i s approximately  W = W2.  W2, s o t h a t  equal  Substituting  integrating  t o P ( Z ) , where P makes P = 0 i n 2.3a f o r  P = P(Z) i n the equation  gives, after  t o assume t h a t  f o r S and  J days:  S = SO - Z.W° . S I 7  where Z.W  07  i s evaluated at the beginning  of the reproductive  p e r i o d and  50  = V.RO.J.(0.7.C-E)/(C-E.U) ,  51  = F.C.(1.-0.7.U) .  But  a c c o r d i n g t o the approximate  Z0 .W0 3  0  7  equation  Z0  9 + 1  ( l . - e x p ( - G X . J ) ) / ((C-E.U).GX) .  where Z O  3  treatment  i s the i n i t i a l  o f growth,  Z.W - =  size of the gth cohort.  0  7  Using  2.1e, i t f o l l o w s t h a t  = S0.X/W0 - Z0 .S1.X/W0°9  This constitutes a difference coefficient  o f ZO  9  i n equation  3  .  2.5  equation  f o r ZO . 3  2.5 i s l e s s t h a n  the e q u a t i o n has a s t a b l e c o n s t a n t  solution  I f the 1 i n magnitude,  ZO* a n d , a c c o r d i n g t o  47  this  approximate  solution  there  i s a corresponding  of the phytoplankton-zooplankton  coefficient stable  theory,  i s greater  cyclic  expected.  2.3 S i m u l a t i o n The  coefficient  rate,  model  conditions  any  for thresholds  view  under  simulation  by c o m p u t e r  simplest  fixed  first  metabolic  rate  simplified 0.06  case  metabolic The  1  generation  stable  cannot  be  between  according  t o 2.4  2.2  terms.  without This  i sin  surprising in  The p r e d i c t i o n has been the  simplified  and the f u l l constant  were  model  mortality  a n d GX  s o l u t i o n s were a s GX  the slow-variable simulated  f o r the poor  a  cycle  theory.  generation  agreement  i s portrayed. over  rate  2.1 f o r rate and  fixing  GX  i n the  the  0.02 t o  with  in qualitative  However,  quantitative  a n d thos,e  (Fig 9).  c a n be s e e n  of copepod  predicted  Part,of  the  i n F i g 10a, where  The p h y t o p l a n k t o n  the period  from  approached  times  good  by  ranging  increased,  i s not p a r t i c u l a r l y  explanation  obtained  the mortality  F o r F = 0.4  cyclic  constant  solution to  (P1=E=GW=0).  decreasing  agreement  cyclic  1 b u t somewhat  o f no t h r e s h o l d s ,  2.2.  with  being  model  F and v a r y i n g  agreement  from  and a  to the basal  predation  of both  s e t of simulations  time  simulated  amplitude  a stable  results.  simulation  rate  model  day" ,  i s unstable  of n u t r i e n t - l i m i t a t i o n  of Chapter  phytoplankton-zooplankton the  predicts  or quadratic  the r e s u l t s  of S t e e l e ' s  tested  I f the  Results.  Steele's  with  constant  2.2.  cyclic  F.  theory  keeping  ZO  i s proportional  approximate  need  system  1 i n magnitude,  s o l u t i o n t o 2.2 h a v i n g  This  metabolic  than  stable  density  growth,  due  i s  far  partly  48  Figure  9.  Comparison of g e n e r a t i o n times p r e d i c t e d by (solid of  line)  the system  and 2.2  those obtained i n numerical (dots).  equation solutions  2.4  49  300  r15  200 U  O)  100  50  100 150 200 TIME (days)  250  300  350 400  150 200 250 TIME (days)  300  350 400  400n 3004 200A U CJ)  3.  100H  Figure  10.  Stable (a)  cyclic  solutions  F=0.4, GX=0.05;  o f the system  2.2 f o r :  ( b ) F = 0 . 2 , GX=0.05.  50  to the  l i m i t a t i o n s of  to the  perturbation  the  i m p o s e d by  of e a c h r e p r o d u c t i v e I n s p i t e of  10b,  the  the  this,  predicted in  the  cycle (Fig  by  the  f u r t h e r q u a l i t a t i v e agreement between  the  simulation  cycle  i s not  10c).  involves  found.  n a u p l i i at  When F i s d e c r e a s e d  l o w e r P and Also,  and  low  2 i n the  h a v i n g been d e s t a b i l i z e d by aperiodic  as  shown i n F i g  higher  ZO,  as  i n c r e a s i n g F to  approach to a rather naupliar  b e h a v i o u r c o r r e s p o n d s i n the  s o l u t i o n of p e r i o d  p e r i o d i c and  was  theory.  r e s u l t s i n an  This  of  a f f e c t e d , but,  involving a l t e r n a t e l y high  to a s t a b l e ZO*  end  slow-variable  simulation  release  partly the  cycle period  simulated  the  a p p r o x i m a t i o n and  period.  a p p r o x i m a t e t h e o r y and t o 0.2,  slow-variable  curious  recruitments  approximate  difference  i n c r e a s i n g F.  The  aroused considerable  theory  equation  2.5,  phenomenon  s o l u t i o n s to d i f f e r e n c e equations  s i m p l e e c o l o g i c a l m o d e l s has  0.6  of  in  interest  (eg  May,1975). The eliminate Steele's  validity  of  n u t r i e n t s has full  model 2.1.  (10 jug C ( e q ) . ! " ) , 1  a stable cyclic that  obtained  r e m a i n s low  the  f a s t - v a r i a b l e assumption  been t e s t e d For  by  F = 0.4,  ( F i g 11a),  f r o m s y s t e m 2.2 ( o f o r d e r B)  and  of  GX  low  = 0.05  However, i t can  be  oscillations  i n R,  (Fig  f o r t h i s can  be  e q u a t i o n 2.1a  and  the  fact that,  identical  R  is ways.  parameters P and  Z appear  found i n the  f o r F=0.2, t h e  to  cycle,  v i o l a t e d i n a number of with other  B  model a p p r o a c h  Throughout the  unchanged, l a r g e , d i v e r g i n g explanation  full  and  1  f a s t - v a r i a b l e approximation  e x a m p l e , i f F i s r e d u c e d t o 0.2,  l i b ) . The  day"  which i s almost  ( F i g 10a). the  to  computer s i m u l a t i o n  n u m e r i c a l s o l u t i o n s of t h e  solution  quite accurate. For  using  cyclic  nutrient  solution  .51  TIME (days) Figure 10c.  Stable c y c l i c solution of the system 2.2 f o r F=0.6, GX=0.05.  O  50  Figure 11a.  100  150 200 250 TIME (days)  300  350 400  Stable cyclic solution of the system 2.1 for F=0.4, GX=0.05 and B=10.  53  r20  400  KI  100 150 200 TIME (days)  r15  300 cr <D  200  U  cn  3  CC  100  t_  o  D_  0  50 . 100  Figure l l b , c .  150 200 250 TIME (DAYS)  300  350 400  Behaviour of the system 2.1 f o r F=0.2, GX=0.05 and B=10 with:  (b) D=100. (unstable o s c i l l a t i o n s )  and (c) D=175. (stable  cycle).  54  p r e d i c t e d under t h e f a s t - v a r i a b l e a p p r o x i m a t i o n i n v o l v e s low phytoplankton the phytoplankton  concentrations.  N u t r i e n t u p t a k e by  i s n o n - l i n e a r i n R w i t h a maximum v a l u e o f A.P.  Thus, i f P i s t o o s m a l l , p h y t o p l a n k t o n mixing  ( F i g 10b)  uptake cannot match t h e  i n p u t V.RO a n d t h e e q u i l i b r i u m s o l u t i o n R ( P , Z )  from a low v a l u e to a high value  ( o f o r d e r B) d e t e r m i n e d ( o f o r d e r RO) d e t e r m i n e d  switches  by p h y t o p l a n k t o n by m i x i n g  f a s t - v a r i a b l e assumption f o r n u t r i e n t s i s of course  losses.  The  no l o n g e r  v a l i d and, i n t h e computer s i m u l a t i o n , t h e i n s t a b i l i t y with the phytoplankton-zooplankton  uptake,  associated  i n t e r a c t i o n appears t o  dominate. This e x p l a n a t i o n suggests cycles  f o r l o w F.  A sufficient  a number o f ways t o r e c o v e r decrease  i n the rate of n u t r i e n t  i n p u t , V.RO, o r i n c r e a s e i n maximum g r o w t h r a t e , A, w i l l nutrient build-up.  stable  C h a n g e s c a n a l s o be made i n o t h e r  w h i c h i n c r e a s e P, c o m p e n s a t i n g f o r t h e d e c r e a s e  prevent  parameters  i n F.  For  e x a m p l e , P i s s c a l e d by t h e g r a z i n g h a i f - s a t u r a t i o n c o n s t a n t , When D i s i n c r e a s e d f r o m 100 t o 175 ug C . l " , 1  Landry),  t h e s o l u t i o n of the f u l l  approaches the s t a b l e c y c l i c  D.  ( t h e v a l u e u s e d by  m o d e l f o r F=0.2 ( F i g 1 1 c )  s o l u t i o n p r e d i c t e d by t h e f a s t -  v a r i a b l e m o d e l 2.2 ( F i g 1 0 b ) . An  obvious  i n c r e a s e B.  way t o v i o l a t e  I f Steele's value  adopted, t h e time  the f a s t - v a r i a b l e assumption f o r B o f 96 ug C ( e q ) . l  approximate treatment  and  c a n n o t be a p p l i e d .  a n d D = 100 ug C . l " , 1  R are obtained  is  s c a l e f o r c h a n g e s i n n u t r i e n t becomes  comparable t o t h a t of changes i n p h y t o p l a n k t o n  0.05  _ 1  i s to  and t h e p r e c e d i n g  In f a c t ,  f o r F=0.4,GX =  rapidly diverging oscillations  i n numerical  in P  s o l u t i o n s on i n c r e a s i n g B t o 96.  55  A  stable  ^jg P  cyclic  C.l" ,  this  and low  R/B.  1  Another these from the  solution solution  approach  copepod  age  P  = A.R.P/(B+R)  G  = 0.7.G.C.f(P)  where  G  model  with  in  analytic  results  o b t a i n e d by d r o p p i n g  + U.F.G  2.6a 2.6b 2.6c  biomass.  possesses a n o n - t r i v i a l  equilibrium  solution  by  = m/(0.7.C) 0.7.V.R0/(m-0.7.U.F)  A.R.P/(B+R)  = V.R0.m/(m-0.7.U.F).  Taylor-series  linear  expansion  SP SG  about  this  equilibrium  system  / • \ SR  high  interpreting  - m.G  i s copepod system  them  t o 175  by c o n s i s t e n t l y  t o be u s e f u l  i s t o compare  D  - G.C.f(P)  (R,P,G) d e t e r m i n e d  A  appears  increasing  structure:  - A.R.P/(B+R)  This  =  which  a more m a t h e m a t i c a l l y t r a c t a b l e  = V.RO  G  r e c o v e r e d by  being characterised  numerical results  R  f(P)  c a n be  I  G.  8R SP  SG  where t h e c o n s t a n t m a t r i x GV h a s t r a c e  yields the  56  I  It  c-  =  -A.B.P/(B+R)  i s well-known  that A  stability  of  thresholds  is  the  (PI  stabilizing the  and  the  the  second  B  that  0),  1/B  magnitude,  to  be  the  negative.  local  of  type  II  functional  the  term  i s negative  held  second  held  1/D D  B  remains  while  the  tends  to  increased first  D  is  also  first  term thus  increased  be  shown  term  stabilize  results  the  constant;  If  i t can  simulation  is the  system.  constant,  like  If  this  response  represents  constant,  term  absence  is positive;  and  limitation.  2.6  the  the  the  2.7  increasing  with  In  in  d e s t a b i l i z e the  that  concludes  proceeding,  obtained  so  far.  recognition  solutions  to  thresholds  and rate.  solutions  do  space  which  for  the  functional  the  analysis  i t seems By  of  use  fast  Steele's  metabolic  Of  for  that  increases  the  reported  system.  above  for  model.  This  outweigh  2.7  term  in  decreases  so  .  second  nutrient  while  is consistent  Before  first  parameters  term  f'(P))  condition  2. c,,  the  parameters  tends  other  Steele's  the  =  is  e f f e c t of  like  increasing  This  necessary  -  >  The  other  decreases  in  a  G.C.(f(P)/P  destabilizing effect  (Holling,1965).  with  A  (R,P,G)  of  +  for  of  the  advisable a  series  to of  model  have  been  predicted  are  capita  simulation  stabilizing  course,  of  Steele  the  stable in  mortality  has  restricted  the  to  shown a  e f f e c t s of the  region  rate  in  nutrient saturating  based  on  cyclic  the  that  period.  results  approximations  scales,  per  limited  summarise  time  d e s t a b i l i z i n g e f f e c t s of  response  nutrient  slow  constant  but  the  and  Computer  exist  of  absence and  fixed  these parameter limitation type  II  copepods.  demanded  of  his  model  that  i t  reproduce  of  57  the q u a l i t a t i v e cycle  features characteristic  i n the North  Sea,  starting  conditions corresponding The  to the  guarantees  initiation solution  = 0.4,  D = 100  In f a c t ,  pg C . l "  and  1  s t a b l e c y c l e o f F i g 11a, conditions, 12a).  the  During  produced.  resulting  GX  = 0.05  t h e i r weight  reduce t h i s this  rapidly  a very  a p p e a r s t o be term;  any  limited  d u r i n g the prolonged  impressive rate  but  reducing  from h a p p e n i n g , i t i s o b v i o u s l y n e c e s s a r y intense grazing pressure.  s i z e of t h i s  the primary  stabilizing  One  way  of t h i s term  who  on c o h o r t  size  do  thereby This  quadratic predation i n the l a t e r  importance.  to  to  in high  l a r g e second c o h o r t .  r o l e of Landry's  effect  t o s e t a maximum l i m i t (1977),  p e r i o d of  54.  r e g i m e i s of s e c o n d a r y  Mullin  high  phytoplankton  r a t e s when c o p e p o d numbers a r e h i g h ,  r e d u c i n g the  (Fig  phytoplankton  i s t o i n t r o d u c e a p r e d a t i o n term which r e s u l t s  copepod m o r t a l i t y  and  see h i g h  The  the  of copepods i s  manner, t h e c o n s t a n t m e t a b o l i c  b u r s t of  F  initial  s i m u l a t i o n i s most u n r e a l i s t i c  the p h y t o p l a n k t o n .  this  to  1  are used t o g e t h e r w i t h S t e e l e ' s  t o z e r o by day  To p r e v e n t  satisfy  day" , corresponding  P w h i c h f o l l o w s , t h e c o p e p o d s s t a r v e i n an  rather u n r e a l i s t i c  to  1  grow r a p i d l y , q u i c k l y p r o d u c i n g  on  s e t of  i f t h e v a l u e s B = 10 pg C ( e q ) . ! " ,  c o n c e n t r a t i o n d e c l i n e s q u i c k l y and  is  conditions w i l l  I n d i v i d u a l copepods i n i t i a l l y  grazing pressure  s p r i n g bloom.  for a particular  the s p r i n g bloom, a l a r g e c o h o r t  c o n c e n t r a t i o n s and  low  of the  initial  t h a t the t r a n s i e n t approach  t h i s c y c l e from the p r e s c r i b e d i n i t i a l Steele's c r i t e r i a .  seasonal  from a p r e s c r i b e d set of  e x i s t e n c e of a s t a b l e c y c l i c  p a r a m e t e r s i n no way  of the observed  Another  nutrient-  alternative  i n t h e manner o f  made ZO a h y p e r b o l i c f u n c t i o n o f  Steele  58  F i g u r e 12a.  Attempted using  s i m u l a t i o n of s p r i n g bloom  t h e s y s t e m 2.1 w i t h B = 1 0 , F = 0 . 4 ,  D=100 a n d GX=0.05.  59  reproductive A  store,  third  constant  S.  alternative i s to increase  f o r grazing,  thereby  phytoplankton  densities.  who  D  increased  made  (Fig 12b).  of shorter  solution bloom from is  is still more  thresholds of  a  only  about  larger  using  value  The  ingestion-dependent The  resulting  limited  2.4  than  this  predation  loss  1  and constant  than  weight  loss  to the stable following  cyclic  the spring reduced  C by s t a r v a t i o n .  without  B=10  It  adopting results  a n d D=175 a r e  depletion  following  simulation C.l" )  pronounced  In F i g 12c, the  of phytoplankton  final  copepod  to less  term.  F=0.5,GX=0.04,  weight  Landry,  i n d i v i d u a l s being  better  ( 2 5 0 pq  of D  high,  by  i s much  i s less  loss  0.35 pq  copepod  50%.  the result  approach  weight  done  a t low  I f t h e same c h a n g e i s  than  The p e r i o d that  .  to less  or Landry's  simulation  extent  C  t o d o much  presented. the  3 pq  1  pressure  been  depletion  rapid  Unfortunately,  -  above,  Phytoplankton  unrealistically  than  possible  t o 1 7 5 jag C . l  and a  grazing  has a l r e a d y  described  duration  occurs.  This  100  to the simulation  improved and  from  reducing  the h a l f - s a t u r a t i o n  i s reduced  the spring  ( F i g 12d) u s e s  and Landry's metabolic following  to  bloom i s  an  even  combination  rate  (E=0.3,  the spring  F=0.2). bloom i s  25%.  Conelusions. • Perhaps  simulation plankton without  t h e most  model  cycle  i s that  terms.  thresholds This  result  'realistic'  i n the North  invoking  predation  striking  of t h i s  review  i m i t a t i o n s of the  S e a c a n be o b t a i n e d in herbivore  of course  does  of  grazing  from or  not e s t a b l i s h  Steele's seasonal  t h e model quadratic that  Figure  12b.  S i m u l a t i o n o f s p r i n g a n d summer w i t h u s e d f o r F i g 1 2 a e x c e p t D=175.  parameters  F i g u r e 12c.  As f o r F i g 12b e x c e p t GX=0.04.  T I M E (days) Figure  12 d.  S i m u l a t i o n o f s p r i n g a n d summer w i t h E = 0 . 3 , GX=0.05 a n d D=250.  F=0.2,  63  thresholds in  the  or  North  quadratic Sea  their  existence  fact,  there  the  species  of  constant more  ecosystem, based  appears  existence  of  on  to  their  be  better  mortality  a  way  of  Landry's  predation  term.  The  preceding  results  qualitative is and in  analysis  possible.  The  incomplete, the  led  to  parameter  space  various  and  parameter  analyses  in which  is  large  enough,  of  values  of  is  lower,  stable  concluded  B,  that  at  the  results  thresholds  in  not  obtain  to  were  <  of  reasonable  least  some  in  a  chain  the  of  found  are  solutions  B  <  B  100  in nature  predation results  to  simulations.  of  the  by  the  standard  over  one  at  for  grazing  a  wide  a  range  However, and  in  if  i t could Steele's  i n B,  already  and  simply  revealed  small  changes  solutions  a l l by  is critical.  were  when  B  this  of  1  for  the  region  jug C ( e q ) . ! ' .  exist  of  approximate  varied  exist  using  wherever  i n computer  inadequacy  no  thresholds  at  were  for  a  than  cyclic  limited  which  of  for In  is certainly  although  stable  been  insensitive  and  now  advantages  absence  over  model.  evidence  h a l f - s a t u r a t i o n constant  0  value  food  the  parameters  cyclic  least  grazing  the  have  solutions only  simulation  did  i f the  stable  in  parameters  sensitivity  Steele's  at  important  argument  e c o l o g i c a l models,  exist  testify  example,  the  of  analysis  any  copepods  discovery  not  and  assumption  here,  qualitative  For  for  combinations  i n t e r a c t i o n s between  time.  the  conducted  These might  and  illustrate  n u t r i e n t - l i m i t e d regime mortality.  The  phenomena,  complex  the  in  experimental  rate  do  present  i t weakens  truncating  analysis  quadratic  trying  of  not  necessity  (Frost,1975),  capita  satisfactory  are  although  threshold-type  copepods  per  predation  but force.  thresholds  were  He  D be  64  omitted  due  The broader to  low  The  partly  insight  major  the  into  approach food  to  to  Steele's  the  problem  response  rather  than  allowing  stable  cyclic  realistic  high the  unrealistic  basal  of  a  depletion  dismissed  as  energy and  to  loss was  was  rate.  non-negative  did  and  (thereby  and  a  (zero  the  the  transient parameter  sub-region  space allows  conditions  difficulty in  the  of  lies  in  post-bloom  assumption  in  occur  of  a  high  of  and  the  optimal  a  metabolism  However,  reduced growth  the  rate  a  result  later,  feeding  more  that  there  to  optimal  feeding  with  Steele's  filtering  rate  at  a l l food  low  densities.  so  trial was  that basal  simulations,  consequently  detailed analysis of  i s an  filtering  consistent  at  entirely  non-zero  response  of  copepods, important rate,  response  not they  (1974) a s s u m p t i o n  food  by  extreme  metabolism),  Steele's  proportional  simulation  preventing  (When  one  In  standard  metabolic  basal  (1977) c o n c l u d e d  i s roughly  involve  the  thresholds.  realistic  of  occur  to  of  a  response  a  small  in his  impossible.  adopted  loss  Frost  ingestion  both  ingestion  budget  of  obtained  grazing  unrealistic).  component  herbivore  initial  can  due  difficulty  phytoplankton)  weight  a  suggests  rate.  in  rate  region  from  which  of  involves  only  D.  here  matter  earlier,  depletion  threshold of  metabolic  Steele  losses  and  obtaining  the  cycle  explained  this  weight  copepod  this  avoided  proportional copepod  was  weight  metabolic  assuming  to  the  in  of  B  obtained  model  solutions,  phytoplankton  Steele  the  As  Steele's  for  question  here  stability;  approach  nutrient.  period  from  model  including  encountered  pattern  assumed  biological  concentrations,  seasonal  a  values  densities  and  as  65  S t e e l e and F r o s t ' s argument does imply  t h a t an o r g a n i s m  w h i c h does not behave o p t i m a l l y , and i n s t e a d m a i n t a i n s filtering  r a t e a t low food d e n s i t i e s , w i l l  high metabolic  be s u b j e c t  c o s t a n d r a p i d l o s s o f body w e i g h t .  there  t o an a s s u m p t i o n o f no t h r e s h o l d a n d a h i g h  metabolism  in Steele's  l o s s , which w i l l according  Obviously,  to a  This i s  equivalent  (1974) m o d e l .  a high  this  basal weight  be e s p e c i a l l y s e v e r e f o r s m a l l i n d i v i d u a l s  t o t h e W0  7  metabolic  law, cannot continue  indefinitely.  T h e r e a p p e a r t o be two e x t r e m e o u t c o m e s p o s s i b l e . One p o s s i b i l i t y the will  organism's f i l t e r i n g decrease.  (most l i k e l y will  i s t h a t , a s i t s body c o n d i t i o n d e t e r i o r a t e s , r a t e and c o n s e q u e n t l y  Provided  t h i s occurs  i t s metabolic  rate  on a s h o r t - e n o u g h t i m e  scale  f o r small organisms), t h i s p h y s i o l o g i c a l response  have t h e same e f f e c t i n , t h e s i m u l a t i o n m o d e l a s an  behavioural  response or a t h r e s h o l d .  been e n v i s a g e d by S t e e l e t o food  concentrations  important observed  point  (1974),  This p o s s i b i l i t y  when he m e n t i o n e d t h a t  i n the f i e l d  functional crustaceans The filter  grazing  An  e x p e r i m e n t s o f t h e t y p e commonly  (Marshall,1973).  and d u r a t i o n  large, late-stage  copepods  E f f e c t s of food  (up t o 5 d a y s ) ' o f  p r e - c o n d i t i o n i n g on  r e s p o n s e p a r a m e t e r s h a v e been r e p o r t e d  f o r fresh-water  (Buckingham,1978).  second p o s s i b i l i t y  i s that the i n d i v i d u a l continues  a n d l o s e w e i g h t a t maximum r a t e s u n t i l  some maximum p e r m i s s a b l e highly  responses  i s t h a t t h e p h y s i o l o g i c a l r e s p o n s e w o u l d n o t be  c o n d u c t e d o v e r 24 h o u r s o r l e s s u s i n g  concentration  may have  m i g h t o c c u r on s e v e r a l t i m e s c a l e s .  i n short-term  collected  'optimal'  simplified  weight l o s s exceeds  amount, when t h e o r g a n i s m d i e s .  s i n g l e cohort  to  In the  v e r s i o n of S t e e l e ' s model, t h e  66  e n t i r e cohort multi-cohort  will  die simultaneously.  v e r s i o n , as p r e s e n t e d  I n a more  by L a n d r y  realistic  (1976),  or  one  i n v o l v i n g d i f f e r e n c e s i n g r o w t h r a t e b e t w e e n i n d i v i d u a l s , as reported  e v e n i n l a b o r a t o r y c u l t u r e s u n d e r homogeneous c o n d i t i o n s  ( P a f f e n h o f f e r and  Harris,1976),  a decrease i n grazing pressure for  small  grazing  zooplankton,  short-term  transient  behaviour  important  and,  at  to  least  e f f e c t to a  goal.  circumstances,such  aspect.  r e f e r r e d t o above can  Again,  consider  only  The  any  physiological reasonable  actual  response,  Subarctic  experimental  the  grazing  be  Pacific, and  approaches experiments  behavioural  response.  i n d i v i d u a l s (Reeve e t aj. ,1970; s t u d i e s h a v e been c o n d u c t e d  by  reported  Paracalanus parvus females without s t a r v a t i o n m o r t a l i t y c o u l d be  open t o t h e  Steele(1974).  ( 1 9 8 0 ) has  on  weight  measured i n s t a r v e d i n d i v i d u a l s ,  r e s p i r a t i o n r a t e s are  interpretation" discussed  to note t h a t Checkley  the  p h y s i o l o g i c a l e f f e c t of  r a t e c a n n o t be  w h i l e measurements of  as  (the  l a b o r a t o r y s t u d i e s of r e s p i r a t i o n  most of t h e  i n d i v i d u a l s , and  filtering  ensure  short-term  weight l o s s i n starved  Ikeda,1977). late-stage  current  The  T h e r e h a v e been a number of r a t e s and  longer-term  densities  d e f i n i t i o n a p p e a r s t o be a c h a l l e n g i n g  Most of t h e  one  food  somewhere b e t w e e n t h e s e e x t r e m e s , may  i t s experimental  address only  response, the  i n S t e e l e ' s model.  lies  i n other  valuable  r e s p o n s e t o low  starvation mortality) could  which probably  of  the phytoplankton  have a s i m i l a r q u a l i t a t i v e  t y p e s of  behavioural  r e s p o n s e and  l o s s on  on  lead  threshold.  Thus a l l t h r e e  and  may  starvation mortality will  food  problems  It is interesting  mean s u r v i v a l t i m e s  t o be  a significant  4-5  d a y s so  for  that  f a c t o r , at l e a s t f o r  67  s m a l l copepods w i t h o u t important  fat stores.  c o n s i d e r a t i o n f o r the  I t i s o b v i o u s l y not  late  s u c h as C a l a n u s p l u m c h r u s w h i c h b u i l d  stages  of  larger  up e x t e n s i v e  an species  fat stores  (Fulton,1973). The  experimental  p r o c e d u r e s most l i k e l y  question  a p p e a r t o be  those  reviewed  by  P a f f e n h o f f e r and  m o r t a l i t y and in  clearance  1  even lower  m o r t a l i t y has  although  these  been  a t low  P a f f e n h o f f e r and  interesting  t o see  food d e n s i t i e s .  these  and  growth, cycle  reported  food  f o o d d e n s i t i e s were  t o a l l o w g r o w t h a t c l o s e t o maximum r a t e s  (Paffenhoffer,1970; be  (1970),  for studying  Temora l o n g i c o r n u s  ( a b o u t 20 jug C . l " ) ,  sufficient  certainly  Harris(1979),  Increased  f o r C a l a n u s p a c i f i c u s and  still  by P a f f e n h o f f e r  this  r a t e s over the e n t i r e copepod l i f e  laboratory cultures.  densities  reported  to resolve  Harris,1976).  I t would  experiments repeated  at  68  CHAPTER 3 PHYTOPLANKTON AT O.S.P.: DATA ANALYSIS AND MODELLING  3.1 I n t r o d u c t i o n The  aim i n t h i s chapter i s t o develop a r e a l i s t i c  p h y t o p l a n k t o n g r o w t h a t O.S.P. an a n a l y s i s o f d a t a c o l l e c t e d the l e v e l  The u n d e r l y i n g  from t h e l i t e r a t u r e , be  The m o d e l i s b a s e d p r i m a r i l y on f r o m t h e w e a t h e r s h i p s a t O.S.P. a n d  o f d e t a i l c o n s i d e r e d h a s been c h o s e n  observations.  inferred  v a l u e s which  standing stock i n t h i s chapter,  i n s u c h a way t h a t  i t c a n e a s i l y be e x t e n d e d  t o i n c l u d e g r a z e r s i n C h a p t e r 4, where t h e q u a l i t a t i v e raised  i n Chapter 1 w i l l  3.2 D a t a 3.2.1  cannot  The m o d e l i s u s e d o n l y t o p r e d i c t n e t  p r i m a r y p r o d u c t i o n from a f i x e d but i s c o n s t r u c t e d  t o match these  s t r u c t u r e o f t h e model i s a d a p t e d  as a r e c e r t a i n parameter  from t h e d a t a .  model o f  be a d d r e s s e d  hypotheses  quantitatively.  Analysis  D e s c r i p t i o n of t h e Data S e t . Weathership observations r e l a t e d  to phytoplankton f o r the  y e a r s 1964 t o 1976 h a v e been p u b l i s h e d (Stephens,1966;  Stephens,1968;  i n manuscript  Stephens,1970;  Stephens,1977).  V a r i a b l e s measured i n c l u d e • S e c c h i d e p t h , c h l o r o p h y l l uptake r a t e s and n i t r a t e c o n c e n t r a t i o n . were m e a s u r e d s t a r t i n g  i n 1969.  form  a,  1 4  C  C h l o r o p h y l l s b and c  The d a t a r e c o r d c o n s i s t s o f  a p p r o x i m a t e l y 200 d e p t h p r o f i l e s a n d an a d d i t i o n a l o b s e r v a t i o n s , a l t h o u g h n o t a l l v a r i a b l e s were  600 s u r f a c e  necessarily  m e a s u r e d on a l l d a y s o r a t a l l d e p t h s . There a r e c e r t a i n p e c u l i a r i t i e s  t o t h e s a m p l i n g regime  which  69  can  be s e e n i n t h e s c a t t e r p l o t  concentrations given  of surface c h l o r o p h y l l  i n F i g 13.  From 1964 t o 1 9 6 8 , o b s e r v a t i o n s  were made f r o m o n l y one o f t h e two w e a t h e r s h i p s , s e r i e s o f 6 week g a p s i n t h e t i m e  series.  s u r f a c e s a m p l e s were c o l l e c t e d e v e r y regularly.  but the sampling  observations  p e r 6 week c r u i s e .  on e a c h  ina  W h i l e on s t a t i o n ,  s e c o n d d a y , more o r l e s s  A f t e r 1 9 6 8 , o b s e r v a t i o n s were made f r o m  weatherships,  taken  resulting  frequency  both  d r o p p e d t o one t o s i x  U s u a l l y one o r two p r o f i l e s  were  cruise.  3.2.2 C h l o r o p h y l l D a t a . The  c o n c e n t r a t i o n o f c h l o r o p h y l l a h a s been  measured from t h e w e a t h e r s h i p i n d i c a t o r of phytoplankton plotted  a t O.S.P. a n d i s t h e p r i n c i p a l  standing  i n C h a p t e r 1.  I t i s clear  a a t O.S.P. t e n d s t o be u n i f o r m l y i n a l l b u t a few o b s e r v a t i o n s .  low: l e s s than  This  s p r i n g bloom seen i n o t h e r  Sea ( S t e e l e , 1 9 7 4 ) Exceptions  1 ug C h i a . l "  (Semina,1958;  contrast t othe  temperate p l a n k t o n i c  of Georgia  and t h e N o r t h  (Parsons,1965),  Atlantic  the  (Cushing,1959).  c a n be s e e n i n F i g 1 3 :  f r o m 1 t o 4 ug C h i a . l " were o b s e r v e d i n  1 9 6 4 , 1 9 6 5 , 1 9 6 9 , 1 9 7 2 a n d .1975. values occur  1  I n 1964,1965 a n d 1 9 6 9 , t h e s e  high  i n small groups, a p a t t e r n c o n s i s t e n t with a s h o r t -  bloom ( o f low i n t e n s i t y compared w i t h c o a s t a l blooms (eg  Parsons,1965)),  1  feature of the S u b a r c t i c  t o the r u l e of constancy  scattered high values,  lived  values  f r o m F i g 13 t h a t c h l o r o p h y l l  Parsons,1965) and i s i n sharp  ecosystems such as t h e S t r a i t North  The s u r f a c e  h a s been r e m a r k e d upon by many r e s e a r c h e r s  Heinrich,1962; classic  stock.  i n F i g 13 a r e g e n e r a l l y r e p r e s e n t a t i v e o f t h e m i x e d l a y e r  described  Pacific  routinely  or with the advection  of a 'patch'  past the  70  2  0  >  1961  1962  •  •  1964  1963  2  X  CD  +  *  CO  0  t  *  •  1967  1966  1965  2  *  . 1968  •  CE  • +  *  -*  • .  • *  IE CJ  0  *  *** •  1972  1971  1970 ^ A  2  s  d A  A «  *  *  » *  0  v.  • •  1969  •  ^ ** •  1973 Figure 13.  *  .. •  •**  1974  ** *  *  *  *  . 1975  *» • *» » » *» **  »•  1976  Surface observations of c h l o r o p h y l l a from the weatherships at O.S.P. (Triangles denote high v a l u e s , given t o nearest i n t e g e r by d i g i t above.)  71  station.  The  sampling  high observations  frequency  m i g h t be e x p e c t e d The  i s low  to c o i n c i d e with a s h o r t - l i v e d  Any  intensity  or a n n u a l  interest, i s not  High  M a r c h and  September,1969, A p r i l  pattern.  Any  and  been m i s s e d  d i s t i n g u i s h e d by p e r f o r m i n g seasonal v a r i a t i o n  year  to year  annual  such  variations.  v a r i a t i o n s by  One  pattern The  August,1965, January,  years.  seasonal p a t t e r n i n the interest.  i n a time  following Kendall  T h e s e may  some a u t h o r s .  such  could  Annual  s e r i e s can  a v e r a g e and  be  (1973),  seasonal  be  looking (  The  to denote  r e f e r r e d t o as  The  and  inter-  patterns  f l u c t u a t i o n s of 12 month  a s t r a i g h t - f o r w a r d approach to  period.) these  i s t h a t the o c c a s i o n a l h i g h v a l u e s c o n t r i b u t e  overwhelmingly pattern.  a  i n the r e s i d u a l s ( K e n d a l l , 1 9 7 3 ) .  i s used here,  problems prevent  data.  that  that high values  a running annual  d i s c u s s e d here are s h o r t e r term Two  t o r e v e a l such  i n o t h e r months o r  seasonal c o n t r i b u t i o n s to v a r i a b i l i t y  'annual'  f r o m F i g 13  December,1972 and  s e r i e s are  c o n s i s t e n t low-amplitude  word  be  pattern in their  i n J u n e , 1 9 6 4 , J u l y and  b a c k g r o u n d C h i a v a l u e s c o u l d a l s o be of  for  will  O c t o b e r , 1 9 7 5 , r e v e a l i n g no c o n s i s t e n t s e a s o n a l  Gaps i n t h e t i m e  have e a s i l y  observation  f o r t h e o r i e s of  i t is clear  sufficient  pattern.  J u n e , J u l y and  values occur  but  the  bloom.  high observations  seasonal  w o u l d be of  the sampling  i s o l a t e d but  r e g u l a t i o n i n the S u b a r c t i c P a c i f i c  discussed l a t e r . occurrence  are  t h e r e , so t h a t o n l y one  i m p l i c a t i o n s of t h e s e  phytoplankton  i n 1975  t o t h e v a r i a n c e and  T h i s has  will  been o v e r c o m e h e r e by  d o m i n a t e any  looking for a  i n t h e s i x week c r u i s e m e d i a n s , p l o t t e d  second problem i s t h a t the presence  resulting  of  seasonal  i n F i g 14a,c.  irregular  gaps i n t h e  1964  1965  1  :  1966  '  1967  !  1968  1.2  G  EL  "3?  CD  O  C?  CD  0  J  :  F  1  M  1  fl  1  M  1  J  'J  1  fl  1  5  1  0  1  N  1  D  -3 Figure  14.  S u r f a c e c h l o r o p h y l l a (mg.m annual smooth f o r 1964-68. for  1964-68.  ):  (a) C r u i s e medians and  (b) S e a s o n a l f i t p l u s r e s i d u a l s  ho  ~  1  J .' F  1  M  'fl  1  M  '  J  '  J  '  H  '  S  '0  1  N  1  D  1  _3 Figure 14.  Surface chlorophyll a (mg.m annual smooth f o r 1969-76. for 1969-76.  ) : (c) Cruise medians and (d) Seasonal f i t plus residuals  74  time  series  will  contributions two-way the  lead  to variation.  analysis  latter  t o c o n f u s i o n of annual This  of v a r i a n c e with  problem,  suggested  is a  and  similar  empty  seasonal  problem  blocks.  by M o s t e l l e r  A  to the  solution  and Tukey(1977),  to i s to  iterate. The series width  procedure  problem  year) from and  describe  as f o l l o w s .  one y e a r )  seasonal  they  i s applied  harmonics  (period  An  re-smoothed;  seasonal  one y e a r ,  f i t a r e then the r e s u l t i n g  i s repeated  until  no  smooth  or seasonal harmonics.  course  obtained  procedure  one-half  (moving  added  back  residuals  further  and the f i r s t year  and The  a r e added  to the  i n annual  results  f i t i s applied  first).  a model  smooth  This  final  t o assuming  one-third  back  occurs  three  residuals  are refitted.  ( T h e same  window o f  to the annual  change  time  a r e of The  f o r observations of  form:  Y(t)  = Y  where  A  (t) + Y (t) + e(t) s  Y ^ ( t ) r e p r e s e n t s an a n n u a l  (repeated annual squares In  each  year)  and e ( t ) a  and s e a s o n a l components of unexplained the case  components SSQ  series  to the  t o the r e s i d u a l s .  i f the seasonal  i s equivalent  adapted  smoothing  f i t and the s e a s o n a l harmonics  procedure  the  annual  t o the time  are least-squares fitted the seasonal  has been  about  effect, residual  t h e mean.  seasonal  effect,  and choosing  s  so as t o m i n i m i z e  effect  t h e sum o f  residuals.  of C h i a, the f i n a l  accounted  Y (t) a  annual  and s e a s o n a l  f o r 19% and 12% r e s p e c t i v e l y Because  of t h e gaps  of the  i n the time  original  series,  no  75  a t t e m p t h a s been made t o t e s t components.  The e s t i m a t e d  0.12 ug C h i a . l " ug C h i a . l " .  c y c l e amplitude  from t h e second  1964-1968 a n d 1969-1976 s e p a r a t e l y , t h e s e a s o n a l  the f i r s t  p e r i o d h a s much l o w e r  and  e x p l a i n s o n l y 1.5% o f t h e SSQ a b o u t t h e mean Concentrations  potentially  contain  amplitude  ( 0 . 0 4 ug C h i a . l " ) 1  (Fig 14).  i n f o r m a t i o n about the taxonomic  a are given  the procedure a l r e a d y  i n F i g 15.  described  found f o r the C h i b/Chl a r a t i o  explained  composition  S c a t t e r p l o t s of the r a t i o s C h i b/Chl a Seasonal and annual  p a t t e r n s were s o u g h t among c r u i s e m e d i a n s o f t h e s e  explaining  cycle  o f C h i b and C h i c, measured a f t e r 1968,  of t h e p h y t o p l a n k t o n . C h i c/Chl  o f 0.2  When t h e p r o c e d u r e i s a p p l i e d t o t h e  for  and  these  d e v i a t i o n of r e s i d u a l s i s  compared w i t h a s e a s o n a l  of t h e time s e r i e s .  periods  standard  This c y c l e a r i s e s almost e n t i r e l y  1  half  1  f o r s i g n i f i c a n c e of  f o r C h i a. i s of very  ratios  The s e a s o n a l small  o n l y 4% o f t h e SSQ a b o u t t h e mean.  using 'cycle'  amplitude,  The a n n u a l  smooth  44% o f t h e SSQ and v a r i e d f r o m a l o w o f 0.28 i n 1969 t o  a h i g h o f 0.83 i n 1975 ( F i g 1 6 ) . The  seasonal  cycle obtained  f o r C h i c/Chl  a ( F i g 17)  explained  1 1 % o f t h e SSQ a b o u t t h e mean, w h i l e  the annual  explained  30%.  a was 1.6,  The a v e r a g e v a l u e  of C h i c/Chl  smooth  c o m p a r e d w i t h 0.6 f o r C h i b / C h l a , b o t h r a t i o s s h o w i n g an i n c r e a s i n g t r e n d throughout t h e p e r i o d of o b s e r v a t i o n s . 3.2.3  Data. Primary  p r o d u c t i v i t y h a s been r o u t i n e l y m e a s u r e d a t  O.S.P. s i n c e 1 9 6 4 . t o 1800 h ( l o c a l C.l^.hr" ). 1  Incubations  were u s u a l l y c o n d u c t e d f r o m noon  t i m e ) a n d a v e r a g e u p t a k e r a t e s r e p o r t e d a s jug  U p t a k e r a t e s p e r u n i t C h i a h a v e been c a l c u l a t e d  76  5  2  (a) •  cr •IE CJ  CO  •  0  *'  •  .*V ... «:  »•*  •  #  * •  1.969  .-•  1970  i  *  1971  3  1973  *i  t  **  <  2  0  r. * '  f  *  1972 3__  1974  1975  1976  5  a) •  *  * • •  CE _ J  •E CJ \ CJ  .  0  *  ,  »•  •  •  1969  1970  1971  *  1972  • •  •  IE CJ  • *  15.  .»»  *  5  0  *  V  _ l  Figure  *  *  Observations of  *  •  1973  »  *  1974  *  *  t  *•  1975  ( a ) C h i b / C h l a and  1976  (b) C h i c / C h l a  t h e w e a t h e r s h i p s a t O.S.P. ( T r i a n g l e s a s i n F i g  13.)  from  Figure 16.  Surface Chi b/Chl a: smooth f o r 1969-76. for 1969-76.  (a) Cruise medians and annual (b) Seasonal f i t plus residuals  Figure 17.  Surface Chi c/Chl a: smooth f o r 1969-76. f o r 1969-76.  (a) Cruise medians and annual (b) Seasonal f i t plus r e s i d u a l s  79  u s i n g t h e c o i n c i d e n t m e a s u r e m e n t s o f C h i a and w i l l t o h e r e by t h e s y m b o l  P.  V a l u e s of t h i s  s a m p l e s , d e n o t e d by P ( 0 ) F i g 18.  Chi a  _ 1  .hr  ),  _ 1  21 pg C.pg  1968.  the time s e r i e s .  are p l o t t e d i n  i n P(0)  increases  A number o f a n o m a l o u s v a l u e s ( f o r e x a m p l e ,  Chi a ^ . h r "  o f 20 l y s o l a r  referred  for a l l surface  I t c a n be s e e n t h e r e t h a t t h e s c a t t e r  markedly a f t e r  total  (ug C.ug  ratio  be  on 1 3 t h December, 1970,  1  r a d i a t i o n ) a l s o appear  w h i c h saw  a  i n the second h a l f  H i s t o g r a m s of t h e f r e q u e n c y d i s t r i b u t i o n  P(0), square-root scaled  t o reduce skewness,  1964-1968 and 1969-1976 a r e p l o t t e d  daily  of  f o r the p e r i o d s  i n F i g 19.  v a l u e s a r e most i m p r e s s i v e i n t h e s e c o n d h a l f  While the h i g h o f F i g 18,  the  h i s t o g r a m r e v e a l s a l a r g e number o f v e r y low v a l u e s f r o m 1969 and a t e n d e n c y  of  f o r the gaussian-shaped d i s t r i b u t i o n  on,  f o r 1964-1968  t o become a l m o s t u n i f o r m i n 1 9 6 9 - 1 9 7 6 . S e a s o n a l and a n n u a l p a t t e r n s b a s e d on c r u i s e m e d i a n s o f  P(0)  were e x t r a c t e d u s i n g t h e p r o c e d u r e d e s c r i b e d a b o v e f o r chlorophyll.  The  seasonal pattern  f o r 1964-1968 i s a s e x p e c t e d  w i t h low v a l u e s f r o m December t o M a r c h to August  ( F i g 20a,b).  and a b r o a d peak f r o m  T h i s c y c l e e x p l a i n s 47% o f t h e SSQ  t h e mean, w i t h t h e a n n u a l smooth e x p l a i n i n g an a d d i t i o n a l The  about 24%.  s e a s o n a l c y c l e o b t a i n e d f o r 1969-1976 ( F i g 2 0 c , d ) e x p l a i n s  o n l y 20% o f t h e SSQ  about  d e v i a t i o n of r e s i d u a l s  t h e mean, and  (0.71)  the h i g h standard  i s not s u r p r i s i n g  i n c r e a s e d s c a t t e r noted above.  The  s u r f a c e o b s e r v a t i o n s of  are not i n c l u d e d  1 4  C  i n view of the  m i d - w i n t e r peak i n t h e  s e a s o n a l c y c l e o b t a i n e d f o r t h i s p e r i o d may The  June  w e l l be an  u p t a k e r a t e s f r o m 1969  i n the r e m a i n i n g a n a l y s i s because  d i s c r e p a n c i e s a l r e a d y n o t e d , and o t h e r s d i s c u s s e d  artifact. to  of the later.  1976  80  *  •  0  *  1961  5  1962  •  A  **+* • • *  cr IE CJ  *  1964 •  1963  IE  _j  •• *f  ;  *  +  *  *  *  ?  0  *  1965  CD CJ  *  *  1967  7  A  *  '  T  **  •  •  *  *  1968 A  •  •  •  fi  ?1  A  • *  V  1966 ?fi  "RR  21  V A  • *  CD  *  .*• *  .  * •**  *A  0  •*  * »  * * »  •  ••  •  •  +  •  • •  1969  CD  *  •  1970  • ***  1971  »  -  *^  •  1972  7  RR A A  *  *  •  • *  * * *  0  •  * U  1973  F i g u r e 18.  *  *** •  * »  1974  *  * • *  *  1975  . »  *  .  1976  S u r f a c e o b s e r v a t i o n s of p r o d u c t i v i t y p e r u n i t C h i a from t h e w e a t h e r s h i p s a t O.S.P. ( T r i a n g l e s  as i n F i g  13.)  81  Figure 19.  Frequency histograms f o r P(0) (square-root scaled) : (a) 1964-68, (b) 1969-76.  0.2 (a)  >—  CJ  LU Z3 a  L U  cr  ^  0 . 1  LU  CE LU CT  0  0  0.2  i .0  2.0  4.0  0.2 (b)  >—  CJ  LU  a  LU  cr  ^  LU  0  . H  CE  _J  LU  cr. 0  0  0.2 P(0)  1 .0 (MG C/MG CHL  .2.0 . fl.HR)  4.0  CO 1964  1  1965  '  O Q O 1966  '  3967  1968  R  A  (b) .  CD  JSZT ~  o  0  CD  CD  —  Nv  D J  1  F  1  M ' fl ' M  1  J  '  1  J  fl S -0 1  l  Figure 20. . P(0) (mg C.mg Chi a ~ . h r ~ ) 1  1  and annual smooth f o r 1964-68. plus residuals f o r 1964-68.  1  N  1  D  (a) Cruise medians (b) Seasonal f i t  84  As  noted  i n C h a p t e r 1, m a c r o - n u t r i e n t s a r e a b u n d a n t a t  O.S.P. a n d p r i m a r y p r o d u c t i o n daily  solar  reported active  radiation  I  and o n e - h a l f  the t h e o r e t i c a l clear  ( F i g 21) s u g g e s t s some r e l a t i o n s h i p scatter.  A linear  total  sky r a d i a t i o n f o r  A plot  e  despite the large  following  ( 1 9 7 0 ) a s t h e minimum o f o b s e r v e d  same d a y , a n d i s d e n o t e d h e r e by I .  c  Total  Photosynthetically  a t t h e s u r f a c e h a s been c a l c u l a t e d  Parsons and Anderson  the  light-limited.  i s m e a s u r e d on t h e w e a t h e r s h i p s a n d  i n t h e M o n t h l y R a d i a t i o n Summary.  radiation  radiation  i s apparently  of P(0) a g a i n s t  between P ( 0 ) and I  regression  0  o f P ( 0 ) on I  0  gives  P(0)  the  =• 0.81 + 0.003.1  0  ,  slope being s i g n i f i c a n t l y  level,  although the regression  SSQ a b o u t t h e mean. and  consistent distinct  i t i s not c l e a r  0  within  different  cruises  f o r r e g r e s s i o n s o f P ( 0 ) on I  This f a i l u r e P(0) it  to find a consistent  may be p a r t l y due t o t h e l a r g e  can a l s o  statistical  on t i m e  a  were n o t yielded  scales successful.  o n l y two o u t o f  from z e r o a t t h e 10% l e v e l ,  w i t h 12 o u t o f 18 s l o p e s b e i n g n e g a t i v e . obtained  i n winter  Attempts t o f i n d a  seasonal co-variation  slopes s i g n i f i c a n t l y  how t h i s  between P ( 0 ) and I  relationship  R e g r e s s i o n s o f P ( 0 ) on I 18  o n l y 14.6% of t h e t o t a l  t e n d t o be h i g h  0  s h o u l d be i n t e r p r e t e d .  from t h i s  f r o m z e r o a t t h e 5%  explains  B o t h P ( 0 ) and I  l o w i n summer s o t h a t  relationship  different  0  Similar  results  were  by month. short-term effect  scatter  of I  G  on  i n o b s e r v a t i o n s , but  be e x p e c t e d on t h e o r e t i c a l g r o u n d s i n t h e a b s e n c e o f  86  any a  scatter.  whole  may  inhibition available  be l i g h t - l i m i t e d ,  i n depth much  six  and twelve  the  upper  one  of 3 standard  taken  varies  vary  widely  This  means  which  t h e sample  depending that  direct  profiles  was  surface  light  regarded  as a  except  1 m  requires  coefficients intensity,  a s an  using  (under  2  i s not be u s e d  the  at the  In f a c t ,  light  t o examine  f o r the standard  (standard)  their  that  r a t i o can  at the time. to obtain  the  relationship  as f o l l o w s . light  The p r o f i l e  The  screens, and the can then  disregarding  to the corresponding  the strong assumption  screen)  valid.  P vs I experiment, index  from  light  coefficient  intensity  a r e known.  light  the extinction  a standard  intensity  to  simulated  s e t o f mesh  d e p t h - i n t e g r a t i o n of p r o f i l e s  can s t i l l  single  belonged  a s t w o - f o l d o r more  taken.  p h o t o s y n t h e s i s and l i g h t  transmission  measured  standard  on t h e e x t i n c t i o n  p r o d u c t i o n under  depths  the l i g h t  made a t  (0,3,7,15,25,60m) and  Consequently,  sample  a t between  were  few e x c e p t i o n s ,  a s much  (Parsons,1965).  with  of samples  (and presumably  at the incubated  from  between  a  a  to winter  The  with  under  a t O.S.P.  photo-  of the light-dependence  (0,2,5,12,25,50m),  depth  as  potentially  measurements  i n c u b a t i o n s on d e c k  not coincide  total  which,  uptake  were  intensity  depth  C  1 4  rates  The S e c c h i  layer  The i n f o r m a t i o n  can  consist  Uptake  coefficent)  need  The  sets:  t h e mixed  there.  a t O.S.P.  depths.  s i x depths,  screens.  summer  O.S.P.  t o an u n d e r s t a n d i n g  (0,6,10,25,33,75m). in-situ  over  p h o t o - s a t u r a t i o n or even  from  production  profiles  growth  at the surface.  profiles  more  phytoplankton The  phytoplankton  c a n be e x p e c t e d  contribute of  While  the stated  screen.  the phytoplankton  be  This population  87  is  homogeneous w i t h d e p t h ,  or a t l e a s t  that the r e l a t i o n s h i p  between p r o d u c t i o n p e r u n i t C h i a and l i g h t  intensity i s  independent of depth.  i n late- f a l l ,  and  T h i s seems p l a u s i b l e  s p r i n g when t h e m i x e d l a y e r e x t e n d s  sample.  However, i n t h e l a t e  seasonal  thermocline  below t h e deepest  summer a n d e a r l y  i s shallowest, a distinct  p o p u l a t i o n c a n be e x p e c t e d  photosynthesis. literature  from the s i m p l e  (for a review,  see J a s s b y  t o the complex  oc.i . e x p ( - I / I  has  been u s e d h e r e .  M A X  simple  (1962):  This equation observations  h a s been s a t i s f a c t o r i l y  Other formulae  parameters t o the s i x observations  h a v e been  light  found  intensities  r e q u i r e three parameters t o more t h a n  i n each p r o f i l e  two seems u n w i s e ,  The t h e o r y p r o p o s e d by S t e e l e ( 1 9 6 2 ) of parameters i n e q u a t i o n  i n terms of l i g h t  fitted  involving photo-inhibition  f o r p h o t o - i n h i b i t i o n , and f i t t i n g  interpretation  A  3.1  ( J a s s b y and P i a t t , 1 9 7 6 ) , but t h e s e  seasonal v a r i a t i o n  type i n  )  s e t s of f i e l d  say t h e l e a s t .  of t h i s  and P i a t t , 1 9 7 6 ) , r a n g i n g  give a superior d e s c r i p t i o n at sub-optimal  account  of  ( e g B a n n i s t e r , 1979.) .  ( H a m e e d i , 1 9 7 7 ; C h a n , p e r s comm).  to  h o m o g e n e i t y must be  T h e r e a r e a number o f f o r m u l a e  P =  to  shade-adapted  f o r the light-dependence  v e r s i o n o f t h a t p r o p o s e d by S t e e l e  other  when t h e  f r o m O.S.P. h a s been u s e d h e r e t o e s t i m a t e  parameters i n a formula  to  C  with suspicion.  Each p r o f i l e  the  1 4  below t h e t h e r m o c l i n e , and r e s u l t s  o b t a i n e d under t h e a s s u m p t i o n of v e r t i c a l regarded  fall,  winter  f o r the  3.1 a n d t h e i r  a d a p t a t i o n and  changing  88  carbon:chlorophyll ratios w i l l  a l s o prove  useful.  E q u a t i o n 3.1 i s u s e d h e r e t o p r e d i c t d a i l y the  b a s i s of d a i l y  proposed  radiation  totals.  u p t a k e r a t e s on  The e q u a t i o n was  first  f o r i n s t a n t a n e o u s r a t e s , b u t i t was shown t o be  applicable to daily variation  in light  t o t a l s on t h e b a s i s o f an i d e a l i s e d intensity  ( S t e e l e , 1 9 6 2 ) . As n o t e d  diurnal  earlier,  i n c u b a t i o n s a t O.S.P. were u s u a l l y made f r o m 1200 t o 1800 h r local  time and average  1  *C u p t a k e r a t e s p e r h o u r o v e r t h e  • i n c u b a t i o n p e r i o d were r e p o r t e d .  T h e s e have been m u l t i p l i e d by  12 t o g i v e a c r u d e e s t i m a t e o f d a i l y incubation period reasonable.  uptake.  i n c l u d e s d u s k , t h i s p r o c e d u r e i s most  Where i t d o e s n o t , d a i l y  a s m a l l u n d e t e r m i n e d amount. diurnal variation  Where t h e  Other  uptake  i s u n d e r e s t i m a t e d by  sources of e r r o r ,  i n photosynthetic a c t i v i t y  such as  and p a r t i c u l a r l y  a f t e r n o o n d e p r e s s i o n (Yentsch and R y t h e r , 1 9 5 7 ; M c A l l i s t e r , 1 9 6 3 ) may be more i m p o r t a n t . To f i t e q u a t i o n 3.1 t o t h e p r o f i l e s ,  Pj  =  0C.I . t r , . e x p ( - I . t r / I o  D  ;  M A X  i t was r e w r i t t e n a s  )  where P; i s t h e u p t a k e r a t e p e r u n i t C h i a a t d e p t h i , I surface l i g h t the  filter  the  0  i n t e n s i t y and t r ; t h e t r a n s m i s s i o n c o e f f i c i e n t of  used  f o r depth  The q u a n t i t i e s A = <x.I  i .  0  and B =  I/M>»X were e s t i m a t e d f o r e a c h p r o f i l e a n d t h e i r d e p e n d e n c e on 0  I  0  I  considered later.  distributed,  ln(P ) ;  I f errors  the s t a t i s t i c a l  i n P; were l o g - n o r m a l l y  m o d e l c o u l d be w r i t t e n a s :  = ln(A) + ln(trj ) - B.tr  ;  + e;  (e  ;  i.i.d.  N(0,a ) ) 2  89  resulting This  was  i n a convenient tried  observations residuals. profiles  with  i t was  found made  first  square-root  to  error  does  least-squares  that  shallow  resulted  As  estimate then  A,  (cf F i g  A.  The having  by P  M A X  parameter  units  o f mg  occasions.  light  quickly  enough  inhibition  A third  attempt  was  approach, but  This  scaling  i s used  distribution  a s an e m p i r i c a l measure  attached  was  to shallow  which  and deep  19a). estimates  procedure  the three  estimate , was  using  used.  A  observations  this  was  also  Chi a values  resulted  retained.  noted  _  1  1  estimate observed  profile. efficiency,  , and has been  are given  depths  i n a higher  a photosynthetic  . l y "  line  and the o r i g i n  The h i g h e s t  f o r each  Steele's  straight  at successively shallower  oc r e p r e s e n t s C.mg  obtained  was a l s o  deepest  but unless  Early  ignored.  a Poisson  being  contribution to the  surface  P; .  was  intensity)  least-squares  here  Observations  the o r i g i n a l  observations  f i t t o t h e raw d a t a  where  were  from  on p a r a m e t e r  included,  , denoted  many  weight  through  intensity)  individual  not decrease  the o r i g i n a l  used  ( 1 ) , an ad-hoc  fitted  were  b u t was  a check  equation was  scaling  i n equal  observations  through shallow  (high  the non-linear  observations  (Barnes,1952),  ;  fitted  to ignore  observations  using  normalize  (low l i g h t  a disproportionately large  deep  again  deep  a n d B.  d i s p r o p o r t i o n a t e l y t o t h e SSQ o f  of r e s i d u a l s , and, e s p e c i a l l y  made,  P  that  f o r ln(A)  log-normal.  a non-linear  exaggerated,  of  a tendency  t o be  regression  of curves  The o b s e r v a t i o n  observations  to  found  contributed  revealed  When  SSQ  then  t h e mean  tried,  b u t i t was  Inspection  entirely.  linear  i n terms  measured  of lux-hr  on rather  90  than  l y and  conversion  can  be p r o b l e m a t i c a l .  p r o p o s e d a u n i v e r s a l v a l u e of 4.10" .48  mg  C.mg  Chi a ^ . l y  v a l u e o f 0.5  mg  C.mg  phytoplankton. C.mg  On  Chi a ^ . l y  .  the other  may  1  lmg  Chi a ^ . l u x - h r "  C.mg  laboratory populations  suggests  or  1  that a  be assumed f o r most  h a n d , a t h e o r e t i c a l maximum o f 2  i s p r o p o s e d by  1  C.mg  Steeman N i e l s e n ( 1 9 7 8 )  Chla^.ly  v a l u e s , g r e a t e r than for  1  mg  4  Steele(1962)  Bannister(1974),  Chi a ^ . l y  1  and  high  , h a v e been  ( C h a n , 1 9 7 8 ) , and  i n the  mg  reported  field  (Hameedi,1977) . A p l o t of A  ( e s t i m a t e d u s i n g t h e ad hoc  f o r t h e p e r i o d 1964-1968 i s g i v e n the best  f i t to a s t r a i g h t  slope corresponding estimates plot,  to  od=0.34 mg  considerable  scatter,  i n c l u d e s the  other.  An  i n F i g 22.  through  ot = 0.25  of A o b t a i n e d  with  line  mg  the  equivalent plot  95%  i t has  1  3.1  similar  B e c a u s e of  confidence  limits  a  The  1  yielded a  1  I  l i n e drawn i s  Chi a " . l y .  C h i a' .ly' . 1  The  the o r i g i n ;  C.mg  using equation C.mg  procedure) versus  the  f o r each  f o r t h e p e r i o d 1969-1976 showed  value  no  A  r e l a t i o n s h i p a t a l l b e t w e e n A and quickly  became a p p a r e n t on  Fig  10  23,  given has  f o r t h e p e r i o d s 1964-1968 and  expressed  altered  frequency  earlier,  and  the  23, a l l C  remaining  1 4  incoherent  versus  1969  depth  In  are  1 4  C  i n c r e a s e d s c a t t e r and  of s u r f a c e v a l u e s  observations after  this  Stephens(1977)  some o f t h e  p a t t e r n of P  for  individual profiles.  of P;  In view of the  distribution  reason  1969-1976.  reservations concerning 1969.  The  i n s p e c t i o n of  randomly s e l e c t e d p r o f i l e s  observations after  Fig  I„ .  reported  w i t h d e p t h as shown i n  have been i g n o r e d and  d i s c u s s i o n a p p l i e s o n l y t o the p e r i o d 1964-68.  This  the  91  Figure  22.  S c a t t e r p l o t of A v s I drawn i s l e a s t - s q u a r e s  f o r 1964-68.  (Line  f i t through o r i g i n . )  92  ioo  i  i  I  r  I  (b) Figure 23.  Depth p r o f i l e s of P f o r (a) 1964-68 and (b) 1969-76. ( S t r a i g h t - l i n e segments connect  observations.)  93  reduces  t h e number of p r o f i l e s  f r o m 200  restricts  t h e u s e f u l n e s s of t h e d a t a  examining  s e a s o n a l p a t t e r n s and  Rather  than p l o t % vs I  cx , an e s t i m a t e  <X = A / I  0  can  0  t o 45 and  set, particularly  annual  and  severely for  anomalies.  e s t i m a t e a s i n g l e mean v a l u e  be c a l c u l a t e d  f o r each p r o f i l e .  of A  A  p l o t o f m o n t h l y means o f oc ( F i g 24) pattern  i n oc  w i t h a maximum i n w i n t e r and  laboratory experiments, phytoplankton  cx has  composition  and  as t e m p e r a t u r e ,  n u t r i e n t s and  1977).  values  The  suggests  low  been f o u n d  may  be v i o l a t e d .  remaining  m o n t h l y means a r e  compared w i t h o t h e r  mg  conditions  C.mg  viewed  i s shallow at t h i s  homogeneity of the  Chi a ~ . l y x  _ 1  time  phytoplankton  low v a l u e s ,  or e q u a l  such  e_t a l ,  S e p t e m b e r must be  l e s s than  In  with  (Parsons  Even d i s c o u n t i n g t h e s e v e r y  N i e l s e n ' s v a l u e of 0.5  The  to vary  light quality  w i t h some s u s p i c i o n as t h e t h e r m o c l i n e t h e a s s u m p t i o n of v e r t i c a l  seasonal  minimum i n summer.  with environmental  i n A u g u s t and  and  a strong  the  t o Steeman  which i s i n turn  recent o b s e r v a t i o n s mentioned  low  above.  i n t e r p r e t a t i o n o f e s t i m a t e s of B i s n o t  as A  straightforward. show a l i n e a r studies MAX  x  ratio,  was  was  c o n s t a n t , a p l o t of B vs I  However, Stee'le(1962) found 3.  to co-vary  S t e e l e and  b e t w e e n 1 and  2,  light  with I  c  so  Baird(1961)  i n the North  that found  Sea  through  r a t i o , and  lower  in  attributed  t o l i g h t a d a p t a t i o n by p h y t o p l a n k t o n  intensity  should  q u o t e s a number of  In a t h e o r e t i c a l d i s c u s s i o n , Steele(1962)  carbon:chlorophyll 2.  M A  l a y b e t w e e n 2 and  this co-variation seasonal  M A  relation.  i n which I  v a l u e s of t h i s summer.  If I x  to  changes i n the  used a f i x e d v a l u e  f o r I„/I x MA  of  F i g u r e 24.  Monthly averages  of  OL.  (Line f i t t e d by  eye.)  95  Estimates  of B a r e p l o t t e d  against I  there  i s no s u g g e s t i o n o f a l i n e a r  would  be e x p e c t e d  large  t o d e s c r i b e B as c o n s t a n t .  lies  if I  i n t h e range  lower  than  x  M A  i n F i g 25.  0  relation  between  B and I , as 0  was c o n s t a n t , t h e s c a t t e r  suggested  The a v e r a g e  by S t e e l e  values reported in other  While  i s rather  v a l u e o f B, 1.2,  and B a i r d ( 1 9 6 1 )  and i s  r e f e r e n c e s quoted i n  Steele(1962). Theory explained light  suggests  that  by a more c a r e f u l  adaptation. ratio  adaptation  to changing  light  averaging  response.  / I  M  A X  , since  light  of t h e t i m e - c o u r s e o f  phytoplankton  this  requires  intensity.  Cells  (Steele,1962)  A seasonal average  be  instantaneous  c o u l d adapt  through light  can m a i n t a i n  some  to  long-term I (t),  intensity,  s  w i t h a window o f w i d t h  E s t i m a t e s of the seasonal a d a p t a t i o n parameter,  B ,  20  given  s  A  by  B  of  B itself.  =  s  I  been  Nielsen these linear  S  / I ^ x  =  B . I  Light  S  / I  0  showed no l e s s - s c a t t e r  ,  adaptation through  reported to occur and P a r k , 1 9 6 4 ) .  time  scales  fitted  on t i m e  changes  scales  A s i m p l e model  of r a d i a t i o n  than  i n C:Chl  of a few d a y s  I  M A )  <  a  ratios  (Steeman  results  on  t o vary as a  on t h e p r e c e d i n g d a y s .  to the p r o f i l e  estimates  for light-adaptation  would a l l o w the parameter  combination  model was the  0  that  using a running average  A  has  I  intensities  was c a l c u l a t e d days.  of  i n F i g 25 m i g h t  consideration  It i s unlikely  a constant  seasonal  the s c a t t e r  as a l i n e a r  Such a  regression in  form:  1/B(t) = W / I  0  + b .I 3  (t) 0  = b  (t-3)/l  0  0  + b ,I ( t - l ) / I 4  (t)  ,  0  0  (t) + b . I s  0  (t-2)/l  0  (t)  •  • •  •  •  • LO  -Q—  0  •  Figure 25.  •  • LLTLTJ  U  (LY/DRY)  S c a t t e r p l o t of B vs I  Q  for  • •  •  •  mm  Q—Q-  P.R.R.  •  •  • • o  •  •  • •  •  •  •  1964-68.  -B-  360  97  but of  t h i s d i d not produce a s i g n i f i c a n t 1/B. I n F i g 26, a p l o t  instantaneous linearly a  reduction i n the variance  adaptation  with I . line  through  0.001, c o r r e s p o n d i n g and  versus  M A X  (I /l 0  The l i n e  0  straight  of P  M A  x  0  i s given.  c o n s t a n t ) , P^AX  i n F i g 26 r e p r e s e n t s  the o r i g i n  Under should  forI /I 0  I t i s interesting  Fig  26 a t l o w l i g h t  intensities  lie  above t h i s  through  line  the e x i s t e n c e of a lower minimum v a l u e  of I  C:Chl a - r a t i o , The O.S.P.  M A X  (less  M A  x  which i s encountered  120 l y / d a y )  This  t o Steele(1962) a t these  from a n a l y s i s of  f r o m a b o u t 0.45 mg C.mg C h i a ^ . l y "  reasonable increase  1  evidence  in I  or less  are not s u f f i c i e n t term o r long-term  I  M A  x /Io  1  1 4  light C  adaptation  intensities.  profiles at Photosynthetic cycle  ranging  from October t o A p r i l  to light  intensity  w i t h an the data  'instantaneous',  t o changes i n s o l a r  t o 0.3  There i s  i n c r e a s i n g I„ , a l t h o u g h  t o d i s t i n g u i s h amongst  Some o f t h e s c a t t e r ratio  with  M A X  tend t o  a minimum  f r o m May t o S e p t e m b e r .  of a d a p t a t i o n  and P  M A X  fitting  i s consistent with  oc i s l o w a t O.S.P. , w i t h a s e a s o n a l  mg C.mg C h i a ^ . l y "  by  that the points i n  c a n t h e r e f o r e be s u m m a r i s e d a s f o l l o w s .  efficiency  b e t w e e n 0.9  to l i g h t adaptation; that i s , a  , and a c c o r d i n g  results obtained  than  the o r i g i n .  limit  f i t to  a n d h a s a s l o p e o f 0.01 ±  t o an a v e r a g e v a l u e  3.1 t o p r o f i l e s .  increase  the best  1.2, c o n s i s t e n t w i t h t h e a v e r a g e v a l u e o b t a i n e d  equation  the  I  short-  irradiance.  i n F i g 25 c a n be e x p l a i n e d by a l l o w i n g A  t o vary  seasonally.  Monthly averages of B and  A  B  s  The  are plotted high values  f i g u r e o f 2.0.  i n F i g 27. i n February This  B o t h show t h e same s e a s o n a l t o May a g r e e r o u g h l y  with  trend. Steele's  i s a p e r i o d o f deep mixed l a y e r s and  99  1.  J  \  A  A A  A  jjj  A  A?  0.  A?  J  F  M  A  M  J  J  A S  O N  MONTH gure 27a.  Monthly averages of estimates of l i g h t adaptation parameter B. (Line f i t t e d by eye.)  D  J  F  MA  M  J  J  A  S  ON  D  MONTH Figure 27b.  Monthly averages of estimates of l i g h t adaptation parameter B . g  eye.)  (Line f i t t e d by  101  increasing  light  intensity.  Throughout the  y e a r , monthly a v e r a g e s of B a r e  low and  growth a t or near the s u r f a c e .  The  S e p t e m b e r may vertical In  v i e w o f t h e low  very  prevented  by  B must be  the data.  a search  l o s s of t h e handicap  1 4  C  and  t o t h e b r e a k d o w n of t h e a s s u m p t i o n  regarded  The  of  variation  f r o m 1969-1976 has  1966-1968 and and  seasonal  suggested  rather  than  together w i t h the s i x t o 1968,  has  also  i n these parameters.  The  been a c o n s i d e r a b l e  respects.  1966.  The  frequency  s u r f a c e and  annual  i n F i g 29.  c y c l e remove 42% and The  37%  i s low  during  t h e r e a f t e r ( F i g 28).  s u m m a r i e s were c a l c u l a t e d a s  are given  a b o u t t h e mean.  d e e p s a m p l e s were  of o b s e r v a t i o n s  increases substantially  v a r i a b l e s and seasonal  the  Data.  after  Seasonal  these  r e g i m e d u r i n g 1964  N i t r a t e c o n c e n t r a t i o n s i n both recorded  B,  as b e i n g  high scatter,  f o r annual  data  i n these  Nitrate  1  i n August  number o f p r o f i l e s c o n s i d e r e d and  week g a p s i n t h e s a m p l i n g  at.l"  t o maximum  low v a l u e s  i n e s t i m a t e s o f OL and  scatter  i n oc and  confirmed  3.2.4  correspond  the  homogeneity.  unexplained cycles  a g a i n be due  remainder of  The  annual  respectively  for other  smooth  and  of the t o t a l  SSQ  s t a n d a r d d e v i a t i o n o f r e s i d u a l s i s 2.1 jug  compared w i t h a s e a s o n a l c y c l e r a n g i n g  f r o m 7 t o 15  ug  at.l" . 1  In  a d d i t i o n , average n i t r a t e values  2 0 , 2 0 - 4 0 , 4 0 - 8 0 , 80-130 and p r o f i l e and extracted for  the annual  130-200 m were c a l c u l a t e d  smooth and  average seasonal  (using individual p r o f i l e s  each depth  range.  The  i n the depth  resulting  ranges  0-  f o r each  cycle  r a t h e r than c r u i s e s e a s o n a l c y c l e s and  medians) the  102  20 •  oo X X  1965  1966  cr cr:  •  1967  20  *  *  *• *  '•  •  '  ' **. *  *  .  •  . * *•  •  •  0  *  1969  1970  1971 *  *  0  *  1973 Figure  28.  if*  •  i  ** •  #**  •  *  1972  20 r  *  1968  ??  CD  LU h—  ,  *•  0  cr  21  .  .  *  *» '••  •» •.•  .  v."  **•  «  1974  1975  Surface observations the weatherships  of n i t r a t e  a t O.S.P.  1976 concentration  from  104  annual  smooths a r e p l o t t e d  20m a n d 20-40m a r e b o t h  i n F i g 30.  similar  The s e a s o n a l c y c l e s f o r 0-  to that obtained  s a m p l e s a n d a l l c a n be e x p l a i n e d a s a r e s u l t phytoplankton  o f u p t a k e by  d u r i n g s p r i n g a n d summer, w i t h a minimum o c c u r r i n g  a b o u t t h e t i m e o f maximum  stabilisation  of the water column.  first  change i n s e a s o n a l c y c l e w i t h depth  quite  interesting.  occurs  nitrate  uptake  concentration  summer, b u t a s t h e t h e r m o c l i n e  s h a l l o w s t o a b o v e 40m, an i n c r e a s e i n n i t r a t e This  The  comes a t 40-80m a n d i s  A small decline i n n i t r a t e  i n s p r i n g and e a r l y  occurs.  f o r surface  concentration  i s p r e s u m a b l y due t o a c o m b i n a t i o n (phytoplankton  of  reduced  a t t h e s e d e p t h s a r e no l o n g e r  upward i n t o r e g i o n s o f h i g h l i g h t  intensity),  vertical  mixed  mixing  from below and p o s s i b l y r e c y c l i n g . The 41%, The  s e a s o n a l c y c l e s f o r t h e t h r e e upper depth  3 3 % a n d 8% r e s p e c t i v e l y  of t h e t o t a l  SSQ a b o u t t h e means.  s e a s o n a l c y c l e a t 80-130m e x p l a i n s o n l y 6% o f t h e SSQ; t h e  principal February  interpretable  f e a t u r e i n t h e c y c l e i s t h e minimum i n  and March, t h e time  o f maximum m i x e d l a y e r d e p t h ,  n u t r i e n t d e p l e t e d water from t h e p r e v i o u s these depths. the t o t a l  summer  when  i s mixed t o  The s e a s o n a l c y c l e a t - 1 3 0 - 2 0 0 m e x p l a i n s o n l y 3% o f  SSQ w h i l e t h e a n n u a l  considerable variation is  ranges e x p l a i n  not of a seasonal  smooth e x p l a i n s 4 9 % .  in nitrate  nature  The  c o n c e n t r a t i o n a t these  and presumably r e f l e c t s  of l a r g e water masses w i t h d i f f e r e n t  depths  the passage  n u t r i e n t h i s t o r i e s past the  station. 3.2.5 N i t r a t e C o n c e n t r a t i o n a n d P r o d u c t i o n . The  average seasonal c y c l e i n n i t r a t e c o n c e n t r a t i o n a t the  s u r f a c e h a s a minimum o f 7 jug a t . l  -  1  but lower  v a l u e s o f 2.7, 1.9  105  Figure 30.  N i t r a t e concentrations (mg at.m ) from depth J  p r o f i l e s . Layer averages and annual smooths, seasonal f i t s plus r e s i d u a l s . (a)  0 - 20m.  (b)  20 - 40m.  (c)  40 - 80m.  (d)  80 -  (e)  130 - 200m.  130m.  1966  1967  1  1968  ' 1969  ' 1970  ' 1971  1  F i g u r e 30a.  1972  1  1973  1  1974  ' 1975  ' 1976  Ill  and  1.1 jug a t . l "  1  reported  from  yjgat.l"  (McCarthy  1  were o b s e r v e d  t o depths  n u t r i e n t - d e p l e t e d waters  o f 50 t o 75 m.  are usually  Values  l e s s than 1  and Goldman,1978) and even t h e lower  values  f r o m O.S.P. w o u l d n o t n o r m a l l y be c o n s i d e r e d t o l i m i t phytoplankton at  growth.  The p o s s i b i l i t y  phytoplankton  O.S.P. h a v e u n u s u a l l y h i g h h a l f - s a t u r a t i o n c o n s t a n t s f o r  growth and t h i s would a f f e c t phytoplankton-zooplankton The  time  greatly  the treatment of  interactions.  s e r i e s a v a i l a b l e here a r e not s u i t e d t o t e s t i n g f o r  n u t r i e n t dependence of p h o t o s y n t h e s i s . of  remains that  experimental evidence  However, i n t h e absence  concerning nutrient  uptake  k i n e t i c s at  O.S.P. , a c h e c k was made f o r u n u s u a l l y l o w v a l u e s o f P ( 0 ) a t these  t i m e s -of l o w n u t r i e n t c o n c e n t r a t i o n .  with effect May  of l i g h t  t o October  intensity,  of any d e c r e a s e  concentrations  ( F i g 31).  suggestive only. slight  v a l u e s of P(0) over  the period  were p l o t t e d a g a i n s t n i t r a t e c o n c e n t r a t i o n .  i s no e v i d e n c e  any  To a v o i d c o n f u s i o n  i n P ( 0 ) a t low n i t r a t e  Again,  The s c a t t e r  There  these  r e s u l t s must be t a k e n a s  i n F i g 31 i s l a r g e enough t o mask  downward t r e n d a t l o w n i t r a t e c o n c e n t r a t i o n s s o t h a t  v a l u e s o f a h a l f - s a t u r a t i o n c o n s t a n t b e t w e e n 0.1 a n d 2 jjg a t . l " w o u l d p r o b a b l y n o t be d i s t i n g u i s h e d . that,- i f c e l l s  responded t o n u t r i e n t  t h e i r C:Chl a r a t i o in  3.3.1  reservationi s  limitation  by i n c r e a s i n g  ( S t e e l e , 1 9 6 2 ; A n t i a e t a l ,1963),  g r o w t h r a t e m i g h t n o t be r e f l e c t e d  3.3 A P h y t o p l a n k t o n  A second  i n decreased  a  1  decrease  P(0).  Growth Model.  Introduction. The  preceding data a n a l y s i s ,  t o g e t h e r w i t h i n f o r m a t i o n from  0 Figure 31.  5 10 NITRATE (MG RT/MXX3)  15  P(0) vs low surface nitrate values, May to October, 1964-68.  113  t h e l i t e r a t u r e where n e c e s s a r y , model o f p h y t o p l a n k t o n time  will  now be u s e d t o c o n s t r u c t a  g r o w t h a t O.S.P. w h i c h  c a n be d r i v e n by  s e r i e s of o b s e r v a t i o n s of p h y s i c a l v a r i a b l e s .  study of t h e seasonal c y c l e of net primary  A simple  production at  O.S.P. h a s a p p e a r e d p r e v i o u s l y ( M c A l l i s t e r , 1 9 6 9 ) , b u t a more s o p h i s t i c a t e d approach,  following  Jamart e t a l ( 1 9 7 7 ) , i s intended 3.3.2  t h e example of S t e e l e ( 1 9 7 4 ) and  here.  P h y s i c a l S t r u c t u r e and D r i v i n g V a r i a b l e s . B a s e d on t h e a r g u m e n t s p r e s e n t e d  state variables w i l l  will  be m o d e l l e d .  phytoplankton of  will  t h a t i s , a water  column 1 metre  by t h e d e p t h - d e p e n d e n t  growth and g r a z i n g as w e l l as v e r t i c a l m i x i n g .  reviewed  here  t and depth  The v e r t i c a l d i s t r i b u t i o n o f  be d e t e r m i n e d  changes i n t h e p h y s i c a l  of  1, p h y t o p l a n k t o n  be t r e a t e d a s f u n c t i o n s o f t i m e  z but not h o r i z o n t a l p o s i t i o n ; square  i n Chapter  s t r u c t u r e of t h e water  processes  The s e a s o n a l  column a r e  as background t o t h e m o d e l l i n g of v e r t i c a l  mixing  phytoplankton. The  mixed l a y e r  i s deepest  ( g r e a t e r than  the end of t h e p e r i o d o f n e t heat  loss.  100m) i n M a r c h , a t  The f o r m a t i o n a n d  m a i n t e n a n c e o f t h e s e a s o n a l t h e r m o c l i n e i n s u b s e q u e n t months occurs through which  the a l t e r n a t i o n  of p e r i o d s of calm,  produce shallow, t r a n s i e n t  h i g h wind a c t i o n d u r i n g which  t h e r m o c l i n e s , w i t h p e r i o d s of  these t r a n s i e n t  m i x e d downward t o s t r e n g t h e n t h e s e a s o n a l (Denman,1972). heat It  The l a t t e r  sunny w e a t h e r  structuresare  thermocline  shallows through  the p e r i o d of net  g a i n t o r e a c h a minimum o f a b o u t 30m i n A u g u s t t o S e p t e m b e r .  i s then eroded  d u r i n g t h e p e r i o d of net heat  c o n v e c t i o n a l o v e r t u r n a n d by w i n d  mixing.  l o s s , b o t h by  114  There in  a  are  conventional  phytoplankton  process,  as  dependent  partial  depth and  simpler  lower  depth  approach  thermocline  as  et  and  differential  f u n c t i o n of  mixing  model.  i n Jamart  on  thermocline  a  two  high  and  time  represented with  and  can  be  the  being below  solved  mixed  well-mixed  depth-varying  vertical as  diffusion  a  i t .  rates above  The  concentration  above  region,  biological  the  resulting  numerically.  layer  mixing  diffusion  assumed  for phytoplankton  regard or  represent  rates  rates within equation  to  be  a_l ( 1 9 7 7 ) ,  uniform  overwhelms  I t can  time,  i s to  a  ways  as  A  the  in which  processes  physical  (eg  Steele,1974). It  is clear  entirely and  the  that  realistic. formation  both  layer  can  a  day  only  to  by  (The  that  The model.  a  of  that  of  in  calm  phytoplankton  or  are  mixed-layer  almost  a l l Chi  a  150m  of  the  includes a  water mixed  profiles within the  concentration will  be  As  be inaccurate  the  process  accurately  there  model from  the  is  of  i s used  O.S.P.  mixed  simpler  column layer  that  mixed  little  r e p r e s e n t a t i o n i s more  uniform  for  may  certainly  be  mean  uniform  which  will  process.  a  is  periods  thermoclines  unlikely  winter  diffusion  rough  r a t e and  simpler  e m p i r i c a l support  This  but  seems  diffusion  variation  and  transient  scales,  fall  the  neither approach  approximations.,  It also  simple  vertical  top  as  time  believe that the  as  review  alternation  regarded  overturn  accurate,  taken  be  basis.  convective  significant  above  breakdown  day  of  fact  and  longer  a  to  The  on  on  represented  the  seasonally-varying diffusion  satisfactory  reason  from  layer  assumed  show  no  can  be  approach).  i s represented depth  here.  z (t) M  to  be  in  the  within  which  independent  of  115  depth.  Since z  M  v a r i e s over  t i m e , the c o n c e n t r a t i o n below  mixed l a y e r as a f u n c t i o n of d e p t h model, phytoplankton values C(zi , t ) , d e p t h s z;  and  i = 1 . . . 3 0 , i n 5m  are a r b i t r a r i l y  next,  thick  taken  layers.  (The  at the midpoints  mixed l a y e r depth  step.  When z  t h e l a y e r s a b o v e t h e new  of  r a t e s there are very  5m  l a y e r s there are The  left  For  averaged.  i t can  be  simplicity,  daily  solar  radiation  series  profiles,  been d e t e r m i n e d  f r o m STD  E n v i r o n m e n t Canada on m a g n e t i c d a t a  down t o d e p t h  tape,  i n the  e n e r g y r e q u i r e d t o mix  z, P E ( z ) , was  calculated  and  (at v a r y i n g  l a y e r depth  potential  and  depth,  observations.  The  i t is  of growth  l e v e l s of c o m p l e t e n e s s ) from w e a t h e r s h i p  manner.  assumed  b e l o w t h e t h e r m o c l i n e ; the.  These a r e a v a i l a b l e as time  has  the  unconnected.  as a f u n c t i o n of d e p t h ,  secchi depth.  to  and n u t r i e n t  p h y s i c a l d r i v i n g v a r i a b l e s are mixed l a y e r  temperature  day  are  assumed h e r e t h a t l o c a l d e p t h - d e p e n d e n t p r o c e s s e s g r a z i n g dominate the e f f e c t s of m i x i n g  nearest set  M  thermocline, low.  the  their  z ( t ) are  mixed l a y e r depth  c o n c e n t r a t i o n s below the s e a s o n a l  by  nominal  i n c r e a s e s f r o m one  M  In view of the slow changes i n s a l i n i t y  that mixing  the  i s g i v e n t o the  c o n c e n t r a t i o n s C ( z , , t ) f o r z-, l e s s t h a n  equal at each d a i l y  In  concentration C(z,t) i s represented  r e s p e c t i v e l a y e r s . ) The 5m  must be m o d e l l e d .  the  The  provided  mixed by  following  the water  column  as:  PE(z)  A v a l u e of P E ( z )  o f 0.02  l a y e r depth  This  z . M  j.cnr  2  was  chosen t o d e f i n e the  i s comparable to the m i x i n g  energy  mixed provided  116  by a t y p i c a l  wind  would  f r o m s e v e r a l c a l m sunny d a y s .  result  criterion,  storm  transient  (Denman,1973) o r t h e d e c r e a s e  shallow thermoclines created during short  layer depth r e s u l t e d . these t r a n s i e n t  s h a l l o w mixed  which  By u s i n g t h i s  p e r i o d s o f c a l m were i g n o r e d and a s m o o t h e r  day,  i n PE  I f p r o f i l e s had  time s e r i e s  been a v a i l a b l e  f o r mixed  f o r each  t h e r m o c l i n e s c o u l d h a v e been t r e a t e d  l a y e r s but  i t was  necessary to interpolate  gaps o f s e v e r a l d a y s and o c c a s i o n a l l y  s e v e r a l weeks.  as over  Under  c i r c u m s t a n c e s , t h e use o f t h e a b o v e c r i t e r i o n  was  likely  It i s consistent  to result  i n erroneous  interpolation.  w i t h t h e use o f t h e u n i f o r m l y - m i x e d l a y e r a p p r o a c h approximation v a l i d The  i n STD  i n the mixed  profiles.  l a y e r was  To a v o i d s t o r i n g and  the t e m p e r a t u r e below  the mixed  exponentially  5 C w i t h a decay  towards  inspection  l a y e r was  reading-, l a r g e  simulation  i n one  growth  Daily  solar  the mixed  in  layer are  r a d i a t i o n was  the  mixed  insignificant.  radiation,  c a l c u l a t e d as d e s c r i b e d i n the a n a l y s i s of by t h e 1964  almost  temperature-dependent  taken from the Monthly  Summaries and p h o t o s y n t h e t i c a l l y a c t i v e  t h e t i m e s e r i e s were f i l l e d  This  c o n s i d e r e d s u f f i c i e n t as  t h e e f f e c t s of s m a l l e r r o r s  r a t e s below  chosen  year.  p h y t o p l a n k t o n p r o d u c t i o n a t O.S.P. o c c u r s w i t h i n  l a y e r and  run,  assumed t o d r o p o f f  c o n s t a n t o f 20m,  of t e m p e r a t u r e p r o f i l e s  r a t h e r c r u d e r e p r e s e n t a t i o n was all  an  t a k e n as the s u r f a c e  q u a n t i t i e s o f t e m p e r a t u r e p r o f i l e d a t a on e a c h  after  as  less  on t i m e s c a l e s o f s e v e r a l d a y s o r l o n g e r .  temperature  temperature  judged  these  1 4  C  t o 1976  Radiation  I„ ,  profiles.  was Gaps i n  means f o r t h a t  day o f t h e y e a r . The  driving  v a r i a b l e w i t h the l e a s t complete  time s e r i e s i s  117  secchi depth,  SD, w h i c h was u s e d t o d e t e r m i n e  extinction coefficient 1.7/SD ( P a r s o n s  forlight  e t a l ,1977).  the background  according t o the formula  The o b s e r v a t i o n s s u g g e s t  : k  8  =  a  s e a s o n a l c y c l e w i t h a minimum i n summer a n d maximum i n w i n t e r , a s w e l l as c o n s i d e r a b l e year not a v a i l a b l e  t o year  variation.  As o b s e r v a t i o n s a r e  f o r a l l y e a r s , an a v e r a g e s e a s o n a l c y c l e was  e x t r a c t e d and used f o r a l l y e a r s :  SD = 14.4 + 2.8 S I N ( 2 . 77 . ( t + 4 1 ) / 3 6 5 )  3.3.3  3.2  B i o l o g i c a l B a s i s f o r the Model. The  carbon  phytoplankton  (pg C I  -  1  ) .  b i o m a s s i n t h e model i s d e f i n e d a s p l a n t  Standing  stock  i s observed  a s C h i a and  comparison of p r e d i c t i o n s w i t h o b s e r v a t i o n s w i l l  require a  knowledge of t h e c a r b o n r c h l o r o p h y l l r a t i o .  i s necessary i n  any  case  carbon  This  a s o u r a n a l y s i s o f * C u p t a k e r a t e s a l l o w s us t o p r e d i c t 1  p r o d u c e d p e r u n i t C h i a, whereas a dynamic model r e q u i r e s  a growth r a t e i n d a y s " . 1  The s i g n i f i c a n c e o f t h i s p r o b l e m was  e m p h a s i z e d by E p p l e y ( 1 9 7 2 ) ,  but i t remains unsolved  l a c k o f a r e l i a b l e method f o r m e a s u r i n g l i v i n g the  t h e absence of d i r e c t  o b s e r v a t i o n s of C:Chl a r a t i o s , a  theoretical  framework w h i c h w i l l  necessary.  The t h e o r y u s e d h e r e was p r e s e n t e d  allow their prediction i s by  Steele(1962)  d e p e n d s on t h r e e a s s u m p t i o n s :  (1) A t l o w l i g h t  intensities,  C h i a , w i t h some c o n s t a n t not  p l a n t carbon i n  field. In  and  due t o t h e  photosynthesis  i s proportional to  photosynthetic efficiency  s u b j e c t t o change through  light  adaptation.  <x w h i c h i s  118  (2)  The  the  dark  which  maximum  rate  reaction  i s assumed  existence  of  a  of  and,  carbon more  growth  (hr  respiration  at  O.S.P.  (3)  Light The  of  adaptation occurs t h e o r y was at  structure  relation of  the  to  the  and  and  be  studies  - 1  )  This which  through  assumed  to  content  implies  the  is light  i s assumed  of  of  yet  the  the  to be  be  a  fixed  non-limiting  to  C:Chl  More  a  ratio.  by  i s now  known  photosynthetic apparatus  light  and  include  the  of  dark-reaction  Morris,1976). rise  or  to a  theory  Nielsen,1974)  (Myers has  role  (Steeman  However,  level. and  in  enzymes  coherent  population  of  special  collecting, apparatus  give  Steele's  i n the  simplification  Examples  cellular  Steeman  a  i t s proposal.  (Beardall  have  changes  be  significance  f o r a s p e c t s of  Bannister,1974;  through  i n the  the  adaptation at  support  will  adaptation.  RuDP c a r b o x y l a s e  light  time  P-700 m o l e c u l e  molecular  change rate  recognized to  function  light  Nielsen,1974) as  (Nutrients  w  not  H  enzyme  by  ).  Steele(1962) the  u .  does  u  i s governed  the  rate,  adaptation. percentage  by  carbon.  but  dark  cell  to plant  temperature-dependent The  per  specifically,  proportional  maximum  fixation  such  these  theory  of  Recent  Graham,1971;  encouraged  i t s use  here. The P  a.l  =  used of  p  l i g h t - d e p e n d e n c e of .exp(-I/l  i n the  analysis  phytoplankton  =  oi.l  M A X  .exp(-I/I  i s based  on  the  relationship  )  3.1 of  1 4  exposed  M A )  growth  < )/V  C  profiles.  to d a i l y  -  d  The  radiation  daily I  net  growth  i s then  given  rate by  3.3  119  where V i s t h e C : C h l a r a t i o a n d d i s t h e r e s p i r a t i o n (day" ).  The maximum d a i l y  1  =  UMAX  a.I  According  M A X  /(V.e)  (X.I AX/(V.e) M  rate  i s g i v e n by  - d  t o assumption  h o u r l y growth  growth  rate  ( 2 ) , a maximum  (temperature-dependent)  r a t e u ( T ) c a n be d e f i n e d ,  so t h a t  H  = u (T).DL + d  3.4  H  where DL i s d a y l e n g t h i n h o u r s . Respiration  i s assumed t o be a f i x e d  proportion,  v, o f  pn(T);  that i s ,  d  = V.UH(T).24  3.5  E q u a t i o n s 3.4 a n d 3.5 c a n be r e g a r d e d a s d e f i n i n g u (T) H  a n d V, o r a s d e f i n i n g V, g i v e n I Light  intensity  d e p t h , so t h a t coefficient  i s assumed t o f a l l  z  M  A  off exponentially  ( i n which  with  the extinction  k ( z ) b e i n g d e f i n e d by k ( z ) = k  growth  x r given  H  0  the average  M A  and j u ( T ) .  X  I(z) = I exp(-/ k(z').dz'),  (Lorenzen,1980, Megard e t a l ,1980). calculating  M  I  B  +  0.02.C(z)/V(z)  The c o n v e n t i o n a l m e t h o d f o r  r a t e over a mixed  k i s constant) i s to integrate  layer  of depth  3.3 v e r t i c a l l y t o  obtain :  ju =  oc.I  M A X  . (exp(-I /I Ax ) " exp(-I /I,^ M  M  0  ))/(k.z .V) M  - d  3.6  120  where I  = I e x p ( - k . z ) i s the  M  0  the mixed l a y e r . rewritten  light  M  i n t e n s i t y at the  U s i n g e q u a t i o n s 3.4  and  H  M  (k,z ) M  Some r e c e n t M a r r a ( 1 9 7 8 a,b)  laboratory  t i m e s c a l e s of  and  to very  i n the  high  o c e a n may  )) /  M A X  may  not  that  correct.  increase  over  photosynthetic  linearly  It follows  mixed l a y e r are  not  be  by  with  that,  short  enough,  e x p e r i e n c e p h o t o - i n h i b i t i o n or  in-situ  incubations  at  fixed  show t h e s e e f f e c t s .  accurate  a s s e s s m e n t of  short  t h e M a r r a e f f e c t a t O.S.P. w o u l d about c i r c u l a t i o n  t i m e s c a l e s , and  than i s c u r r e n t l y a v a i l a b l e .  same a t t e n t i o n be i n t e n s i t y as  As  Marra e f f e c t , the  i n the  to short-term  i s c u r r e n t l y being  nutrient a v a i l a b i l i t y Parslow,1980).  paid  average growth r a t e the  physiology,  require  true  that  f l u c t u a t i o n s i n ambient  paid  a c r u d e a p p r a i s a l of  mixed  a s s e s s m e n t of  o c e a n may  (Turpin,1980; Turpin,  d e t e r m i n e d as a f u n c t i o n o f  i n the  about p h y t o p l a n k t o n  U l t i m a t e l y , the  i n - s i t u phytoplankton production  light  /I  p h o t o - i n h i b i t i o n occur  levels.  although  r e q u i r e much more i n f o r m a t i o n  the  e  evidence presented  t h i s a p p r o a c h may  times in the  even p h o t o - s a t u r a t i o n ,  l a y e r on  field  exposure periods  circulation  depths w i l l  )-exp(-I  t e n s of m i n u t e s t o h o u r s and  i n t e n s i t y up  phytoplankton  M A X  3.7  l i g h t - s a t u r a t i o n and  r a t e s over short  An  be  - d  suggests that  Marra found that  provided  t h i s can  of  as:  jj = j u ( T ) .DL.e. (1 + 24. V /DL) . ( e x p ( - I / I  light  3.5,  bottom  the  to f l u c t u a t i o n s i n Harrison  and  i m p o r t a n c e of  the  i n t h e m i x e d l a y e r can  average l i g h t  i n t e n s i t y there  be :  121  u = «.I.exp(-I/I  M A X  )/V - d  3.8  where I = I . ( l - e x p ( - k . z ) ) / ( k . z ) . T h i s a l m o s t  certainly  exaggerates  f o r a l l but  0  very  M  the  importance  s h a l l o w mixed l a y e r s ,  equation  3.8  M  of the Marra e f f e c t a s , I/I x  i s much l e s s t h a n  M A  represents a linear  m i g h t be e x p e c t e d  for continuous  through  layer.  the mixed  response very  to l i g h t  rapid mixing  B e l o w t h e m i x e d l a y e r , where d a i l y v e r t i c a l phytoplankton  c e l l s a r e assumed t o be v e r y  growth r a t e at depth  z i s given  cX.l ( z ) . e x p ( - I ( z ) / I  p(z) =  M A X  layer  of  cells  movements of local  by  )/V - d  3.9  been i n t e g r a t e d o v e r  Z j  +2.5)/I  M A X  ) - exp(-I (z,-2 . 5 ) / l  M A X  )) /  (5.k.V) - d  Using  3.10  the above e x p r e s s i o n s , p h y t o p l a n k t o n  throughout  the water column can  daylength,  I , mixed l a y e r depth  depth.  The  now  0  growth r a t e s  be p r e d i c t e d , g i v e n  and  temperature,  In a d d i t i o n , the parameters  V must be  as  to give  oc. ( e x p ( - I (  ju(Zj ) =  and  intensity,  s m a l l , the  In t h e n u m e r i c a l m o d e l , t h i s e x p r e s s i o n has e a c h 5m  1  o t , j u ( T ) , and H  and  either  specified. temperature  E p p l e y ( 1 9 7 2 ) who  dependence of p  gave a s an u p p e r  H  was  bound  reviewed  secchi  by  I  M  A  X  or  122  log  l o  (jJ  M A X  )  = 0.0275.T - 0.070  c o r r e s p o n d i n g t o an 1.88X.  3.11  i n c r e a s e i n growth  Growth r a t e s  for particular  t h i s c u r v e and  fall  temperatures.  Taxanomic d i f f e r e n c e s  Chan(1978),  who  approximately  of t e m p e r a t u r e  O.S.P.  l i e below  i t at s u f f i c i e n t l y i n J U were c l e a r l y h  t h a t maximum g r o w t h  be a l l o w e d f o r by  i n e q u a t i o n 3.11  i s not well-known. and  An  and  high shown  r a t e s of d i a t o m s  retaining  v a r y i n g the  p r o v i d e d by K e n n e d y and o f more c a r e f u l l y  the  sampling  Fulton).  collected  and  ( l e s s t h a n 10 ju) f l a g e l l a t e s Comm.).  and  The  growth  C.  intercept.  been a c c u m u l a t e d  from  as p a r t  the weatherships  However, p r e l i m i n a r y preserved phytoplankton  intensity  has  throughout  of marine  analysis samples  I s h i m a r u and  r a t e of 2 d i v i s i o n s / d a y Coccolithus huxleyi This corresponds  studied  Waters,pers.  as t h a t o f  was  on c e r t a i n  of  diatoms  occasions  Nemoto,1977).  A maximum  growth  r e p o r t e d by P a a s c h e ( 1 9 6 7 ) f o r  u n d e r a 16H:8H l i g h t . d a r k  c y c l e a t 20°C.  to  ( j u ) = 0.0275.T H  (R.  small  C o c c o l i t h o p h o r i d s h a v e been r e p o r t e d a s  dominant p h y t o p l a n k t e r at O.S.P. (McAllister,1961;  the year  (data  n a n o f l a g e l l a t e s as a f u n c t i o n  n o t been a s w e l l  dinoflagellates.  1 0  were  coefficient  s u g g e s t e d 'that t h e p r i m a r y p r o d u c e r s a r e p r e d o m i n a t e l y  light  by  e x t e n s i v e s e t of o b s e r v a t i o n s of  d i n o f l a g e l l a t e s has  of r e g u l a r m i c r o z o o p l a n k t o n  log  ) of  1 0  s e a s o n a l p a t t e r n of p h y t o p l a n k t o n s p e c i e s c o m p o s i t i o n a t  l a r g e r diatoms  has  (Q  t w i c e a s l a r g e a s t h o s e o f d i n o f l a g e l l a t e s a t 20  These d i f f e r e n c e s can  The  10°C  species generally  s h a r p l y away f r o m  found  r a t e per  1.65  3.12  the  123  The  phytoplankton  respiration  constant  fraction  critical  in determining  particularly was in  v of ^ u ( T ) .  The  H  f o r deep mixed l a y e r s .  suggested  by R y t h e r ( 1 9 5 6 ) ,  r e p o r t e d v a l u e s , and  the ocean.  little  as c o n s i s t e n t l y standard  represented seasonal  by  rates,  noted  the l a r g e range  value  seemed t o  and  0.07  I t has  by H a r r i s and  g r e a t e r than  0.1  Piccinin(1977)  by B u r r i s ( 1 9 7 7 ) .  the s o l i d  line  i n F i g 24 w i l l  I „ M  the s o l i d  used.  The  i n F i g 27 w i l l  subject to  v a l u e of V a c c o r d i n g  to  l a y e r , V i s set equal  t o 20 t o r e p r e s e n t a  shade-adapted  The  use  of t h e s e  i n t e r p r e t a t i o n of the p a r a m e t e r s oc and  p h o t o s y n t h e s i s , and o b s e r v a t i o n s as photosynthesis.  I  M A  1 4  x  1  in equation  i n  equation 3.1  has  technique  interpretation  3.3  measured of *C 1  low  light  depth,  the  and  gross photosynthesis  intensities, 1  *C  especially  3.3  hinges  on  some d i s c u s s i o n .  determine  been f i t t e d  much c o n t r o v e r s y : i t i s g e n e r a l l y a c c e p t e d between net  Below the mixed  data which deserves  equation  i f t h e *C The  C  estimates  be  the  20.  The  and  line  f o r the mixed l a y e r ,  X  be  must be g r e a t e r t h a n  an  be  r e c e n t l y been  3.5  population.  0.1  p e r s i s t e n c e i n deep mixed l a y e r s  t h a t the c o r r e s p o n d i n g 3.4  he  be  s i m u l a t i o n s , t h e s e a s o n a l p a t t e r n of <X  used to determine  equations  shown t o  o f t - q u o t e d v a l u e of  t h a t a lower  resulted.  p a t t e r n f o r B g i v e n by  constraint  growth  be  T h e r e h a v e been many m e a s u r e m e n t s o f v s i n c e , but  more c o n s e n s u s has  In  An  although  observed  r e p o r t e d a s between 0.03 and  v a l u e of V w i l l  net p h y t o p l a n k t o n  needed t o e x p l a i n p h y t o p l a n k t o n in  r a t e i n the model i s a  to  gross weathership  gross  u p t a k e r a t e s has  seen  t o measure something  (Parsons  et a l ,1977).  near or below the  u p t a k e r a t e must be c l o s e t o g r o s s  At  compensation  photosynthesis,  124  as n e g a t i v e  1 4  C  uptake r a t e s a r e not t e c h n i c a l l y p o s s i b l e .  of t h e s a m p l e d d e p t h r a n g e a t O.S.P. light 1 4  C  intensities,  especially  uptake r a t e s as gross  less likely  corresponds  i n winter.  r a t h e r than  The i n t e r p r e t a t i o n o f  3.3 c a n be r e g a r d e d  o f oc a n d I ^ x u s e d i n  as p r e d i c t i n g net d a y l i g h t  product i o n , w i t h a reduced value respiration  only  i s then  I f an i n t e r p r e t a t i o n a s n e t  carbon uptake i s p r e f e r r e d , the estimates equation  t o such low  net photosynthesis  t o i n v o l v e major e r r o r .  Much  of v corresponding  to nocturnal  (cf McAllister,1969).  3.3.4 S i m u l a t i o n R e s u l t s . The  d e v e l o p m e n t so f a r a l l o w s t h e p r e d i c t i o n o f  phytoplankton  growth r a t e s over  phytoplankton  standing  g r a z i n g be m o d e l l e d , chapter.  time.  stock over  time  and t h i s w i l l  In t h i s chapter,  The p r e d i c t i o n o f zooplankton  be done i n t h e f o l l o w i n g  phytoplankton  p r e d i c t e d on t h e b a s i s o f t h e o b s e r v e d stock.  requires that  production  will  phytoplankton  standing  It will  be assumed t h a t g r a z i n g i s s u f f i c i e n t  standing  stock  i n t h e m i x e d l a y e r a t 0.4 ug C h i a . l " ,  observed  concentration.  will  1  n o t be a l l o w e d  allowed  to fall  Standing  be  t o keep t h e mean  s t o c k below t h e mixed l a y e r  t o exceed t h i s c o n c e n t r a t i o n and w i l l  b e l o w i t where r e s p i r a t i o n  be  exceeds  photosynthesis. In  F i g 32, t h e time  the y e a r s primary  s e r i e s of p h y s i c a l d r i v i n g v a r i a b l e s f o r  1964 t o 1976 a r e p l o t t e d .  production  on t h e f i x e d f o r oc a n d B  standing  seasonal  patterns  equation  3.12 f o r j u ( T ) a n d a v a l u e H  T h e s e were u s e d t o p r e d i c t stock b a s i s , using the  ( F i g 24,27) d i s c u s s e d f o r v o f 0.05.  above,  One o f t h e  noteworthy r e s u l t s of t h i s and subsequent s i m u l a t i o n s i s t h e l a c k  125  Figure 32.  Time series of t o t a l s o l a r r a d i a t i o n , surface  temperature and mixed layer  depth used to drive simulation model.  YEAR  126  of  year  (Fig  to  33  year  and  variation  Table  II).  Chapter  4;  for  focused  on  seasonal  1976,  to In  scale  avoid the  of  using  patterns  the  the  to  model.  I (t)/B  (2)  I  M  A  X  (t)  =  (I  (3)  I  M  A  X  (t)  =  I (t)/B  V(t)  so  was  =< a  mixed  first  by  The  V  and  fluctuations concentration  and  (C  interest  in  attention will  presented  for  of  different can  be  one  be  year,  now  stock  must  i t follows  vary  in equation  0.4  to ^g  could  time  not  be  day,  be  considered  (short  the  concept  considered adaptation  from  time  of  more on  equations  smoothly  scale)  a  3.4  fixing  carefully. seasonal and  3.5  Chi  concentration over  a . l '  by  I ,  I  does.  1  0  Q  will  time that  3.13  i s assumed  as  time  scale).  carbon  3.13  be  adaptation  + 24.V/DL).e.B)  adaptation  day  the  adaptation),  cases,  light  that  considered:  time  two  = V.0.4),  from at  will  found  intensities  production  phytoplankton  light s  light  first  s  replacing I  of  i t was  effects  cases  . I / ( p»(T).DL.(1  instantaneous  results  profiles,  simulation,  assumed  layer,  and  (seasonal  s  standing  t h a t both  chapter,  this  be  (t-l-)+I„ ( t - 2 ) + I „ ( t - 3 ) ) / ( 3 . B )  0  c o n s i d e r i n g the  the  of  (instantaneous  0  phytoplankton  will  changing  Three  =  scale  C  p r e d i c t e d primary  lMAX<t)  In  1 4  data.  (1)  Before  feature  production  repetition.  a n a l y s i s of  from  on  This  remainder  adaptation  inferred scales  the  in predicted phytoplankton  time.  (case  will  cause  V(t),  given  very  large  undergo  the  the  If  1),  Fixing  in  the  Chi  carbon  a  Figure  33.  P r e d i c t e d d a i l y net production using standard  parameter, s e t (see  text).  128  Table II Predicted annual primary production (gC.m 1964  _2  )  at O.S.P. f o r  to. 1-976. ( D e t a i l s of simulation provided i n text.)  Year  Gross (Daylight)  Net  1964  35.2  20.6  1965  35.5  21.2  1966  35.2.  19.1  1967  35.6  20.5  1968  35.4  21.5  1969  35.6  20.7  1970  35.2  19.8  1971  35.1  18.7  1972  35.3  20.0  1973  35.6  19.2  1974  36.2  21.6  1975  34.6  17.9  1976  34.9  19.6  129  concentration simulations carbon,  rates  are  such  realistic,  to  not  rapid  predicted. that  concentration. Chi  a  (a)  Fix  layer  C (t)  = V  where  V (t)  s  scale  vary  s  the  should  occur  This  can  accomplished  0.4  ug  carbon  rapidly, through  Chi  carbon  are  while 1  as  of  not carbon  i t seems changes  a . l "  these  conservation  g e n e r a l l y low  this  near  Although  growth  more  in Chi  keeping  a average  follows:  concentration, C (t),  in  s  the  by  ( t ) .0.4  3.14  i s given  s  Fix  i s to  be  with  in plant  of  If V  same m a n n e r .  concerned  i n view  seasonally varying  mixed  the  fluctuations  concentrations a  in  explicitly  especially  reasonable  (b)  fluctuate  the  actual  models;  the  by  equation  C:Chl Chi  a  a  ratio,  3.13. V,  using  one  of  concentration  i s then  daily  production  the  given  short  by  Chi  time a(t)  =  C (t)/V(t). s  1976  In  Fig  34,  in  the  first  simulation  The  daily  values  scale. mixed  layer  simulation varies  are  so  35.  smoothly  does  Chi  fluctuations is  a  simple  a  V  of  on  a  a  concentration. primary  explanation  and  Chi  Light  a  time  i s constant  for  markedly On  the  again  an  expanded  concentration  scale, at  in  so  0.4  other  hand,  production  are  much  this.  dominant  time  the  this  jug.l  daily  in  that  adaptation  with  The  predicted for  on  adaptation  instantaneous  fluctuates  i n net  V  seasonal  Chi for  primary  is plotted  plotted.  while  output  Here,  net  also  occurred  corresponding Fig  the  - 1  V .  (=  V (t)) s  The  is plotted  changes  in I  in Q  and  short-term reduced.  There  contribution  to  130  150  J F M R M J J R S ON D MONTH ' Figure 34.  Predicted net production, mixed layer Chi a and mixed l a y e r  C:Chl a r a t i o f o r 1976 using standard parameter set and seasonal l i g h t adaptation.  131  150  CJ  OH  1  1  1  1  1—'—i  ;  r  1  1  J F M fl M J J A S. O N MONTH  Figure 35.  1  D  As f o r F i g . 34,. but with 'instantaneous' adaptation.  light  132  net p r i m a r y layer.  production  Using  equation  layer  i s given  p.C .z  M  s  i n t h e w a t e r c o l u m n comes f r o m t h e 3.7,  d a i l y primary  s  As T and  H  M  - 2 4 . p ( T ) .V.C  z  M  vary  seasonal  smoothly over  adaptation,  fluctuates  f r o m day  adaptation, fluctuate  I  I  W  A  The  t o day  t o day.  For  choice  production  I t has  intensive The  little  although  sampling seasonal  periods  intermediate case  (2), V  C h i a and  III.  determines  daily  e f f e c t on a v e r a g e  primary  primary  C o m p a r i s o n of  adaptation  in I  s  in V  from w i n t e r  the  a r a t i o s o f 20 o r l e s s c a n  results  in  i n much l a r g e r  were o b s e r v e d d u r i n g  the  1968.  i s s m a l l compared w i t h the  five-  t o summer.  in I  the seasonal  of V a r e  w o u l d be c o n s i d e r e d  and  ( F i g 36).  scale therefore  to  R  instantaneous  the  time  O  exponent  not  f r o m 1964  variation  predicted values they  F  s e r i e s f o r C h i a i n F i g 35 w i t h o b s e r v a t i o n s  i s p a r t l y c o m p e n s a t e d f o r by The  this  •  the exponent does  f l u c t u a t i o n s i n Chi a than  fold variation  and  IO/IMAX  f l u c t u a t i o n as e x p e c t e d  shows t h a t i n s t a n t a n e o u s  t o day  S  For  l e v e l s , as shown i n T a b l e  p r e d i c t e d time  I  f l u c t u a t i o n s i n V,  very  3.15  M  and  of t h e a d a p t a t i o n  s i z e of s h o r t - t e r m  ,z  i s the exponent  Q  0  S  w  the major c o n t r i b u t o r to  with I .  varies with I  M A X  f r o m day  production.  DL.  time,  varies with  X  C h i a show r e d u c e d s h o r t - t e r m  day  H  f l u c t u a t i o n s in production  F i g 13  mixed  = C .p (T).DL.e.(1+24.v/DL).(exp(-I /l ^ ) D  the  i n the  by  e x p ( - I / W ) )/k  daily  production  mixed  This v a r i a t i o n  variation  reasonable  for  u n d e r low  H  and  flagellates  l a r g e for diatoms,  be e x p e c t e d  i n p (T)  where C : C h l  light  s  150  CJ  0  ~i JFMRMJJRSQ N D MONTH  Figure 36.  1  1  As f o r F i g  1  34,  1  1  1  1  1  1— —i 1  but w i t h 3-day adaptation.  1—  134  Table I I I _2 Monthly means of predicted d a i l y net primary production (mgC.m ) at O.S.P. using taree l i g h t adaptation time scales.  Seasonal  3-day  Instantaneous  Month January  3.  "5;  6.  February  26.  26.  26.  March  51.  53.  54.  April  73.  75.  77.  May  77.  75.  76.  June  112.  111.  111.  July  99.  100.  102.  August  77.  78.  80.  September  57.  58.  61.  October  47.  47.  49.  November  22.  18.  19.  December  7.  6.  7.  135  conditions.  It  is interesting  and  on  a n a l y s i s of  B,  based  seasonal under  variation  the  assumption  constant  B  increase  from  pattern  (2.0)  attainable  compared  and  with  McAllister,1969). of Cm"  47  g  and  2  upper to  Cm  g  bounds  allow  Cm"  were  for  effects annual  exceeding  300  mg  become mixed  Prod  clearer layer  =  (This  this  of  1976  a^.ly" )  that  the  seasonal  constraints  II  region  primary  a  of  are  on  rather  samples  lower  production,  over  production the  for  net  10  bounds and  of  this  daily  20  34  hr a  does  exp  case  of  instantaneous  (-I /I AX M  )  M  1  has  daylight, McAllister) production oc a n d  extinction  B  which  not  discrepancy  production  in  the  3.16  M  the  peak  as:  S  is for  g  1.25  has  in Fig  61 The  of  to  ( F i g 4)  peak  low  (Sanger,1972;  correction factor  p o s s i b l e sources 3.14  and  1  percentage  in Table  whereas  1  equation  Chi  for  for daylight production.  daily  Cm" .day" ,  when  C.mg  greater  this  net  using  c y c l e of  The  ratio  the  phytoplankton.  for  enclosure  total.  a  in  reduces  (1969) g a v e u p p e r a n d  by  2  also  suggests  totals  for  2  variation  0.4.«.I .(l-exp(-B)-24.v.z .k/(e.(DL+24.V))/(B.k)  parameters the  of  is rewritten  approximation (or  Cm"  mg  result  i n O.S.P.  obtained  His  half  a  respectively  2  periods.  reach  This  estimates  g  profiles,  much  partly  production  31  seasonal  oc (0.5  summer. be  the  p r e d i c t e d C:Chl  McAllister  and  - 2  48  to  previous  *C  shows a  ratios  annual  1  constant  B may a  the  The  ( F i g 37)  C:Chl  net  of  winter  oc  in  The  i n V.  that  were  coefficient.  Of  adaptation.  been  depends  estimated the  made). only  from  parameters  1 4  C  The Note on  that  the  profiles,  taken  gross  from  and the  100 CE I  £ 50~ CJ  0  i  1  1  1—'  i  1  1 — : — i  1  1  1  J F M fl M.J J fl S 0 .N D MONTH Figure 37.  Predicted C:Chl a r a t i o i n the mixed l a y e r for  1976 using OC =0.5  and B„ =  2.0.  137  literature,  y  determines  all.  This  stock  assumption,  a,  i s not  precisely It  for  annual  produced  be  here  about  the  of  x 4  C  criticised  earlier.  the  lower  of  the  the  water  surface  inclusion estimate.  He  which  based  were Given  respiration  also  the  on  low  term  in Table  The  of  choice  y  the  by  direct  done  was  different  32  an  reducing  for  the  gross  critical  in  the  other  annual an  daily  1964-1968  The  ,  than  resulting  a  littleand  about  possible  and  intensity  from  or  or  estimates  - 2  but  was  empirically  the  the  obtain  for  production  light  gross  Cm  to  estimation,  are  incubation  of  g  estimate  depth  to  production  used  in  Chi  literature,  McAllister,  now  There  surface  unit  estimate  data,  weathership  was  observations  is clearly  The  parameter  observations  II,  the  simulation°approach.  value.  per  attributed  weathership  in higher  estimate  and  - 2  from  depth-integrated  used  Cm  values  estimate  at  standing  McAllister's  be  used  through  and  constant  production  cannot  this  is responsible  production  g  the  McAllister  production surface  2  from  and  lower  between  of  48  column,  obtained  disagreement.  the  appear  field.  analysis.  However,  McAllister's  of  Cm"  data  estimation  that  relationship of  i n the  not  H  under  different  daylight production than  of  of  i f i t resulted  parameter  annual  2/3  see  g  profiles  for  ju (T)' does  is predicting  parameter  of  but  d i f f e r e n c e between  35  use  techniques  production  to  the  different  to  integration  data  model  i s measured  that  of  of  due  different  the  since  daylight production  assumption must  surprising,  what  follows  respiration  to  causes  obtained the  product  allow  the  production  earlier  time  period,  techniques. daylight production, very  low  production the  net  annual  almost  prediction  the  of  by net  half.  138  primary  production.  F o r e x a m p l e , i n c r e a s i n g y t o 0.10 ( t h e  c h o s e n by R y t h e r  (1956)), reduces  zero and r e s u l t s  i n n e g a t i v e d a i l y p r o d u c t i o n over  year  ( F i g 38).  I t i s clear  deep mixed l a y e r s , phytoplankton  that f o r oceanic  production almost t o half  environments with  i s essential  production.  Laboratory  to the accurate estimation r e p o r t s of p a r t i a l  shutdown a n d / o r l o w p o s i t i v e n e t p r o d u c t i o n a t v e r y intensities this light  regard.  Again  the time  s c a l e s of f l u c t u a t i o n s  i n t e n s i t y and of t h e p h y s i o l o g i c a l  low l i g h t  response  importance.  i n ambient  of c e l l s  i n the l i f e  i n the  The i n c l u s i o n o f l o n g  s c a l e a d a p t a t i o n such as t h e f o r m a t i o n of r e s t i n g  involved  cell  (Smayda a n d M i t c h e l l - l n n e s , 1 9 7 4 ) a r e i n t e r e s t i n g i n  m i x e d l a y e r may be o f c r i t i c a l time  of the  s u c h a s O.S.P., a b e t t e r u n d e r s t a n d i n g o f  respiration  of n e t p r i m a r y  net annual  value  spores  c y c l e s o f some p h y t o p l a n k t e r s c o u l d a l s o  reduce p r e d i c t e d l o s s r a t e s i n w i n t e r . I n s p e c t i o n of equation increase annual oc.  3.16 shows t h a t a s i m p l e way t o  production to McAllister's  levels  The s e a s o n a l p a t t e r n s o f n e t d a i l y p r i m a r y  i s to increase  p r o d u c t i o n and  C : C h l a r a t i o s p r e d i c t e d by t h e m o d e l f o r <X = 1.0 a n d B = 2.0, w i t h t h e a d d i t i o n a l c o n s t r a i n t t h a t V be l e s s t h a n plotted  i n F i g 39.  production supported  Annual gross production 2  2  and n e t  by t h e O.S.P. d a t a b u t i s c e r t a i n l y w i t h i n t h e r a n g e o f field  or l a b o r a t o r y s t u d i e s , as  above.  A n o t h e r way t o i n c r e a s e p r o d u c t i o n e f f e c t by u s i n g e q u a t i o n B obtained  i s 67 g C m "  A v a l u e o f cx a s l a r g e a s 1.0 i s n o t  i s 45 g C m " .  v a l u e s r e p o r t e d from o t h e r discussed  100, a r e  3.8.  With  i s t o introduce the Marra  t h e s e a s o n a l c y c l e s o f oc a n d  from p r o f i l e a n a l y s i s and used e a r l i e r , p r e d i c t e d  139  150  — 9 0 -!  1—~i  1  J  Figure 38.  F  i  1  M fl M J  1  1  J  MONTH  1  1  S  1  1—-I  0 N D  Predicted d a i l y net production on increasing V to  0.1.  140  Figure  39.  Predicted  d a i l y net production  C:Chl a r a t i o f o r oC =1.0, V  100.  '  and mixed l a y e r  B =2.0 s  with  constraint  141  gross  and  net  annual  respectively. (Fig  40)  and  loss  from  earlier, of  the  Marra  effect, and  levels  on  depth  i s the  to a  were  to  production, extinction annual  annual  net  Secchi  depth  The  included of  the  driving  25  depths.  production need  not  effect  Secchi 2  Table  This  Cm  10  more  g  the  was  than  of  Chi  not  observed  series  for  and  year net  g  lower  (Fig  daily  of and  Cm"  net  2  extinction  respectively  variation  annual  obtained  for •  by  cycles  in in  using  for a l l years. causes  of  observed  s e a s o n a l and  annual  a  annual  gross  35  of  secchi  of  three-fold  extremes  the  The  time  of  (minimum - 2  in  low  time  and  Cm  further  k  Upper  i s impressive, although match  of  predicted  - 2  importance  warranted.  II.  Estimates g  the  seasonal cycles  depths  and  which  against  extreme  61  discussed  observed  source  of  striking  seasonal cycle of  - 2  affects  interpolated  depth  is  Cm  respiration  is negligible.)  or  g  results  values  variable  ( F i g 42).  As  directly  average  i n F i g 33  g Cm"  constant  appears  The  seasonal cycles  f o r maximum  a  these  also  represents a  p r o d u c t i o n were  and  k,  (The  Secchi  coefficients  Secchi  envelopes  on  corresponding to  coefficients)  of  39  production  investigation  t o p r o v i d e an  generate  respectively  basis  coefficient  physical  plot  primary  minimum  the  based  and  exaggerates  ( e q u a t i o n 3.15).  consequently  envelopes used  certainly  coefficient,  density  not  of  54  production level.  experimental  one  and  variability  on  extinction  sufficient year  gross  to  primary  subtraction  (equation 3.2).  the  increases  i n net  almost  simulations are depth  the  but  estimates  Secchi  each  3.8  extinction  production  in  to  increased  equation  The  doubling  i s due  an  theoretical  above  The  production  variations  in  41)  142  270  J ' F ' M ' R ' M ' J ' J ' R ' S "O 'N 'D MONTH Figure  40.  P r e d i c t e d d a i l y net production f o r standard  parameter s e t w i t h Marra e f f e c t  introduced.  143  30  25  - < J >  5-  0 i  Figure  a  <x>®  .  J  • • i  41.  ^ b e f o r e 1964,  i  F  i  1  1  r~—i  M fl M J J fl MONTH  1  r  1  1 —  SO' N D  Observed Secchi depths at O.S.P. vs time of year, D  1964  to 1976.  S o l i d l i n e s represent upper and  lower evelopes and seasonal f i t to observations after  1964.  144  270  - 3 0 ~!  1  J  n  i  1—:—i  F M R M J  1  J  r  MONTH  F i g u r e 42a.  1  R S  1  ~i  1  0 N D  P r e d i c t e d d a i l y net production using  parameter s e t and upper envelope to S e c c h i d e p t h s .  1  standard  145  150  J  F  M  MJ  J.R  SO  ND  MONTH Figure  42b.  Predicted d a i l y net production using  parameter s e t and  lower envelope t o S e c c h i d e p t h s .  standard  146  Secchi depth generally  is. not c l e a r .  insufficient  i n any case  t o account  f o r observed  d o e s n o t show c o r r e s p o n d i n g  variations. detrital  The c o n c e n t r a t i o n o f C h i a i s  McAllister  et al(1961)  seasonal or annual  r e p o r t e d 100-200 mg C m  m a t e r i a l i n the surface waters  some f o u r t o e i g h t t i m e s Recent o b s e r v a t i o n s  S e c c h i d e p t h s and  the estimated  of  a t O.S.P. i n J u l y - A u g u s t , living  plant  carbon.  indicate a seasonal cycle i n d e t r i t a l  w i t h a minimum i n w i n t e r a n d maximum  - 3  carbon  i n summer, ( K . I s e k i and  C.S.Wong,pers. comm.), c o n s i s t e n t w i t h t h e n o t i o n t h a t c h a n g e s i n t h e e x t i n c t i o n c o e f f i c i e n t may be due t o c h a n g e s i n d e t r i t u s levels. Rather is  than  interesting  phytoplankton  that  V=  i n both  M  A  X  empirically  i n equation  3.4, i t  t o c o n s i d e r u s i n g an o p t i m a l i t y p r i n c i p l e f o r by d e m a n d i n g t h a t l i g h t  should proceed mixed l a y e r .  f i x i n g V or I  a d a p t a t i o n of  phytoplankton  so as t o maximise t h e a v e r a g e g r o w t h r a t e i n t h e Taking  equation  oc a n d  a s g i v e n , i t i s e a s y t o show  3.6 a n d e q u a t i o n  3.8, p i s m a x i m i s e d when  CX . I / (e . u . DL . ( 1 . + 24 . V/DL) ) . H  When V was c a l c u l a t e d cycle  i n t h i s way, u s i n g t h e o b s e r v e d  f o r ex, i t d i d n o t e x c e e d t h e minimum v a l u e o f 20 a l l o w e d  i n t h e model  ( F i g 43a).  c y c l e o f V was much l o w e r cX a n d B e s t i m a t e d phytoplankton rates  seasonal  than  seasonal  that obtained using the values f o r  from p r o f i l e s  ( F i g 43b). - A p p a r e n t l y ,  a t O.S.P. a r e n o t a d a p t e d s o a s t o m a x i m i s e  i n t h e mixed l a y e r .  to achieve  E v e n f o r tx=1.0, t h e r e s u l t i n g  As some m a r i n e p h y t o p l a n k t o n  C : C h l a r a t i o s o f 20 o r l o w e r  (eg Chan,1978),  growth  a r e known other  147  100  100  0"T—  J  I  r——i  M  F  r——i  R M J '  Figure  43.  r a t i o using (a)  Predicted  OC .  r——I  1  J  H  S  1  1  1  ,0 ,N D  MONTH seasonal cycle  optimality criterion  estimated  (b) c*:=i.o.  1  i n mixed  l a y e r C:Chl a  a n d c o n s t r a i n t V ^ 20 f o r .  148  selective pressures  not c o n s i d e r e d  g r a z i n g , must p r e v e n t 3.3.5  Primary The  primary  i n the growth model, such  t h e i r d o m i n a n c e a t O.S.P.  P r o d u c t i o n and  Nitrate Depletion.  amount o f c o n f i d e n c e  p l a c e d on t h e  p r o d u c t i o n a t O.S.P. ( F i g 33)  factors.  I f the  1 4  C  profiles  estimated standard  v a l u e s b e t w e e n 0 and  V.  oc and  I f the  easily  1  *C  The  26 g C m "  without  nitrate profiles  ug a t NO  or n i t r a t e  f o r 40  are  As  be  noted  the  above, on V,  obtained using  assuming v a l u e s  offer  f o r ot and  .m~  3  1  the p o s s i b i l i t y  and  the ranges f o r can  B outside  A lower  ( F i g 30)  t o g i v e 25 g C m " .  ( F i g 30).  v e r t i c a l mixing A slightly  The  to gross  annual  and  decrease  assuming a  of n i t r a t e  i n the average seasonal  and  f i g u r e o f 28 g  at least  not  be cycle Cm"  some e f f e c t  r e c y c l i n g as d i s c u s s e d  C:N  from below  h i g h e r e s t i m a t e can  resulting  i n c l u d e s p r o d u c t i o n down t o 80m v e r t i c a l m i x i n g and/or n i t r a t e  problems  T h i s e s t i m a t e does  2  summing a l l d e c r e a s e s  t o 80 m  limit  other  t a k i n g the average annual  i n t h e t o p 40m  recycling.  f o r an  production, although  involved.  f o r g r o w t h b e l o w 40m,  o b t a i n e d by  can  2  be o b t a i n e d by  r a t i o o f 7 mg.rng"  40m  +30%.  to within  ranges.  p r o d u c t i o n can  allow  t a k e n as a c c u r a t e  the  observations are discounted, gross production  interpretation  o f 6.4  as  this period is  d e p a r t i n g from l i t e r a t u r e  i n d e p e n d e n t c h e c k on p r i m a r y of  d e p e n d s on a number of  p r o d u c t i o n depend c r i t i c a l l y  B without  be d o u b l e d  literature  be  i n OC : a p p r o x i m a t e l y  e s t i m a t e s of n e t p r i m a r y  estimated  s i m u l a t i o n of  of t h e s i m u l a t i o n p e r i o d , 1 9 6 4 - 1 9 7 6 , t h e n  g r o s s p r o d u c t i o n may error  initial  f o r 1964-1968 a r e a c c e p t e d  a c c u r a t e m e a s u r e s of p r o d u c t i o n , and characteristic  as  2  of  earlier.  149  The  C:N  r a t i o can p o t e n t i a l l y  v a r y f r o m 3 t o 15  a l t h o u g h h i g h e r v a l u e s a r e most l i k e l y limitation.  The  annual v a r i a t i o n s fluctuations presumably  has n o t been u s e d t o l o o k a t  i n p r i m a r y p r o d u c t i o n , as the  large-scale  c o r r e s p o n d i n g t o a d v e c t i o n of d i f f e r e n t  particular  effects  time s e r i e s  t o o c c u r under n i t r o g e n  i n deep-water n i t r a t e c o n c e n t r a t i o n s ( F i g 30),  past the s t a t i o n i n any  nitrate  (Banse,1974)  , p r e v e n t an  interpretation  of n i t r a t e  y e a r as p h y t o p l a n k t o n u p t a k e .  have been removed i n t h e a v e r a g e  e s t i m a t e s of a v e r a g e  water  (These  annual p r o d u c t i o n based  on n i t r a t e  u n d e r e s t i m a t e s by an unknown amount w h i c h d e p e n d s on  and  recycling  production there  long-term The  decreases mixing  they are lower than the p r e d i c t e d  annual  ( T a b l e I I ) , t h e y a l l o w o n l y t h e weak c o n c l u s i o n t h a t  i s no e v i d e n c e  production  As  decreases  seasonal cycles).  are  rates.  masses  i n the n i t r a t e o b s e r v a t i o n s f o r higher  than p r e d i c t e d  i n the standard s i m u l a t i o n  ( F i g 33).  150  CHAPTER 4 HERBIVOROUS ZOOPLANKTON AT O.S.P.: DATA ANALYSIS AND MODELLING.  4.1 P a r a m e t e r 4.1.1  E s t imat i o n .  D e s c r i p t i o n of Data. The  time s e r i e s of o b s e r v a t i o n s of z o o p l a n k t o n from t h e  O.S.P. w e a t h e r s h i p s e x t e n d s  f r o m 1956 t o t h e p r e s e n t a n d  c o n s t i t u t e s one o f t h e most e x t e n s i v e open o c e a n d a t a s e t s o f i t s type.  F o l l o w i n g an e a r l y  r e p o r t on v a r i a t i o n  w i t h season and  depth of z o o p l a n k t o n c o m p o s i t i o n and abundance  (McAllister,1961),  two  i n manuscript  summaries o f t h e t i m e s e r i e s have appeared  (LeBrasseur,1965; Fulton,1978). a basis  f o r trophodynamic  studies  form  The d a t a have a l s o been u s e d a s (LeBrasseur,1969;  McAllister,1969,1972). The  a b u n d a n c e d a t a a n a l y s e d h e r e come f r o m 150 m  h a u l s u s i n g a 350 jam n e t . will  n o t be c o n s i d e r e d .  Q u a l i t a t i v e d a t a f r o m s u r f a c e tows C h a n g e s i n t h e d e s i g n o f t h e 350 p  o v e r t h e s a m p l i n g p e r i o d a r e d i s c u s s e d by F u l t o n ( 1 9 7 8 ) abundance e s t i m a t e s based  vertical  net  : earlier  on t h e NORPAC n e t have been a d j u s t e d  h e r e t o m a t c h t h e c u r r e n t l y - u s e d SCOR n e t u s i n g F u l t o n ' s correction  factor.  Wet w e i g h t s o f 150m v e r t i c a l throughout  the sampling p e r i o d .  h a u l s a m p l e s were r e c o r d e d T h e s e s a m p l e wet w e i g h t s ,  t o g e t h e r w i t h e s t i m a t e d wet w e i g h t s o f f o u r m a j o r groups  : c o p e p o d s , c h a e t o g n a t h s , e u p h a u s i i d s a n d a m p h i p o d s , were  r e p o r t e d a s t i m e s e r i e s by L e B r a s s e u r ( 1 9 6 5 ) to  taxanomic  1964.  Average  f o r t h e p e r i o d 1956  s e a s o n a l c y c l e s a n d a b u n d a n c e f o r 28 i m p o r t a n t  s p e c i e s d u r i n g t h i s p e r i o d w e r e p r e s e n t e d a n d d i s c u s s e d by  151  LeBrasseur(1965,1969) . Sample a n a l y s i s c h a n g e d a f t e r statistics and  #ind/m  p r e s e n t e d by - F u l t o n ( 1 9 7 8 ) c o n s i s t o f s a m p l e wet w e i g h t i n each  3  LeBrasseur's in  of 5 taxanomic  groups which  identified  w i t h most  comprehensive  are used  This analysis w i l l  in this  thesis  s e c t i o n , a parameter  abundance d a t a  The d a t a a r e  variations  seasonal  i n t h e s e c y c l e s t o be  be r e p o r t e d e l s e w h e r e .  i n two r a t h e r s p e c i a l i s e d  The d a t a  roles.  e s t i m a t i o n technique developed  copepod c o h o r t s i s adapted  classes.  t o a l l o w LeBrasseur's average  c y c l e s t o be r e f i n e d , a n d a n n u a l considered.  organisms  tape and a p r e l i m i n a r y a n a l y s i s  been made i n c o - o p e r a t i o n w i t h J . F u l t o n .  sufficiently  Starting  t o s p e c i e s l e v e l and a s s i g n e d t o s i z e  T h e s e d a t a a r e s t o r e d on m a g n e t i c has  include  4 groups and t h e trachymedusae, A g l a n t h a .  1 9 6 9 , s a m p l e s were a n a l y s e d i n d e t a i l  being  In  1965 a n d t h e summary  In t h i s  for analysing  and a p p l i e d t o t h e s i z e - s t r u c t u r e d  f o r t h e d o m i n a n t h e r b i v o r o u s c o p e p o d s a t O.S.P.  the remainder  of t h e .chapter, t h e data w i l l  be u s e d  i n the  c o n s t r u c t i o n and s t u d y of a model o f t h e p h y t o p l a n k t o n zooplankton 4.1.2  i n t e r a c t i o n a t O.S.P.  Review of Parameter As b a c k g r o u n d  r e s u l t s of Parslow  Estimation  t o the parameter  Techniques. e s t i m a t i o n f o r O.S.P., t h e  et. a l (1979) a n d S o n n t a g a n d P a r s l o w ( 1 9 8 0 ) a r e  reviewed.  I n t h e s e p a p e r s , methods f o r o b t a i n i n g p o p u l a t i o n  parameters  from time s e r i e s of d e n s i t i e s  in life  u s i n g a t e c h n i q u e of systems i d e n t i f i c a t i o n  history  stages  were e x p l o r e d .  u n d e r l y i n g technique has the f o l l o w i n g g e n e r a l s t r u c t u r e . a s e t of o b s e r v a t i o n s Y ( t ; ) ,  i = 1,...,N, a n d a m o d e l  The Given  which  A  a l l o w s p r e d i c t i o n s Y ( t , p ) as a f u n c t i o n of time and parameters ;  152  p  that  r  squared  set  of  errors  parameters  p  (SSQ  (Y(t;)-Y(tj)) ).  =  I  i s sought  which  minimizes  the  sum  of  2  i  In  Parslow  densities and  the  Sj(t  parameters, population  d_Z(a,t)  T  0(a)  i s the  and  This  must  6.  i s the  be  observations  in M  stages  the  stages  (or  consisted of  stages)  times  Tj  and  residence (or groups  Underlying  the  model  of  groups  of  stages).  p r e d i c t s Sj ( t ; ) g i v e n  age-structure  a  set  A of  s t a g e - s t r u c t u r e of which  has  the  the  form  = -e(a).Z(a,t),  d_Z(a,t)  per  the  c o n s i s t e d of  0j i n t h e s e  i s the  +  1,...,M  r e q u i r e d which  Z(a,t)  Sj(t).  ),j =  rates  i s then  where  a_l ( 1 9 7 9 ) ,  parameters  mortality model  et  population  capita  coupled  density  mortality rate  with  a  If a l l individuals  formula spend  the  of  age  for  a  at  time  individuals  f o r mapping same  time  t  of  Z(a,t) T,-  in  and age  a  into  stage  j ,  k Sj ( t )  Z(a,t).da k--i  More  K  complicated  times  are  formulae  allowed an  to  fitting  such  initial  conditions  required,  but  are  result  number  models  performances  were  Z(a,0) not  can  such  vary  are  be  of  and  directly  made  to  a  model the  by  parameter  i f individual  mean v a l u e , to  available.  models  this  et  estimation  residence  The  conditions  al  Sj(t).  (1979),  procedure  Z(0,t)  and  and  in  the are  simplifying  difficulty predict  problem  i s that  Various  which  Parslow  Ij .  observations  boundary  overcome  simpler  considered  i n the  required  about  age-structure  assumptions in a  .  these Four their  compared.  153  Any  model  compartment There  is a  previous being  model, flux  The  fixed are  R  j + |  The  - Rj  +)  is essentially  times.  into  compared  by t h e  with  rate  the observed  after t o be  the residence with  time  into  recruitment  t =  when  Rj(t).  time  we  This  rate  being  model  with  and out of stage  T  k  calculate  cannot  be  j  stage  in S ^ t ) .  R (t) = p.S (t) 2  Sj(t  ) a r e used start some  stage is  Then  c a n be  i  , so t h a t  done  the second  i n the f i r s t  i f observations £  t r y to  into  s c a l e s of changes  densities  calculable have  rates  approximation  is  stages  stage,  t h e time  Clearly,  compared  ( t ) , and the other  the Rj's are c a l c u l a t e d .  appears  i f the residence  variable.  If  )  Bj.'Cj)  driving  later  +  rela-tionship  the f i r s t  transfer  where  only  Rj  from the  f l u x e s o u t , one  an a g e - s t r u c t u r e  Recruitment  c o n d i t i o n problem  be c a l c u l a t e d  used,  stage,  recruitment  a n d two  i n t h e way  b u t an a p p r o x i m a t e  linear  represents  contents.  6j.Sj(t)  differ  connected  boundary  short the  (t) -  l a g model  recruitment  can  t h e compartment  reproduction)  ( t ) = Rj ( t - T,-) . e x p ( -  exactly  which  t o t h e next  models  residence  then  Sj ( t ) b e i n g  of as a  Thus  simpler The  with  ( o r from  recruitment  = Rj(t)  p r e d i c t s Sj ( t ) c a n b e t h o u g h t  in,Rj(t),  stage  mortality.  Sj  which  to construct a t t=0,  a  Rj(t)  observations in  discarded.  times  i n a l l stages  s c a l e s of changes  a r e assumed  in densities,  t o be  the  short  linear  154  transfer  R  (t)  j + 1  can  rate  =  P  j  b e made  driving  assumption  .Sj ( t )  f o r a l l stages,  variable.  simple  rather  problem  was p r o p o s e d cohorts.  follows  a gaussian this  S (t) 1  'linear  solution  assumed  distribution predicts  T  that  k  ( t )  used  transfer' c a n be  f o r time  over S  being  model i s  condition  series  recruitment time.  as a  used.  t o the boundary  by M a n l y ( 1 9 7 4 )  Manly  model  again  and a l l observations  different  distinct  S-. ( t ) ,  with  The r e s u l t i n g  (and l i n e a r )  A  :  into  Rather  by a s s u m i n g  representing each  than a  stage  predict  common  im-  J  mortality time  rate  f o r these  and spread  Although model  stages.  of recruitment  stage-dependent  i s not i n t e r n a l l y  Estimates  are obtained  mortality rates consistent  of the size, f o r each  mean  stage.  c a n be o b t a i n e d , t h e  i f mortality rates  vary  over  stages. The  lag-Manly  recruitment between  into  assumption  not  vary. The  a of  complex  each  model  that  Manly's  stage  models  residence  of the parameter i n turn  of the models.  inferior  stage.  gaussian  found  I t allows the  times  i n each  estimation  designed that  to the others  relationship  mortality rates,  was c o n s i d e r e d  I t was  of  and the lag-model's  of stage-dependent  s i m u l a t i o n model  was m a r k e d l y  assumption  and out of each  individual  performance  of these  more  into  estimation  the  uses  the f i r s t  recruitment  consistent  each  model  to probe  do  using  produced  by  t h e weaknesses  the linear under  stage  technique  f o r data  under  almost  transfer a l l  155  conditions.  The o t h e r t h r e e m o d e l s p r o v i d e d good e s t i m a t e s o f  residence times  f o r a range of sampling  stage aggregation.  The l a g - M a n l y  model p e r f o r m e d  e s t i m a t i o n e r r o r s o f a few p e r c e n t  a t most.  r e s i d e n c e t i m e s by t h e l a g a n d l a g - M a n l y insensitive  t o log-normal  average e r r o r s l e s s than 0.4  i n the observation  stages  E s t i m a t i o n of relatively  i n the data, with  25% f o r c o e f f i c i e n t s of v a r i a t i o n  up t o  errors. r a t e s was much  R e l i a b l e e s t i m a t e s of m o r t a l i t y  were n o t o b t a i n e d f o r any m o d e l . reliable  best, with  m o d e l s was  observation error  The e s t i m a t i o n o f m o r t a l i t y satisfactory.  i n t e r v a l s and d e g r e e s of  The l a g - M a n l y  e s t i m a t e s of average m o r t a l i t y  f o r small sampling  less r a t e i n each  stage  model d i d produce  r a t e s i n aggregated  i n t e r v a l s under z e r o o r low o b s e r v a t i o n  error. When a p p l i e d t o a s e t o f CEPEX d a t a r e p r e s e n t i n g Pseudocalanus all  models.  unsatisfactory  high observation e r r o r s i n the data, or t o a  of t h e models' assumption  are c o n s t a n t over  be o b t a i n e d a s a b y - p r o d u c t  specified.  that population  p r o d u c t i o n , i n c l u d i n g m o r t a l i t y , can of the parameter e s t i m a t i o n  once a v e r a g e i n d i v i d u a l Estimates generated  weights  secondary  technique  s i m u l a t e d d a t a and a d i f f e r e n t  representing a Paracalanus  i n each stage are  i n t h i s way h a v e been c o m p a r e d  with these obtained using a c l a s s i c a l for both  parameters  time.  E s t i m a t e s of secondary  procedure,  r e s u l t s were o b t a i n e d f o r  I t was n o t c l e a r w h e t h e r t h i s was due t o  unexpectedly violation  minutus,  parvus  (Winberg,1968)  s e t o f CEPEX d a t a  population.  Estimates of  p r o d u c t i o n f o r s i m u l a t e d d a t a were f o u n d  t o be much  156  more r e l i a b l e  than m o r t a l i t y  rate estimates.  The l a t t e r a r e  c a l c u l a t e d e s s e n t i a l l y as d i f f e r e n c e s between e s t i m a t e s of t o t a l recruitment  i n t o s u c c e s s i v e s t a g e s , w h i l e secondary p r o d u c t i o n  r e p r e s e n t s a weighted average 4.1.3  Application  t o O.S.P.  of these r e c r u i t m e n t s . Data.  A l t h o u g h t h e s e t e c h n i q u e s were d e v e l o p e d densities  in life  history  time s e r i e s of d e n s i t i e s application problems copepods,  f o r time s e r i e s of  stages, they a r e e q u a l l y a p p l i c a b l e t o in size classes.  However,  their  t o t h e d a t a f r o m O.S.P. h a s p r e s e n t e d a number o f  due t o t h e l i f e  history  s t r a t e g i e s of t h e dominant  t h e s a m p l e methods e m p l o y e d , a n d t h e d e t a i l s o f s a m p l e  analysis. The  life-history  s t r a t e g i e s of Calanus plumchrus  c r i s t a t u s were d e s c r i b e d i n C h a p t e r r e c r u i t e d as n a u p l i i  1.  and Calanus  Both s p e c i e s a r e  i n the s u r f a c e waters  i n l a t e winter or  early  spring.  I n d i v i d u a l s f e e d a n d grow i n t h e s u r f a c e w a t e r s  until  t h e y r e a c h s t a g e V and have a c c u m u l a t e d  a lipid  reserve.  These copepods then l e a v e t h e s u r f a c e w a t e r s and o v e r - w i n t e r below  200m.  Reproduction takes place a t depth  Calanus plumchrus  ( F u l t o n , 1 9 7 3 ) and n a u p l i i  surface waters the f o l l o w i n g  spring.  There  i n t h e case of  are recruited to i s some s u g g e s t i o n i n  the  O.S.P. d a t a t h a t C. c r i s t a t u s a d u l t s may a p p e a r  the  s u r f a c e waters  briefly in  i n spring.  These c y c l e s a r e q u i t e c o m p a t i b l e w i t h t h e e s t i m a t i o n t e c h n i q u e s d i s c u s s e d above, cohort. 150m  as they r e s u l t  However, t h e v e r t i c a l  in a distinct  annual  h a u l s a t O.S.P. s a m p l e t h e t o p  o n l y so t h a t t h e d e p a r t u r e of o v e r - w i n t e r i n g copepodids  surface waters appears  from  i n the time s e r i e s as a d e c l i n e which can  157  potentially  be c o n f u s e d  e s t i m a t e s of m o r t a l i t y o b s e r v a t i o n s taken  with mortality.  broad be  after  the departure  i n an u n a c c e p t a b l e  c o h o r t s of t h e type observed  simply represented  distort  commences a r e d i s c a r d e d . way t o c h o o s e t h i s  size-class  time,  l o s s of i n f o r m a t i o n f o r  a t O.S.P.  i n the lag-Manly  of t h e l a r g e s t o b s e r v e d observed.  badly  r a t e u s i n g t h e Manly model, u n l e s s a l l  E v e n a s s u m i n g t h e r e was some a p r i o r i t h i s would r e s u l t  This w i l l  O v e r - w i n t e r i n g can  model a s r e c r u i t m e n t o u t  i n t o another  which i s not  T h i s does not e l i m i n a t e t h e p o t e n t i a l  confusion  between o v e r - w i n t e r i n g and m o r t a l i t y however, w h i c h w i l l l a t e r a s u n c e r t a i n t y i n r e c r u i t m e n t and m o r t a l i t y  appear  rate estimates.  There i s a l s o a problem i n t h e i n t e r p r e t a t i o n of r e c r u i t m e n t in  the spring.  The s m a l l e s t s i z e c l a s s e s i d e n t i f i e d a s  C. p l u m c h u s o r C. c r i s t a t u s h a v e a n o m i n a l (Naupliar  l e n g t h of  s t a g e s a r e n o t i d e n t i f i e d and a r e p r o b a b l y  r e t a i n e d by t h e 350 jam n e t ) .  efficiency.  of r e c r u i t m e n t p r o d u c e d here t h e r e f o r e a p p l y size  s i z e o r sample  below w h i c h a l l i n d i v i d u a l s a r e i g n o r e d , and  above which a l l a r e sampled w i t h c o n s t a n t  across this  poorly  I t i s assumed h e r e t h a t a s i z e  t h r e s h o l d e x i s t s , due e i t h e r t o n e t a p e r t u r e analysis techniques,  1mm.  threshold.  Estimates  to recruitment  I t i s possible that size classes  above t h i s t h r e s h o l d a r e sampled w i t h v a r i a b l e e f f i c i e n c y , o r that  some i n d i v i d u a l s a r e r e c r u i t e d  above t h e t h r e s h o l d .  To t h e e x t e n t  i n t o t h e t o p 150m a t s i z e s that either  e s t i m a t e s p r o d u c e d h e r e may be d i s t o r t e d .  occurs, the  Estimates  of  secondary  p r o d u c t i o n p r o d u c e d h e r e do n o t i n c l u d e m o r t a l i t y o f s m a l l i n d i v i d u a l s b u t t h i s u s u a l l y makes a n e g l i g i b l e c o n t r i b u t i o n t o total  secondary  production  (Sonntag  and  Parslow,1980).  158  1  The the  a b o v e p r o b l e m s c a n be h a n d l e d by s i m p l e  l a g or lag-Manly models.  creation  o f a new e s t i m a t i o n  adjustments of  A t h i r d problem has r e q u i r e d t h e model.  The a n a l y s i s o f  O.S.P. z o o p l a n k t o n s a m p l e s h a s i n v o l v e d a l a r g e number o f technicians  over s e v e r a l years.  ensure c o n s i s t e n c y (Fulton,1978),  of counting  can  size classes,  identification  As a r e s u l t ,  amongst  several d i f f e r e n t sets  i n v o l v i n g f r o m one t o f i v e d i f f e r e n t d i v i s i o n s ,  be f o u n d even w i t h i n one y e a r ' s d a t a f o r one s p e c i e s .  attempt t o force these data in order  t o apply  resolved  Any  s e t of s i z e - c l a s s e s i n v o l v e much  t h r o u g h t h e b l u r r i n g o f some s i z e  the d i s c a r d i n g of poorly An  i n t o a standard  t h e l a g or lag-Manly models w i l l  l o s s of i n f o r m a t i o n , and  and s p e c i e s  h a s been t a k e n t o  t h e s i z e c l a s s e s u s e d have v a r i e d  t e c h n i c i a n s and over time. of  While care  classes  samples.  a l t e r n a t e a p p r o a c h h a s been d e v e l o p e d w h i c h a v o i d s a n y  l o s s of information  i n using  the a v a i l a b l e data.  assume a s e t o f f i x e d r e s i d e n c e  times  Rather  than  Tj f o r a u n i q u e s e t o f s i z e  c l a s s e s , a c o n t i n u o u s r e l a t i o n s h i p between a g e a n d l e n g t h , a = A ( l ) , i s assumed a n d p a r a m e t r i s e d . gaussian recruitment, allow  together  with  The a s s u m p t i o n o f  the age-structure  t h e p r e d i c t i o n o f an a g e - d i s t r i b u t i o n Z ( a , t )  over  w h i c h c a n t h e n be t r a n s l a t e d i n t o a s i z e - d i s t r i b u t i o n For  each sample, i n t e g r a t i o n of the p r e d i c t e d  between a p p r o p r i a t e  l i m i t s provides  model, time,  Z'(l,t).  size-distribution  p r e d i c t i o n s of d e n s i t i e s i n  s i z e c l a s s e s which e x a c t l y match t h e s i z e c l a s s e s used i n t h e a n a l y s i s of that  sample.  A power l a w i s c h o s e n f o r A ( l ) : a = a .(l 1  l f  -lX)  159  where 1  i s t h e t h r e s h o l d s i z e below which i n d i v i d u a l s a r e not  R  sampled. weight l ,  i s proportional to l  ^ = 1.  2  and X  The p a r a m e t e r s a± a n d y a r e e s t i m a t e d . and growth  suggest  i s l a r g e r than  1.  i n g e s t i o n may be p r o p o r t i o n a l t o l e n g t h ) ,  As e x p o n e n t i a l g r o w t h i s a p p r o a c h e d ,  must a l s o be p a r a m e t r i z e d .  scatter  follows.  (eg Cushing,1976),  t h e l a c k of s i z e - c l a s s  due t o o v e r - w i n t e r i n g d i s c o u r a g e d  beyond a c o n s t a n t The  exact  r e s o l u t i o n and a n y a t t e m p t t o go  mortality rate.  parameter e s t i m a t i o n procedure i s t h e r e f o r e as  Given  initial  guesses a t t o t a l  o f r e c r u i t m e n t <T, mean t i m e  recruitment, R , T  i s g i v e n by  Z(a,t) = R .exp(-(t-a-p) /(2.a  ) - ©.a) / (7277.(7)  T  can  2  values  fora  i  be d e d u c e d .  and  #,  F  maximum o b s e r v e d  U  = IR  the l e n g t h - d i s t r i b u t i o n 1 , and a t overR  f o r e a c h s p e c i e s b a s e d on minimum a n d  lengths.  c l a s s e s w i t h nominal s i z e s size class limits  2  The l e n g t h s a t r e c r u i t m e n t ,  wintering, 1 , are fixed  spread  o f r e c r u i t m e n t jj a n d m o r t a l i t y r a t e  e, t h e a g e - d i s t r i b u t i o n Z ( a , t )  Given  y  W h i l e some d e p e n d e n c e o f  r a t e on s i z e m i g h t be e x p e c t e d  i n the data,  confusion  (eg S t e e l e  The m o r t a l i t y r a t e l a w e ( a ) , o r e q u i v a l e n t l y  approaches zero.  mortality  (dW/dt) p r o p o r t i o n a l t o  I f g r o w t h i s a s m a l l e r power o f l e n g t h  Frost(1977)  0(1),  3  Note t h a t i f  On e a c h d a y , g i v e n a s e t o f M 1 ,1 , . . . , 1 , a c o r r e s p o n d i n g A  i s d e f i n e d by  2  M  size s e tof  160  lj  = (l]+l>,  IM  =  )/2  j= l,...,M-l  IF  r'J  Predicted densities Sj(t) = J with observed  Z'(l,t).dl  can then  be c o m p a r e d  d e n s i t i e s s j ( t ) and t h e p a r a m e t e r s c o r r e s p o n d i n g t o  minimum SSQ e r r o r s o b t a i n e d u s i n g t h e M a r q u a r d t a l g o r i t h m (Marquardt,1963). 4.1.4 S t a t i s t i c a l In  Considerations.  view of the l a c k of comparable e s t i m a t e s of the  p a r a m e t e r s s o u g h t h e r e , an a t t e m p t an  indication  of s t a t i s t i c a l u n c e r t a i n t y i n e s t i m a t e d  As a p r e l i m i n a r y s t e p , t h e n a t u r e zooplankton replicate  was made t o p r o v i d e a t l e a s t  of v a r i a b i l i t y  A n a l y s i s o f 10  w i t h i n a s h o r t t i m e p e r i o d (3 h o u r s ) on  10th June, 1978, i n d i c a t e d  that the variance i n species  i n c r e a s e d w i t h t h e mean a t l o w d e n s i t i e s mean s q u a r e d  i n the  s a m p l e s f r o m O.S.P. was i n v e s t i g a t e d .  samples taken  parameters.  at high d e n s i t i e s  coefficient  of v a r i a t i o n  normalising  transformation  ( P o i s s o n ) and w i t h t h e  (log-normal)  o f 0.2.  counts  w i t h an a s y m p t o t i c  F o l l o w i n g Barnes (1952), the  Y' = s i n h " (/3.JI)//3 1  with  j8= .2, was e m p l o y e d a n d t h e SSQ e r r o r s b e t w e e n  p r e d i c t i o n s a n d o b s e r v a t i o n s was m i n i m i z e d . obtained without from year  transforming the data  transformed  (Estimates  showed g r e a t  initially  variability  t o year and i n s p e c t i o n of p r e d i c t e d timestreams  showed  a t e n d e n c y t o f i t one o r two a n o m a l o u s h i g h o b s e r v a t i o n s  closely  and  ignore the rest.)  161  The.exact s t a t i s t i c a l  distribution  of e s t i m a t e s  obtained  through  non-linear, least-squares techniques  c a n n o t g e n e r a l l y be  found.  H o w e v e r , an a p p r o x i m a t e d i s t r i b u t i o n  c a n be o b t a i n e d by  approximating function  t h e n o n - l i n e a r f u n c t i o n o f p a r a m e t e r s by a  i n t h e neighborhood of t h e parameter e s t i m a t e .  this approximation, . . . £• i i d N ( 0 , ( 7 e  2  ),  i f the s t a t i s t i c a l  Under  m o d e l i s Y,- = f j (p) + £\  P* i s t h e t r u e p a r a m e t e r s e t a n d p i s t h e  l e a s t - s q u a r e s e s t i m a t e , p -p* h a s a m u l t i v a r i a t e distribution  linear  with covariance  normal  matrix -i  e v a l u a t e d a t p (Benson,1978). clearly  The a c c u r a c y  d e p e n d s on t h e r e l a t i v e  magnitude of t h e l i k e l y  R e s u l t s f o r Calanus plumchrus. I t was p o s s i b l e t o a p p l y  to  error in  i n i.  p and t h e s c a l e of n o n - l i n e a r i t i e s 4.1.5  of t h e approximation  s i z e - s t r u c t u r e d data  1969,1970,1973,1975,1976 structure  the parameter e s t i m a t i o n  techniques  f o r C. p l u m c h r u s i n t h e y e a r s and 1977.  i n f o r m a t i o n was e i t h e r  allow parameter e s t i m a t i o n .  I n 1971 a n d 1 9 7 2 , s i z e -  not present  or i n s u f f i c i e n t t o  I n 1974, t r a n s i t i o n  water  appeared  i n mid-summer a n d t h i s , c o m b i n e d w i t h a 6-week d a t a g a p , made i t i m p o s s i b l e t o f i t a c o h o r t model s e n s i b l y . Six total  p a r a m e t e r s were e s t i m a t e d  T h e s e were  r e c r u i t m e n t R , mean r e c r u i t m e n t t i m e ju, s t a n d a r d d e v i a t i o n T  of r e c r u i t m e n t l e n g t h power  T,  i n each y e a r .  estimated  cr , i n d i v i d u a l y and m o r t a l i t y  residence time r a t e ©.  i n place of the constant  i n s a m p l e s T, a g e -  The t o t a l r e s i d e n c e a , i s g i v e n by t  time,  162  T h e s e s i x p a r a m e t e r s c o u l d be d i v i d e d i n t o two g r o u p s b a s e d on the p r o p e r t i e s of the approximate c o v a r i a n c e m a t r i c e s . c o n s i s t e d o f u,  0  and  T.  One s e t  These v a r i a b l e s , a l l r e l a t e d t o  t i m i n g , were n o t h i g h l y c o r r e l a t e d w i t h one a n o t h e r  or with  members o f t h e s e c o n d s e t , c o n s i s t i n g o f R , 0 a n d  Jf.  T  for this  Estimates  s e c o n d s e t were v e r y h i g h l y c o r r e l a t e d w i t h one  c o r r e l a t i o n c o e f f i c i e n t s o f 0.98 b e i n g common. essentially  one d e g r e e o f f r e e d o m i n v o l v i n g  T h e r e was  these  parameters  w h i c h c o u l d n o t be r e s o l v e d by t h e e s t i m a t i o n t e c h n i q u e these  time The  actually  from  series.  parameter estimates a r e given  confidence  another,  i n t e r v a l s given  the l i m i t s  i n Table  The 9 5 %  i n t h a t t a b l e f o r u, <J a n d T a r e  of a 95% c o n f i d e n c e  parameters determined  IV.  by F^„.  ff  c o r r e l a t i o n amongst R , e a n d T  region for a l l 6  statistics..  Because of t h e h i g h  y, i n d i v i d u a l confidence  intervals  a r e u n i n f o r m a t i v e a n d i n s t e a d p r o j e c t i o n s o f t h e 95% c o n f i d e n c e r e g i o n s on t h e ( R , e ) a n d ( 0 , y ) p l a n e s T  h a v e been p r e s e n t e d i n  F i g 44. N o t e t h a t , b e c a u s e o f t h e c o n f u s i o n o f ©, R became n e c e s s a r y  t o c o n s t r a i n 9 t o be n o n - n e g a t i v e  In f a c t , as z e r o m o r t a l i t y constraint  T  6 ^ 0.005 d a y "  rates are unlikely 1  and  y, i t  i n some  years.  i n nature, the  was i m p o s e d a n d c o n f i d e n c e  regions f o r  ( R , 9 ) and ( 6 , ) a r e t r u n c a t e d a c c o r d i n g l y . T  The  mean r e c r u i t m e n t t i m e o c c u r s  apparently e a r l i e r  recruitment  d e v i a t i o n of r e c r u i t m e n t constant Residence  in April  a n d e a r l y May  i n 1975 a n d 1 9 7 6 .  The  into the smallest size c l a s s  a t a b o u t 30 d a y s ,  with a larger  t i m e s a b o v e 150m a r e r e m a r k a b l y  spread  with  standard is fairly  i n 1975.  c o n s i s t e n t from  year-  163  Table IV. Parameter estimates f o r Calanus Year  (days)  CT(days)  1969  121.±27.  27.± 9.  56.±34.  50.  0.005  1.4  1970  126.±21.  31.± 5.  58.±23.  238.  0.005*  1.0  1973  112.±18.  24.± 7.  66.±31.  318.  0.040  2.2  1975  93.±30.  48.±12.  65.±65.  72.  0.005*  0.45  1976  96.±29.  34.± 8.  64.±37.  141.  0.005*  1.0  1977  105.±10.  31.± 4. "66. ±20.  498.  0.027  1.6  *  T(days)  plumchrus.  R (ind.m~ ) T  3  These v a l u e s were f r o z e n t o a l l o w  e(day  _ 1  )  convergence.  164  165  Figure 44b.  Projection of approximate 95% confidence regions for Calanus plumchrus parameter estimates on (y,0) plane.  166  to-year considering the large confidence  intervals.  The s a m p l e d  p o p u l a t i o n o f C. p l u m c h r u s a b o v e 150m a t O.S.P. c a n t h e r e f o r e be c h a r a c t e r i z e d as a broad  cohort, recruited  over  2 months o r more  f r o m M a r c h t o May, o f i n d i v i d u a l s w h i c h s p e n d a b o u t 60 d a y s i n the  s u r f a c e l a y e r and l e a v e d u r i n g June and J u l y . A d e t a i l e d account  the S t r a i t  of Georgia  deep v e r t i c a l maturation  of t h e l i f e  history  o f C. p l u m c h r u s i n  was g i v e n by F u l t o n ( 1 9 7 3 ) .  hauls allowed the f u l l  life  In that  cycle,  a n d r e p r o d u c t i o n , t o be f o l l o w e d .  study,  including  Fulton  reported  t h a t maximum egg p r o d u c t i o n o c c u r r e d on M a r c h 5 t h a n d maximum abundance of n a u p l i i  i n surface waters  abundance of c o p e p o d i t e 18th.  H i s schematic  exceeding  stage V occurred  diagram suggests  2 months i n w i d t h ,  during February,  on M a r c h 1 7 t h . 62 d a y s l a t e r ,  t h a t a broad  i s recruited  The peak on May  cohort,  i n t o t h e t o p 150m  March and A p r i l and d e p a r t s  as stage V  c o p e p o d i d s i n May, J u n e a n d J u l y . Comparison of h i s r e s u l t s w i t h t h e e s t i m a t e s produced is  difficult  as n a u p l i i  recruitment estimates on  the importance  a r e n o t s a m p l e d a t O.S.P. a n d o u r  refer  t o e a r l y copepodids.  A l s o , depending  o f m o r t a l i t y , h i s peak t o peak t i m e s may n o t be  c o m p a r a b l e t o t h e mean r e c r u i t m e n t a n d r e s i d e n c e t i m e s here u s i n g a dynamic model. clear  here  estimated  I g n o r i n g the second problem, i t i s  that recruitment of n a u p l i i  i n the S t r a i t  of Georgia  p r e c e d e s r e c r u i t m e n t o f e a r l y c o p e p o d i d s a t O.S.P. w h i c h i s n o t surprising. nauplii increase Georgia  In fact,  would occur i n primary  i t m i g h t be e x p e c t e d  that recruitment of  l a t e r a t O.S.P. a s w e l l , a s t h e s p r i n g production occurs e a r l i e r  (Parsons,1965).  i n t h e S t r a i t of  The p e r i o d o f 62 d a y s b e t w e e n peak  167  n a u p l i a r and  peak CV a b u n d a n c e i n t h e S t r a i t  same a s t h e e s t i m a t e d s t a g e V a t O.S.P. slower  residence times  The  uncertainty arises  cohort  and  T  knowledge of m o r t a l i t y  from a c o m b i n a t i o n  i s u n f o r t u n a t e , as interest.  of t h e c o n f u s i o n  s i z e c l a s s e s and  have the e f f e c t  in larger  of d e c r e a s i n g  size classes.  broad  c o h o r t w i d t h , t h i s e f f e c t can  total  r e c r u i t m e n t and  the  consequently  the  a  of a  broad  r a t e i n the  t h e number of  However, because of be c o u n t e r a c t e d  X,  increasing  the  The  observation errors,  For example, i n c r e a s i n g the m o r t a l i t y  model would n o r m a l l y  d u r a t i o n and  V  over-wintering departure,  width.  i n view of  there.  r a t e s w o u l d be o f g r e a t  s m a l l number o f c o a r s e  individuals  surprising,  standing stock  l a r g e u n c e r t a i n t y i n R ,0  relatively  to  I f these are t e c h n i c a l l y comparable, growth i s  phytoplankton  m o r t a l i t y and  i s the  from e a r l y c o p e p o d i t e  a t O.S.P. w h i c h w o u l d h a r d l y be  much l o w e r  of G e o r g i a  the  by i n c r e a s i n g  which i n c r e a s e s the  relative  r e l a t i v e abundance, of l a r g e r  size  classes. N o t e t h a t i n 1973 for p o s i t i v e & while be p o s i t i v e . plane  and  i n the o t h e r y e a r s , 0  o r i e n t a t i o n and  o c c u r s amongst a l l y e a r s .  r e g i o n s can parameter 200  and  450  the confidence be o b t a i n e d Y.  For  ind/m  are estimated. 2.  t h e optimum p a r a m e t e r s e t  However, t h e c o n f i d e n c e  show s i m i l a r  i m p l i e d by  1977,  3  i n 1977  Higher  regions  i n the  T  low  fixing  v a l u e s of R , T  t o b e t w e e n 25 and  v a l u e s of R  t o 0.1  Much s m a l l e r  a n d e by  T  ( 0 ,V  )  so t h a t some o v e r l a p  A r a n g e f o r 0 of 0.0  for R  V = 1,  must be c o n s t r a i n e d t o  position,  regions.  occurs  day"  1  is  confidence  the growth  law  ranging  from between  60  3  ind/m  and 6 a r e o b t a i n e d  in for  1969, Y=  168  4.1.5 R e s u l t s f o r C.  cristatus.  P a r a m e t e r e s t i m a t e s c o u l d be o b t a i n e d 1969,1970,1974,1975,1976 and 1977. again omitted C. c r i s t a t u s fall  intrusion  series  The y e a r s  f o r l a c k of s i z e - s t r u c t u r e i s present  later  for this  species.  the e a r l y presence  As  t h a n C. p l u m c h r u s , a  i n 1973 i n t e r r u p t e d t h e t i m e  The t e c h n i q u e  of t r a n s i t i o n  1971 a n d 1972 were  information.  i n the year  of t r a n s i t i o n water  f o r the years  was a p p l i e d t o 1974 b u t  water c l e a r l y  distorts  parameter  est imates. I n g e n e r a l , t h e C. c r i s t a t u s behaved' as those uncertainty.  time  s e r i e s were n o t a s  f o r C. p l u m c h r u s a n d e s t i m a t e s  Estimates  o f u, <T  and T  show g r e a t e r  are given  T h e r e i s c l e a r l y much l e s s c o n s i s t e n c y f r o m y e a r r e c r u i t m e n t time 1977 due  for this  t o d a y 142 i n 1 9 7 0 . t o t r a n s i t i o n water  of r e c r u i t m e n t  i s similar  in residence times, with  i n mean  f r o m day 67 i n  recruitment  i n 1974 i s  t o t h a t f o r C. p l u m c h r u s , r a n g i n g  T  Great  ranging.from  variability  from  i s shown  83 a n d 89 d a y s i n 1976  1 9 6 9 , t o 213 d a y s i n 1 9 7 7 . No r e a l l y  c o n s i s t e n t p i c t u r e of the l i f e  C. c r i s t a t u s emerges f r o m t h i s a n a l y s i s . be  V.  i n t r u s i o n . ) D i s c o u n t i n g 1974, t h e s p r e a d  20 d a y s i n 1977 t o 42 d a y s i n 1 9 7 0 .  and  late  i n Table t o year  species, with u ranging (The v e r y  'well-  s a i d t h a t a broad  cohort  month p e r i o d a t some t i m e  some t i m e b e t w e e n J u n e a n d O c t o b e r . approximate confidence variability  intervals,  may be a s t a t i s t i c a l  regard to the v a r i a t i o n  E x c l u d i n g 1974, i t can  i s r e c r u i t e d over  between F e b r u a r y  h i s t o r y of  approximately  a 2  and June and l e a v e s a t  In view of t h e l a r g e  t h i s apparent artifact.  i n residence time,  annual  Especially  with  i t i s interesting  that  169  T a b l e V. Parameter estimates  f o r Calanus  Year  yt/(days)  <7(days)  r(days)  1969  89.±45.  24.±17.  89.± 6 1 .  1970  142.±25.  42.±13.  119.±100.  1974  188.±29.  18.± 9.  1975  100.±35.  34.±19.  1976  99.±22.  34.±  1977  67.±27.  cristatus.  T -3 R (ind.m )  9.5  e(day~ ) 1  0.008  0.5  97.  0.034  1.4  53.± 3 9 .  16.  0.005*  3.0  134.±171.  34.  0.005*  0.9  8.  83.± 4 1 .  118.  0.020  0.5  20.±15.  213.± 8 4 .  12.  0.027  1.7  * These v a l u e s were f r o z e n t o a l l o w convergence.  *  *  170  observations very  low  spatial 8mm)  o f C.  except  cristatus  may  as  late  stages  they  ( M a r l o w e and  P r o j e c t i o n s of t h e ) planes  T  these  95%  are given  t h e 150m  In  not  these  before  individuals. and by  a cohort eye.  The  i n F i g 45.  and  T  T  1969  simultaneous individuals resulted  resulting  1976,  time  1  in  those  between  y  and  has  a l s o been  0  hauls. (© ,Y  the  plumchrus,  in a l l years.  or G  )  and  high However,  i n 1970,1975 a n d . 1 9 7 7 . individuals  appeared  small cohort,  and  to freeze  t h i s w o u l d be e x p e c t e d  of  ^,  i n the  r a n g e 0.5  V(at  0.5)  O ( a t 0.005 d a y " ) i n 1974.  The  necessary  years  to freeze 1  intermediate  water  t o 2.0,  sized  intrusion of  #.  d a y " , e x c l u d i n g 1974. 1  where ( e,y  and  in  1974  Otherwise,  were o b t a i n e d .  r e g i o n s encompass a wide range of  i n s m a l l e r r a n g e s i n 1969  to  0.  following a transition  t o 0.07  results  (7-  s e r i e s showed a c l e a r e r l a g i n t h e  a p p e a r a n c e of s m a l l and  the c o n f i d e n c e day"  f o r C.  i n an a n o m a l o u s l y h i g h e s t i m a t e  estimates  to  large  a s p a r t of a s e p a r a t e  o b t a i n c o n v e r g e n c e , i t was and  be due  s t a r t i n g w i t h s m a l l s i z e c l a s s e s was d i s t i n g u i s h e d  reduce c o n f u s i o n  in  stages  c o i n c i d e n t w i t h or e a r l - i e r than  T h e s e were r e g a r d e d  often  (Nemoto,1957).  r e g i o n s on  As  a p p e a r a n c e of l a r g e r s i z e c l a s s e s and  To  T h i s may  c r i s t a t u s are  y e a r s , numbers of l a r g e ( c a 8mm) 100,  are  daylight vertical  highly correlated with R  day  fall  i t i s quite possible that  confidence  c o r r e l a t i o n s were f o u n d b e t w e e n R y was  late  M i l l e r , 1 9 7 5 ) and  a r e p o o r l y s a m p l e d by  (R ,0  of C.  show a c t i v e s w a r m i n g b e h a v i o u r  • D i u r n a l v e r t i c a l m i g r a t i o n by reported  summer and  for rare, high observations.  aggregation,  and  in late  Fixing 1976,  ©,  Again,  f r o m 0.005  t h e v a l u e of but  ) c o r r e l a t i o n s are  has  low.  low  little The  X effect  estimates  171  Figure 45.  Projection of approximate 95% confidence regions f o r Calanus c r i s t a t u s parameter X  estimates on (a) plane.  (6,R  ) plane  and (b)  172  Figure 45a.  173  174  of r e c r u i t m e n t for  i n 1970 a n d 1976 a r e a p p r o x i m a t e l y  1969 a n d 1 9 7 7 .  Estimates  low c o m p a r e d w i t h t h o s e  of recruitment  10 t i m e s  those  f o r C. c r i s t a t u s a r e  f o r C. p l u m c h r u s , e x c e p t  i n 1976.  4.1.7 O t h e r s p e c i e s . The  next  m a j o r c o n t r i b u t o r t o h e r b i v o r e b i o m a s s a t O.S.P. i s  a l s o a l a r g e copepod, Eucalanus bungi. species  i s not as c l e a r  from o b s e r v a t i o n s .  a p p e a r i n t h e 150m v e r t i c a l s i z e c l a s s e s i n March. through July.  The l i f e  hauls  h i s t o r y of t h i s  Individuals typically  i n intermediate  There i s a f a i r l y  t o t h e l a r g e s t ( 6 . 5 t o 7 mm)  clear  ( 2 . 5 t o 5 mm)  progression  s i z e c l a s s e s by J u n e o r  I n J u l y o r A u g u s t , l a r g e numbers o f s m a l l  ( 1 . 5 t o 2.5mm)  i n d i v i d u a l s a p p e a r , and t h e l a r g e i n d i v i d u a l s d i s a p p e a r . occurs  through  to intermediate  i n d i v i d u a l s disappear  are mostly  consistent with a l i f e  history  E u c a l a n u s w h i c h i n v o l v e s o v e r - w i n t e r i n g b e l o w 150m a s e a r l y  copepodite  stages,  r a t h e r slow  growth i n the euphotic  m a t u r e by J u n e o r J u l y a n d p r o d u c t i o n grows t o o v e r - w i n t e r i n g s i z e by l a t e complicated 2.5 mm)  o f a new g e n e r a t i o n fall.  individuals  in spring.  size,  been r e p o r t e d  In  which  This picture i s (1.5 t o  W h e t h e r some i n d i v i d u a l s  o r w h e t h e r some r e p r o d u c t i o n  b e l o w 150m i n t h e s p r i n g , i s n o t c l e a r .  Lewis,  zone t o  by t h e a p p e a r a n c e o f s m a l l numbers o f s m a l l  winter at t h i s  has  but these  by t h e e n d o f O c t o b e r .  These o b s e r v a t i o n s for  s i z e c l a s s e s (3 t o 5 mm)  Growth  takes  overplace  A spring breeding  f o r Eucalanus i n c o a s t a l waters  period  (Krause and  1979). any c a s e ,  estimation  i t was n o t p o s s i b l e t o a p p l y  techniques  t o Eucalanus data.  from e a r l y t o l a t e c o p e p o d i t e  stages  the parameter  The g r o w t h o f E u c a l a n u s  i s i n t e r r u p t e d by t h e o v e r -  175  wintering period.  Both halves  of the growth p e r i o d u s u a l l y  involve only  2 s i z e c l a s s e s , so t h e r e  information.  Another problem i s that  at a range of l e n g t h s , spring  recruitment  probably  i n d i v i d u a l s may  so t h a t o v e r - w i n t e r i n g  class over-winter  departure  and  may a f f e c t a number o f s i z e c l a s s e s .  of E u c a l a n u s a r e t y p i c a l l y contribution  i s a l a c k of s i z e  to standing  Numbers  l o w (5 - 15 i n d / m ) a n d i t s 3  s t o c k and secondary p r o d u c t i o n i s  s m a l l c o m p a r e d w i t h t h a t o f C. c r i s t a t u s a n d  C. p l u m c h r u s . (Oithona,  Other herbivorous  c o p e p o d s a t O.S.P. a r e o f s m a l l  Pseudocalanus) or intermediate  (Calanus  pacificus,  M e t r i d i a p a c i f i c a ) s i z e and r a r e l y c o n t r i b u t e s i g n i f i c a n t l y t o standing still  stock  by w e i g h t .  be o f i n t e r e s t ,  Their population  of course,  but data  not.adequate f o r parameter e s t i m a t i o n . that t h e i r  generation  two  so t h a t  size classes.  f o r these  species  were  Part of the problem i s  time i s s h o r t e r compared w i t h  i n t e r v a l s b u t t h e main r e a s o n too coarse,  parameters would  sampling  i s t h a t t h e s i z e c l a s s e s used a r e  i n d i v i d u a l s are often assigned  o n l y t o one o f  I n t h e case of O i t h o n a and P s e u d o c a l a n u s , ,  o n l y t h e l a r g e s t i n d i v i d u a l s a r e r e t a i n e d by t h e 350 um mesh n e t . 4.1.8  Secondary product ion Estimates  of secondary p r o d u c t i o n  above t e c h n i q u e , The  estimates.  once a w e i g h t - l e n g t h  weight or carbon content  a v e r a g e wet w e i g h t o f 4 mg was r e p o r t e d V.  to l , t h i s observation 3  from  and l i t e r a t u r e v a l u e s  used t o d e f i n e a w e i g h t - l e n g t h  mm C. p l u m c h r u s s t a g e  using the  relationship i s prescribed.  of i n d i v i d u a l copepods  O.S.P. h a s n o t been m e a s u r e d d i r e c t l y initially  c a n be o b t a i n e d  relationship.  were  An  by F u l t o n ( 1 9 7 3 ) f o r 4.5  I f i t i s assumed t h a t W i s p r o p o r t i o n a l corresponds t o W = .044.1 . 3  Fulton  176  (undated)  g i v e s a g e n e r a l w e i g h t - l e n g t h r e l a t i o n s h i p f o r copepod  populations  i n the S t r a i t  a wet w e i g h t Ishii and  of Georgia  : W = .068. 1 " 2  o f 2.7 mg f o r a 4.5 mm C. p l u m c h r u s .  yields  Taguchi and  ( 1 9 7 0 ) r e p o r t wet w e i g h t s r a n g i n g f r o m 2.21 t o 4.55 m g / i n d  l e n g t h s o f 5 a n d 4.5 mm r e s p e c t i v e l y  V, r e s u l t i n g Use data  which  4 5  f o r C. p l u m c h r u s  stage  i n c o e f f i c i e n t s o f .02 t o 0.05 mg.mnr . 3  o f any o f t h e s e f o r m u l a e w i t h O.S.P. s i z e - s t r u c t u r e d  resulted  An a t t e m p t  i n s e v e r e o v e r - e s t i m a t i o n o f s a m p l e wet w e i g h t s .  was made t o d e r i v e a l e n g t h - w e i g h t  relationship  a p p r o p r i a t e t o O.S.P. f r o m t h e s i z e - s t r u c t u r e d a t a a s f o l l o w s . For each  sample,  a l l t h e c o p e p o d d a t a was r e v i e w e d , a n d t h e  number o f i n d i v i d u a l s c o u n t e d 9 1mm  i n each  s i z e c l a s s a d d e d t o one o f  'super' s i z e c l a s s e s a c c o r d i n g t o i t s nominal  multiple in each  linear  r e g r e s s i o n o f s a m p l e wet w e i g h t s  size.  A  on numbers N?  o f t h e s e s u p e r - s i z e c l a s s e s was t h e n p e r f o r m e d  : that i s ,  t h e c o e f f i c i e n t s Wj i n  w  =  IN,  .WJ  were e s t i m a t e d .  An o b v i o u s c r i t i c i s m  weights of organisms  i s t h a t W*  i n c l u d e s wet  o t h e r than copepods : w h i l e samples h a v i n g  i n t e r m e d i a t e t o h i g h wet w e i g h t s a r e p r e d o m i n a n t l y ( 9 0 % ) c o p e p o d s , t h e c o e f f i c i e n t s Wj may t e n d t o be s l i g h t l y  high.  These c o e f f i c i e n t s a r e i n t e r p r e t e d as average  f o r each  weights  1mm s i z e c l a s s a n d a r e p l o t t e d a g a i n s t l e n g t h on a l o g - l o g i n F i g 46.  A p a r t from anomalous r e s u l t s  size classes,  the c o e f f i c i e n t s  fall  f r o m two,, a l m o s t  scale empty,  close to a l i n e having the  s l o p e 2.45 s u g g e s t e d by F u l t o n ( u n d a t e d )  a n d an i n t e r c e p t  177  -4  -5  -6  +  t  -7  t  -8  -9  -10  0  Figure 46.  0 5  10  15  20  ln(li)  Regression c o e f f i c i e n t s w\  2 5  (±1 standard error)  vs corresponding lengths 1^ on log-log scale. Line corresponds 6 mm  to W = 0.033.1  and 7 icm s i z e classes.  2.45  , xgnores  178  corresponding the value coarse  t o W = . 033 . 1  2 45  found  nature  thinner While  Given the  t o any b i a s i n a s s i g n i n g  i t may be r a s h t o c o n c l u d e  t h a t O.S.P. c o p e p o d s a r e  their  the lower  (Parsons  c o a s t a l c o u s i n s on t h e b a s i s o f t h i s  food c o n c e n t r a t i o n s and temperatures be e x p e c t e d  to result  e t a l ,1977), i t i s not c l e a r  f o r a g i v e n body l e n g t h s h o u l d  statistical secondary  of Georgia.  of t h i s c o e f f i c i e n t  than  weights  0.033 i s h a l f  o f t h e s i z e c l a s s e s u s e d f o r t h e O.S.P. d a t a a n d  O.S.P. m i g h t r e a s o n a b l y rates  The c o e f f i c i e n t  by F u l t o n f o r t h e S t r a i t  the s e n s i t i v i t y lengths,  .  relationship  result.  at  i n lower  growth  t h a t lower  body  result.  I n any c a s e , t h e  i s used here i n t h e e s t i m a t i o n of  p r o d u c t i o n b a s e d on O.S.P. d a t a .  Estimates  of secondary  estimates are given estimates  p r o d u c t i o n and 95% C.I.  i n Table V I .  The u n c e r t a i n t y i n t h e s e  i s c l e a r l y much l e s s t h a n  recruitment parameters,  as found  u n c e r t a i n t y i n m o r t a l i t y and  f o r simulated data  confidence  by S o n n t a g  and  Parslow(1980).  and  1976 f o r C. p l u m c h r u s , b u t t h e l i m i t s a r e s m a l l enough i n  other years  Large  f o r these  l i m i t s are obtained  t o c l e a r l y d i s t i n g u i s h a year  a s 1969 f r o m y e a r s  of h i g h p r o d u c t i o n such  that, while highest recruitment estimates obtained reduced  o f low p r o d u c t i o n a s 1970 o r 1 9 7 7 .  secondary  such Note  f o r C. p l u m c h r u s were  f o r 1973 a n d 1 9 7 7 , h i g h e r m o r t a l i t y estimated  f o r 1975  r a t e s i n these  years  p r o d u c t i o n t o 4 t h a n d 2nd r a n k  respectively. Estimates  of secondary  production  f o r C. c r i s t a t u s showed a  similar  range t o those  pattern  i n t h e a b s o l u t e o r r e l a t i v e c o n t r i b u t i o n s o f t h e two  species to t o t a l  f o r C. p l u m c h r u s .  secondary  production.  T h e r e i s no d i s c e r n i b l e  Both  a r e u n u s a l l y low i n  Table  VI.  Secondary p r o d u c t i o n e s t i m a t e s  Year  C_. plumchrus  (g wet wt.m  £. c r i s t a t u s  21+9  -2  )  Total.  1969  26±10  1970  119±45  1973  57±17  —  1974  --  51±18  1975  35±35  20+11  55±37  1976  69±54  146±61  215±81  1977  114±22  15+3  129±22  56+25  47±13 165+51 --  180  1969  a n d 1975 a n d b o t h a r e h i g h i n 1970 a n d 1 9 7 6 .  I n 1977,  C. p l u m c h r u s i s v e r y h i g h a n d C. c r i s t a t u s v e r y l o w . While  smaller species a r e not i n c l u d e d , these  a r e g e n e r a l l y assumed t o d o m i n a t e s e c o n d a r y interesting  to treat  estimate of t o t a l  available.  p r o d u c t i o n and i t i s  t h e i r combined t o t a l s as a c o n s e r v a t i v e  secondary  (>4 f o l d ) v a r i a t i o n  over  production.  the 5 years  There i s c o n s i d e r a b l e  f o r which t o t a l s a r e  I f these estimates a r e converted  v a l u e o f .05 f o r t h e c a r b o n : w e t w e i g h t ,1977), e s t i m a t e s obtained.  ranging  For comparison,  t o carbon  ratio,  using a  (Parsons  f r o m 2.4 t o 10.7 g C . m ^ . y r "  et a l 1  are  trophodynamic c o n s i d e r a t i o n s l e d  M c A l l i s t e r ( 1 9 6 9 ) t o a 'most l i k e l y '  e s t i m a t e o f 13 g C . m - . y r ,  w i t h a minimum v a l u e o f 2 g C . i r r . y r 2  respiration  l a r g e copepods  2  _ 1  ,  _ 1  d e p e n d i n g on assumed  rates.  4.2 B i o m a s s M o d e l f o r Z o o p l a n k t o n . 4.2.1 I n t r o d u c t i o n . In Chapter zooplankton involving  1, s i m p l e b i o m a s s m o d e l s o f t h e p h y t o p l a n k t o n -  interaction  a t O.S.P. were c o n s i d e r e d .  g r a z i n g t h r e s h o l d s a p p e a r e d t o be c a p a b l e  t h e s e a s o n a l c y c l e a t O.S.P.. i n a q u a l i t a t i v e results  of the data a n a l y s i s of Chapters  A model of mimicking  sense.  Using the  3 a n d 4, an a t t e m p t  will  be made t o a s s i g n v a l u e s t o t h e p a r a m e t e r s i n a m o d e l o f t h i s k i n d , a n d compare p r e d i c t i o n s a n d o b s e r v a t i o n s The  quantitatively.  q u a n t i t a t i v e models c o n s i d e r e d here a r e n u m e r i c a l  b a s e d on t h e p h y t o p l a n k t o n to a l l o w a r e a l i s t i c  p r o d u c t i o n model of Chapter  treatment  vertical distribution.  of phytoplankton  models, 3, s o a s  c o m p o s i t i o n and  The l i m i t a t i o n s o f c o m p u t e r s i m u l a t i o n  181  and  sensitivity  analysis  when  discussed  i n Chapter  possible,  the approximate  Chapter the  2 have  models  4.2.2  been  wherever  Formulation In  2.  To overcome  used  to direct  model  Chapter  Phytoplankton  m(day" ) 1  metabolism  G  = e.F  The  single  1 and  simulation  G  zooplankton are  (mg C m " ) ,  i n g e s t e d per day, F  biomass  with  regarded  efficiency  T  e.  , is A  as a combination  in unspecified  as i n  2  proportion.  constant  of b a s a l Then  - m.G  T  rate  of  Model.  variable,  carbon  c a n be  and m o r t a l i t y  4.1  phytoplankton  growth  as f a r as  of Chapter  and i n t e r p r e t  Grazing  biomass  to zooplankton  rate,  results  considered, herbivorous  by a  loss  limitations  possible.  represented  converted  a r e u n c e r t a i n were  these  analytical  of a Biomass  the f i r s t  1.  parameters  carbon  calculated  i n each  as  5m  layer  i n Chapter  changes  according  3 and the c a l c u l a t e d  to a  grazing  loss. The  grazing loss  response  of Chapter  concentration only  must  depth  i s based  1 but the v a r i a t i o n depth  must  the d i s t r i b u t i o n  now  of  be t a k e n  of zooplankton  upon  the  functional  phytoplankton into  account.  represents a  relative  feeding If extreme  must  time  fast-variable  scales  approximation  of zooplankton  Not  grazing activity  be c o n s i d e r e d , b u t , a s t h e n o n - l i n e a r f u n c t i o n a l  itself the  with  i n t h e model  movement  with  response  (Holling,1959), and s a t i a t i o n  of  be c o n s i d e r e d .  the distribution assumptions  which  of zooplankton could  be u s e d  i s denoted in this  by q ( z ) , two  model a r e :  182  ( i ) z o o p l a n k t o n a r e on a v e r a g e throughout (ii)  t h e t o p 150m  uniformly distributed  so q ( z ) = 1/150.  z o o p l a n k t o n a r e on a v e r a g e  distributed  i n proportion to  -"SO  phytoplankton carbon The  average  so q ( z ) = C ( z ) / / Jo  d i s t r i b u t i o n q ( z ) g i v e s no  v e r t i c a l movement o f i n d i v i d u a l q(z), of  two  extreme assumptions  v e r t i c a l movement and (a) Z o o p l a n k t o n  faster  the water  response  i s g i v e n by  _  f ( C ) , where C =  —* f(C)  The  = /  r'  Jo  c a p a b l e of extended  set  /  r'  so  C(z).q(z)dz.  amount i n g e s t e d i n t h e  copepods,  v e r t i c a l movement. low,  While  As  the  i t seems u n l i k e l y  I n f a c t , as a g r a z i n g t h r e s h o l d v a l u e CO i n t h e m o d e l , i t seems u n l i k e l y  proportional  water  f(C(z)).q(z)dz.  f e e d a t t h e v e r y low c o n c e n t r a t i o n s b e l o w t h e  time at depths  response  so  phytoplankton c o n c e n t r a t i o n i s always  considered  so f a r a s  I f the f u n c t i o n a l  d o m i n a n t g r a z e r s a t O.S.P. a r e l a r g e  they would  the  v e r t i c a l movement o c c u r s on t i m e s c a l e s much  c o l u m n i s t h e n g i v e n by  spend  :  f ( C ) , t h e amount i n g e s t e d i n  _  s l o w e r t h a n t h o s e of s a t i a t i o n .  presumably  scales  Each z o o p l a n k t e r then sees  i s concerned.  c o l u m n i s g i v e n by  (b) Z o o p l a n k t o n  layer.  given  c o n c e r n i n g the r e l a t i v e time  s a t i a t i o n are p o s s i b l e  the  v e r t i c a l movements o c c u r on t i m e s c a l e s much  constant food l e v e l s  The  F o r any  p h y t o p l a n k t o n c o n c e n t r a t i o n ( w e i g h t e d by q ( z ) ) ,  the f u n c t i o n a l at  i n f o r m a t i o n about  zooplankters.  t h a n t h o s e of s a t i a t i o n .  average  C(z')dz'.  where C ( z ) < CO.  will  mixed  be  t h a t copepods  would  T h e r e f o r e , q ( z ) has  been  to (C(z)-C0) . +  the dominant h e r b i v o r o u s copepods are  c a p a b l e o f v e r t i c a l movement, o n l y t h e l a t e s t a g e c r i s t a t u s copepodids  were o b s e r v e d  that  presumably Calanus  t o undergo e x t e n s i v e d a i l y  183  v e r t i c a l m i g r a t i o n (Marlowe  and M i l l e r , 1 9 7 5 ) .  For  individuals  u n d e r g o i n g d i u r n a l v e r t i c a l m i g r a t i o n , the extreme would (eg  assumption  be a p p r o p r i a t e i f s a t u r a t i o n o c c u r r e d on a l o n g t i m e  gut-emptying  time).  For organisms  w h i c h do n o t  d i u r n a l m i g r a t i o n , o r become s a t u r a t e d on s h o r t e r assumption  (b) i s more l i k e l y  (a)  scale  undergo  time  t o be a p p r o p r i a t e and  scales,  i t has  been  a d o p t e d h e r e , so t h a t t h e g r a z i n g l o s s r a t e a t d e p t h z i s g i v e n by  f(C(z)).q(z).G  4.2.3  C h o i c e of Z o o p l a n k t o n The  biomass  parameters  w h i c h must be s p e c i f i e d  in this  model of z o o p l a n k t o n a r e the f u n c t i o n a l  parameters  i , M  D and CO,  unrealistic  rate parameter,  to expect to s p e c i f y  number o f r e a s o n s . variable  The  simple  response  t h e g r o w t h e f f i c i e n c y e and t h e c o m b i n e d  r e s p i r a t i o n and m o r t a l i t y  biomass  Parameters.  m.  I t would  these parameters  be  precisely  r e p r e s e n t a t i o n o f h e r b i v o r e s by a  is itself  for a  single  a crude a p p r o x i m a t i o n : the g r a z e r s at  O.S.P. a r e composed o f i n d i v i d u a l s c o v e r i n g a r a n g e o f s p e c i e s and  s i z e s , and  accordingly  The  parameter  i n t h e model must i n some s e n s e be r e g a r d e d  as  averages.  These parameters range  c a n be e x p e c t e d t o v a r y  ( S t e e l e and F r o s t , 1 9 7 7 ; F r o s t , 1 9 7 9 ) .  values inserted approximate  these parameters  h a v e y e t t o be m e a s u r e d a t O.S.P. f o r a  of s i z e c l a s s e s of t h e dominant h e r b i v o r e s .  m e a s u r e d f o r t h e same s p e c i e s i n c o a s t a l ,1969; T a g u c h i and  Ishii,1972;  locations  Ikeda,1972;  Some have been ( P a r s o n s e t a_l  F r o s t , 1 9 7 9 ) but  e x t r a p o l a t i o n of t h e s e r e s u l t s t o the v e r y d i f f e r e n t c o n d i t i o n s a t O.S.P. i s n o t s t r a i g h t f o r w a r d  the  food  (Buckingham,1978).  V a l u e s o b t a i n e d f o r o t h e r s p e c i e s of g r a z e r s under  low  food  184  c o n d i t i o n s may by o t h e r  p r o v e more a p p r o p r i a t e .  authors  controversy  attached  p a r a m e t e r s and given  (Steele,1974;  the  h e r e i s not  an  t o g i v e an  reported  i n the  results.  of t h e  by any  .  A r a n g e o f 10  i s given  of a b o u t 18%  ( P a f f e n h o f f e r and  i n g e s t i o n and  by  - 40%  the  by  of  Paffenhoffer  for  experiments  elongatus  i m p l i e d by  4.1  varies with  experiments,  Ikeda  ( s t a g e V)  feeding  (with approximately  1/100  respiration  r a t e s of  e t a l ,1977; S t e e l e , 1 9 7 4 ) . )  reported  respiration  r a t e s of  T a g u c h i and  5%/day f o r C.  1-  to 3  t h e body w e i g h t (The  r a t e i n copepods i s g e n e r a l l y  (Parsons  one  Respiration rates  p l u m c h r u s ) were much h i g h e r , a b o u t 9-18%/day.  dependence of  r a t e and  w i t h r a t e s up  individuals.  to  I n a s e r i e s of  (1977) found r e s p i r a t i o n  plumchrus  for actively  as  (1974) found t h a t the p r o b l e m s i n  i m p o r t a n c e of a b a s a l m e t a b o l i c  starved Paracalanus  of C.  (1976) f o r  similar  for Pseudocalanus  Maximum  r e s p i r a t i o n e x p e r i m e n t s make i t d i f f i c u l t  relative  higher  gross  t h e p r o p o r t i o n of m w h i c h i s r e g a r d e d  Steele  2%/day f o r s t a r v e d C. times  values  the  (1977).  which i s p r o p o r t i o n a l to i n g e s t i o n i n copepods. careful  is  f o r t h i s parameter i n  P a r s o n s et. al  growth e f f i c i e n c y  basal metabolism.  assess  means, but  Harris,1976).  d e p e n d s on  interpretation  discussion  i s known as  p a c i f i c u s in laboratory c u l t u r e s , while  gross  these  r a n g e of p a r a m e t e r  ingested  o f a b u t 30% were r e p o r t e d  The  still  literature.  r a t i o of growth t o food  yielded values  The  discussed  is  for measuring  review  indication  marine zooplankton  C.  techniques  exhaustive  growth e f f i c i e n c y , K  values  Buckingham,1978) t h e r e  i n t e r p r e t a t i o n of  intended  The  to the  I n a d d i t i o n , as  Ishii  size accepted  (1972)  c r i s t a t u s and  9%/day  185  for  C. p l u m c h r u s u n d e r c o n d i t i o n s o f a c t i v e  feeding.  Given the  l a r g e s i z e o f t h e d o m i n a n t h e r b i v o r e s a t O.S.P., a l o w b a s a l metabolic  rate  i n t h e range  e f f e c t i v e g r o s s growth  1-5%/day seems r e a s o n a b l e .  efficiency  will  amount d e p e n d e n t on t h e r a t i o o f r a t i o n the low b a s a l m e t a b o l i s m will  result  suggested  i n g r o s s growth  The  be l o w e r t h a n e by an t o body w e i g h t .  Given  a b o v e , a v a l u e f o r e o f .5  e f f i c i e n c e s o f .4 o r l e s s a t t h e l o w  f o o d c o n d i t i o n s f o u n d a t O.S.P. The  parameter  body w e i g h t .  i  M  r e p r e s e n t s maximum r a t i o n a s a f r a c t i o n o f  Ranges of 40-60%/day f o r s m a l l e r copepods and 10-  20%/day f o r l a r g e r c o p e p o d s were f o u n d by P a r s o n s V a l u e s o f 2 0 - 6 0 % / d a y were f o u n d by P a r s o n s  et. a l ( 1 9 6 7 ) .  e t a_l (1969) f o r  C. p l u m c h r u s f e e d i n g on n a t u r a l p h y t o p l a n k t o n a s s e m b l a g e s i n . t h e Strait  of Georgia.  P a f f e n h o f f e r and H a r r i s  high r a t e s f o r Pseudocalanus and  in culture  (1976) found  ( c a 1 5 0 % body  very  weight/day)  P a f f e n h o f f e r (1970) r e p o r t e d v a l u e s f o r C. p a c i f i c u s o f c a  100%/day f o r s t a g e V c o p e p o d i d s Apparent  maximum i n g e s t i o n  r e p o r t e d by T a g u c h i  a n d 300%/day f o r s t a g e V n a u p l i i .  r a t e s f o r s t a g e V C. c r i s t a t u s  and I s h i i  ( 1 9 7 2 ) were v e r y l o w , c a 5%/day.  A g a i n t h e v a l u e c h o s e n must r e p r e s e n t an a v e r a g e classes. copepodite  A value f o r i  M  across  size  o f .5 o r l e s s may be r e a s o n a b l e  for late  s t a g e s o f t h e l a r g e c o p e p o d s , b u t a v a l u e o f 1.0 may  be n e e d e d i f i n g e s t i o n by n a u p l i i  and e a r l y copepodids i s  included. Thresholds from t h e S t r a i t range  from  i n zooplankton  f e e d i n g on n a t u r a l  o f G e o r g i a were f o u n d by P a r s o n s  50-190 ug C . l "  constant) i n a similar  1  w i t h values of D  range.  assemblages e t a l (1967) t o  (half-saturation  F r o s t (1972) f i t t e d h i s  186  observations and  on C. p a c i f i c u s w i t h a t y p e  h a l f - m a x i m u m r a t i o n was o b t a i n e d  ranging  f r o m 50 t o 150 jug C . l " , 1  In a l a t e r  study,  Frost  corresponding  b a s e d on r e l a t i v e l y  a t food  concentrations  d e p e n d i n g , on f o o d p a r t i c l e  (1975) found t h a t t h e c l e a r a n c e  reduced a t food c o n c e n t r a t i o n s ration,  I f u n c t i o n a l response  yielding  t o 15-45 ug C . l " . 1  short-term  size.  r a t e was  l e s s t h a n 1 5 % o f maximum A l l these  results  i n c u b a t i o n s of i n d i v u a l s  were  captured  from t h e f i e l d . In a s e r i e s of l o n g - t e r m thresholds  culturing  down t o a b o u t 25 ug C . l "  (Paffenhoffer,1970) Harris,1976).  1  experiments,  was f o u n d f o r C. pac i f i c u s  or Pseudocalanus elongatus  Their  results  ( P a f f e n h o f f e r and  suggested a h a l f - s a t u r a t i o n constant  f o r P s e u d o c a l a n u s i n g e s t i o n o f 25 t o 50 p g C . l " .  The  1  conclusions concerning  of Buckingham long-term  (1978) a n d M a y z a u d a n d P o u l e t ( 1 9 7 8 )  adaptation  the d i f f e r e n c e s between t h e s e incubations  r e p o r t e d above.  concentrations  t o f o o d c o n c e n t r a t i o n may e x p l a i n  r e s u l t s and t h e s h o r t - t e r m Adaptation  t o low food  may a l s o e x p l a i n t h e h i g h c l e a r a n c e  e x a m p l e , o v e r 200 m l / d a y f o r a 10 pg d r y w e i g h t r e p o r t e d by P a f f e n h o f f e r  no h i n t o f  rates (for  Pseudocalanus)  (1970) and P a f f e n h o f f e r and H a r r i s  (1976).  Clearance  constant  D a n d t h r e s h o l d CO, a r e w e l l - k n o w n t o d e p e n d on t h e s i z e  and  value  r a t e s , or e q u i v a l e n t l y the h a l f - s a t u r a t i o n  of food p a r t i c l e s  S t e e l e and F r o s t , 1 9 7 7 ) . phytoplankton  e_t a l ,1967;  Frost,1975;  Given t h e s m a l l s i z e of t h e  c e l l s and t h e l a r g e s i z e of t h e dominant  a t O.S.P., l o w c l e a r a n c e be  (Parsons  expected there.  C. p l u m c h r u s f e e d s  rates or high values  However, F r o s t efficiently  o f CO a n d D m i g h t  (1978) h a s r e p o r t e d  on s m a l l c e l l s ,  herbivores  that  due t o an u n u s a l l y  187  small that  inter-setule these  levels  zooplankton  observed  constant,  spacing  f o r a copepod  do grow  a t O.S.P.  and c e r t a i n l y  o f i-ts s i z e .  and reproduce  suggests  grazing  that  The  a t t h e low  fact  food  the h a l f - s a t u r a t i o n  thresholds,  must  be  relatively  low. Estimates analysis Various  problems  4.2.4  first  parameters values  to  m  M  Results  = 1.0  cycle.  the amplitude  in  simulation  The c y l e  47;  as d i s c u s s e d  results  observations Chapter  small  bloom  zooplankton  1.  stock  focused  this  first  t o 0.05  growth  varied  i s typical  follows  settles  i n May.  down  at ca  layer  from  .5 mg  clear  of the  and in Fig  year  to  that the  consistent either  occurs  C h i a/m with  similar  with model  constant,  in April  June,  varies approximately  1964  of the q u a l i t a t i v e  remaining  After  e =  1  on t h e f o r m  a n a l y s i s of a  duration  C.l" ,  the period  somewhat  of phytoplankton  parameter  i n 1976 i s g i v e n  I t i s immediately  20 d a y s  = 20 pq  run over  i n t h e t o p 150m of peaks  CO  i n the mixed  or the approximate  then  was  are not q u a l i t a t i v e l y  Instead  peak  be  of C h i a  later,  of about  concentration standing  stock  in a l l years.  simulation  0.0  on t h e p h y t o p l a n k t o n  1  This  standing  variation  i n these  i n the range  D = 40 u g C . l " ,  1  herbivore  the  i s based  1976, but a t t e n t i o n w i l l  year,  4.1.  and D i s c u s s i o n .  day" ,  1  while  uncertainty  by  in Section  i n F i g 33 a n d t h e f o l l o w i n g z o o p l a n k t o n  .05 d a y " .  seasonal  obtained  abundances  low e s t i m a t e s ,  simulation  used  : i =  i n a high  were  obtained.  Simulation The  .5,  resulted  but r e l a t i v e l y  were  1  a t O.S.P.  of t i m e - s e r i e s of s i z e - c l a s s  estimates, day" ,  of m o r t a l i t y rate  and a  a sharp  the c h l o r o p h y l l 3  and  primary  zooplankton production.  Figure 47.  Predicted mixed layer Chi a and herbivore biomass f o r 1976 using standard parameter set (see t e x t ) .  189  It  may  seem  phytoplankton  possible  production  and  v a r y i n g carbon  the  approximate  the  case  and,  approximated 1.  This  by  so  that  of  below  the  G  = M  =  this  given  as  by  account,  irrelevant.  that  noted  off quickly  mixed  feeding  z (t)  layer  takes  i n the  mixed  mixed  as  in Chapter  the  3,  by  layer  in  Chapter  a  mixed  layer, mixed  single  carbon the  rate  by in  ignoring be  G (t) M  the events  replaced  the  by  4.2a 4.2b  from  small  changes  on  the  relatively  i n mixed  grounds  layer  that  depth  changes  have  i n mixed  been layer  slowly).  stability  : provided system  low  i n the  depth,  growth  can  very  -is g i v e n  on  not  closely  the  layer  s i m u l a t i o n model  is  3,  place  then,  rendered  i s always  below  represented  i s the  M  be  in Chapter  falls  be  This  considered  the  can  can  has  M  approximate  model  slowly,  into  M  occur The  as  as  the  - G .f(C)  resulting  neglected depth  layer,  (e.f(C)-m).G  (Terms  1  of  depth-distribution  r ( t ) is....the p h y t o p l a n k t o n  calculated  mixed  r(t).C  If  takes  below  zooplankton  If  M  layer  because,  C(t).  = G(t)/z (t). mixed  simple  phytoplankton  concentration  Chapter  a l l zooplankton  concentration,  which  complexity  s i m u l a t i o n model  as  carbon  added  composition  the  production  almost where  a of  model  phytoplankton  layer,  C  a  is possible  phytoplankton and  : Chi  fact,  the  model,  analysis in  that  seasonal  state  should  analysis changes track  of  Chapter  i n r ( t ) and the  1 applies z (t) M  to  occur  quasi-equilibrium cycle  190  f(C) G  = m/e  =  constant  = r(t).C.z (t).e/m M  The  phytoplankton  zooplankton the  water  this  The  fact,  system  production  this  production  primary  in a  CO  down  during  t o have  resulting  in  early  from  F i g 47 i s  the spring. occurred in a  which  i s returned  D and  i n the spring  i n a much  biomass  Chi a  peak'  observed  has a very  problem  here  lies  not with  sharp  the a b i l i t y  the quasi-equilibrium cycle after  has begun,  bloom.  smaller  the average still  this  t o t h e same  the resulting  not c o n s i s t e n t with  to  of the  = 20 c o r r e s p o n d s  However,  too  transient  by d e c r e a s i n g  result  The  peak.  the basic  net primary  approximation  of approach  reduction  = 22, D  ( F i g4 8 ) .  is still  to track  period,  conclusion  increased  D=40 a n d d o e s  i n 1976  unrealistic  negative  with  The r a t e  ( F i g 14) and t h e z o o p l a n k t o n  In the  to result  combination  March-April  and  while  the slow-variable  appears  of the year.  o f C a s C0=20, bloom  on  t o respond,  t o e q u i l i b r i u m c a n be  spring  be c o n s t a n t ,  the q u a s i - e q u i l i b r i u m c y c l e , half  parameter  cycle  in production  be e x p e c t e d  value  in  from  the second  might  vary  has broken  f o r zooplankton  departure  system  depends  approximation  increase  quickly  should  and the s i m p l e s t  M  spring  stock  'prediction'  r ( t ) and z ( t )  that  in  standing  should  column.  This for  concentration  but with production  the very  low a n d o f t e n  i n the preceding  the q u a s i - e q u i l i b r i u m disappears,  standing  stock  falls  below  the threshold  level  standing  stock  d e c l i n e s e x p o n e n t i a l l y to very  the spring slightly  winter.  During  phytoplankton and  of  zooplankton  low v a l u e s  by  Figure 48.  E f f e c t of decreasing D on F i g 47.  192  March. lies  When p o s i t i v e p r i m a r y p r o d u c t i o n  f a r from t h e q u a s i - e q u i l i b r i u m c y c l e and a t r a n s i e n t  p h y t o p l a n k t o n bloom i s i n e v i t a b l e . conditions the  i n the f o l l o w i n g year are determined  entirely  t h e model's parameters and p h y s i c a l d r i v i n g v a r i a b l e s . i n long-term simulations,  a f f e c t e d by t h e c h o i c e consider only  the seasonal  of i n i t i a l cycle  only  the f i r s t  conditions.  In  year i s  Similarly, to  i n 1976, i t i s s u f f i c i e n t  to simulate  1975 a n d 1 9 7 6 . The  stock  problem a r i s i n g  i n winter  strategy  immediately brings  of the major s p e c i e s  they, or t h e i r  spring.  t o mind t h e l i f e  overwinter  offspring, return  standing  history i n Chapter  b e l o w 150m a n d  t o the mixed l a y e r i n t h e  l o s s e s through m o r t a l i t y and r e s p i r a t i o n a r e  I t i s p r e c i s e l y these l o s s e s which a r e r e s p o n s i b l e f o r  s p r i n g bloom i n t h i s  simple  model and t h e r e c r u i t m e n t  z o o p l a n k t o n biomass t o t h e s u r f a c e prevent t h i s  layer  i n the spring  of  should  bloom.  R e c r u i t m e n t h a s been i n t r o d u c e d biomass i n p u t , normally standard  i n zooplankton  The o b v i o u s a d v a n t a g e o f t h i s b e h a v i o u r t o t h e c o p e p o d s  that winter  reduced.  from t h e d e c l i n e  o f t h e d o m i n a n t c o p e p o d s a t O.S.P., d i s c u s s e d  A l l three  either  the  t o t h e model's b e h a v i o u r i n  i n t h e s e c o n d h a l f o f e a c h y e a r means t h a t  conditions  particular,  is  the i n i t i a l  s p r i n g , t h e c l o s e approach of the system t o t h e q u a s i -  initial  1.  Note t h a t w h i l e  i n each year a r e c r i t i c a l  equilibrium cycle  by  i s resumed, t h e system  deviation  parameter e s t i m a t i o n  i n t o t h e model as a d a i l y  d i s t r i b u t e d o v e r t i m e w i t h mean u ,  cr a n d t o t a l  integrated recruitment,  G . R  The  r e s u l t s o f s e c t i o n 4.1 c o r r e s p o n d t o cr  b e t w e e n 20 a n d 40 d a y s , JJ b e t w e e n 70 a n d 110 d a y s a n d G  R  between  193  40 a n d 400 mg wintering three  R  i s represented  from  The  The d e p a r t u r e  2  dominant  extends  G  Cm' .  June  to  Cm"  and m  2  t h e p r e d i c t e d time  in  F i g 49.  phytoplankton present.  compared as  G(t)  =  above.  1  t o 300,  has e s s e n t i a l l y  ju=90, resulted  biomass  given  disappeared,  i n C h i a due t o s m a l l i n carbon:Chl  changes i n a  ratio are i n May h a s  but the increase  i n zooplankton  biomass  i s early  o b s e r v a t i o n s a t O.S.P. i n June  Biomass  during  this  declines  and s t a b i l i z e s  the departure period  markedly  a t about  period.  The q u a s i -  i s given  by  half  cyle  and spread  As G  R  p r e d i c t e d without  was  found  although  zooplankton  even  a  i n a marked peaks  mg  insensitive  t o changes i n  the ranges  bloom  low r e c r u i t m e n t reduction  over-wintering.  within  the spring  starts  level,  ( F i g 50)  G  given  to R  reappear  = 25  mg  i n the chlorophyll  o f F i g 47.  predicted phytoplankton  a n d 0.6  t o be  of recruitment  i s decreased,  results  The  w  of the v a l u e  seasonal  gradually,  0.3  (7=30,  with  biomass  biomass  half  timing  2  together  i n zooplankton  M  The  Cm" ,  period  r(t).C.z (t).e/(m+m )  just  the  Ifa l l  w  the departure  d a y " ' f o r d a y 170  and v a r i a t i o n s  i n F i g 47 o v e r  level  m.  peak  with  equilibrium  and  fluctuations carbon  rate,  of C h i a and h e r b i v o r e  bloom  o v e r - w i n t e r i n g commences  the  or  series  The p r o n o u n c e d  disappeared,  of F i g 47,  = .05  w  The s p r i n g  small  loss  f o r over-  October.  values  in  although  by an a d d i t i o n a l  species are considered,  parameter  =200 mg  of i n d i v i d u a l s  C h i a.m'  standing  i n the mixed  3  equilibrium  phytoplankton  zooplankton  parameters  :  stock  layer.  varies  between  The q u a s i -  concentration i s determined  by  1 .5  J F  M fl M . J  J  fl  S  0  N D  TH Figure  49.  E f f e c t of i n t r o d u c i n g strategy  in Fig  47.  over-wintering  1 .5 cn  >K  Figure 50.  E f f e c t of reducing spring recruitment to -3  25 mg Cm  on .Fig 49.  196  C = CO + m . D / ( e . i - m ) M  The  peak z o o p l a n k t o n  carbon  b i o m a s s i s a b o u t 1100 mg C m " .  : wet w e i g h t r a t i o o f 2 0 : 1 , t h i s  weight.m" , or approximately i n t e r m e d i a t e year  given approximately uncertainty  a t O.S.P.  Zooplankton  by e/m t i m e s  i n parameter values  standing  primary  standing  production.  t h a t t h e magnitudes of  Chapter 1 w i t h H e i n r i c h ' s c o n t e n t i o n  circle.  that the l i f e  t h e d o m i n a n t g r a z e r s a t O.S.P. were r e s p o n s i b l e . of s i m p l e , q u a l i t a t i v e models suggested  observations. history So  level  range  e x p l a n a t i o n and a s i m p l e  t o be c a p a b l e  regulation at I t started i n h i s t o r i e s of Consideration  that the l i f e  i n c o r p o r a t i n g t h r e s h o l d s and n e g l e c t i n g l i f e a qualitative  broad  values.  O.S.P. may a p p e a r t o have gone a l m o s t f u l l  at  The  d i s c u s s e d above i s such t h a t i t  t h i s p o i n t , the d i s c u s s i o n of phytoplankton  were n o t a s u f f i c i e n t  to  stock i s  stocks a r e c o n s i s t e n t with the very  of p o s s i b l e p a r a m e t e r At  20 g wet  3  w o u l d be m e a n i n g l e s s t o c l a i m more t h a n observed  represents  150 mg wet wt.m" , c o r r e s p o n d i n g  2  an  Assuming a  2  histories  biomass model  histories  appeared  of e x p l a i n i n g t h e  I n t h e s i m u l a t i o n model i n t r o d u c e d h e r e ,  s t r a t e g i e s a n d t h r e s h o l d s a p p e a r t o be f a r , the spring recruitment  both  life  necessary.  of zooplankton  biomass has  been d i s c u s s e d a s i f i t were a s t e p c h a n g e i n t h e s y s t e m s t a t e which b r i n g s the system c l o s e r  t o q u a s i - e q u i l i b r i u m before  primary  I n t h e s i m u l a t i o n model,  production  recruitment  increases.  of h e r b i v o r e s  takes p l a c e over  may a l l o w a phenomenon, r e p o r t e d by B r a u e r a predator-stocking effect, simple  neutrally  t o occur.  an e x t e n d e d p e r i o d a n d and Soudack  For s i m p l i c i t y ,  (1979) as consider a  s t a b l e L o t k a - V o l t e r r a type model w i t h a  constant  197 3 7  recruitment  C = r.C  rate for herbivores  -  a.C.G  G = e.a.C.G - m.G  The  non-trivial  + R  e q u i l i b r i u m , given  C =  (m-R.a/r)/(e.a)  G =  r/a  can  easily  be  shown t o be  zooplankton recruitment contributes  t o the  interaction.  stable  That i s , d u r i n g  i n the  stability  of  s p r i n g , the the  I t i s possible that  necessary to ensure that  the  cycle during  of  this period  by  asymptotically  course C i s non-negative).  the  added :  feeding  threshold  should  recruitment  included,  the  the  (provided period  of  of  recruitment  itself  phytoplankton-zooplankton  a feeding  threshold  system t r a c k s the  is  not  quasi-equilibrium  increasing primary production. t u r n out  t o be  unnecessary  If  with  d i s c u s s i o n w o u l d i n d e e d h a v e come  full  circle. This  was  t e s t e d under the  s i m u l a t i o n m o d e l by  most f a v o u r a b l e  r e p l a c i n g the  threshold  w i t h a t y p e I f u n c t i o n a l r e s p o n s e w i t h an clearance and  rate.  The  resulting  somewhat d e l a y e d , b u t  as o v e r - w i n t e r i n g herbivore  identical  histories,  and  the  maximum  a s e c o n d l a r g e r peak i n C h i a  d e p a r t u r e commences(Fig 51).  Both l i f e  in  f u n c t i o n a l response  s p r i n g bloom i s r e l a t i v e l y  b i o m a s s t i m e s e r i e s show l i t t l e  'observations.  conditions  The  Chi  resemblance to  a continuing  small  develops a  and the  stabilising  198  effect to  as provided  reproduce  by a g r a z i n g  the observed  t h r e s h o l d , appear  constancy  t o be  of phytoplankton  necessary  standing  stock.  4.3  A Cohort  Model  for  Zooplankton.  4.3.1 I n t r o d u c t i o n . There model size  a r e a number  of herbivore structure.  collected such  have  a model  vacuum.  been  While  specification  dynamics  One  need  of reasons  good  a t O.S.P.,  reason  analysed  to size  t h e more  detailed  o f a much  greater  the painful  task  o u t o f an o r d e r - o f - m a g n i t u d e  r e p r e s e n t a t i o n of r e c r u i t m e n t  the  simple  detailed  For  model.  level,  example,  predicts observed. recently small weight  of F u l t o n Finally, model  more  even  peak  may  i n the 'best  zooplankton  i s some  and s p e c i e s .  than  introduced  reason  to believe  that  biomass  ( F i g4 9 ) , the biomass  stages  much  metabolic  i n zooplankton  earlier  are dominated  of the dominant  a higher  more  model.  in April,  biomass  and  in a  the simple  case'  in  4.1  at the  early  reduction  size  at  parameter  o b s e r v a t i o n s , even  recruited  and a  i t will  'average'  over  c a n be  the zooplankton  have  require the  of s e c t i o n  biomass  that  biological  on t h e r e s u l t s  time,  should  in a  so  a n d a more  At t h i s  copepods  level,  data  crude,  reproduce  faithfully  s p e c i e s and  and o v e r - w i n t e r i n g departure  (1973), there  an  range  i s necessarily  r e p r e s e n t a t i o n , based  size-structured biomass  model  some  of parameters,  of choosing  The  biomass  will  number  value  observations  and s p e c i e s  model  realistic  the zooplankton  n o t be c o n s t r u c t e d o r t e s t e d  avoid  the  a more  involving  i s that  least  realistic  to consider  model  than  is  by  copepods.  rate  a  per unit  These body  c a n be e x p e c t e d i f  i - • i  ^"n  i  i  i  1  1  1  1  1  ~T  J F M R M J J R ' S Q N D MONTH  Figure 51.  1  Predicted.Chi a and grazer standing stock for biomass model with Type I functional response and standard spring recruitment.  200  this  i s taken  4.3.2  Model The  account.  Formulation.  s i z e - c l a s s m o d e l c o n s t r u c t e d h e r e i s b a s e d on  of Landry is  into  (1976) and  Steele  (1974).  r e c r u i t e d as a s e r i e s o f day  normally  distributed  a b u n d a n c e and  over  weight,  s, and  E a c h day  so t h a t Z , ( t , s )  i n d i v i d u a l s of s p e c i e s i l e f t on day  Wj(t,s)  on day  i s the weight  initial  conditions corresponding  Z (s,s)  = R] . e x p ( - ( s - p i ) / ( 2 .CT-,) ) /  ;  Wj(-s,s) =  where W°  For each s p e c i e s , a  classes, with i n i t i a l  time.  2  the  c l a s s has  models  cohort  numbers  an a s s o c i a t e d  i s t h e number o f t i n t h e day  class  recruited  of e a c h i n d i v i d u a l .  to recruitment  The  are  (JTjT.tTi)  2  W°,  i s the weight  of the e a r l i e s t ,  actively-feeding naupliar  stage. Again, and  f o l l o w i n g Landry  b a s a l metabolism  ( 1 9 7 6 ) and  Steele  (1974), i n g e s t i o n  a r e assumed t o be p r o p o r t i o n a l t o W - ,  so  0 7  that, at phytoplankton  concentration  C,  W i ( t , s ) = ( e . f ( C ) - b ; ) .W;- ( t , s ) 0 7  ;  ;  I n t e g r a t i o n of f e e d i n g o v e r t h e same manner as mortality  rate  e  ;  the phytoplankton  i n the biomass model. i s a s s u m e d , so  A  profile  i s done i n  species-specific  that  Z- ( t , s ) = - ©,-.Z j ( t , s ) . (  The  two  d o m i n a n t l a r g e c o p e p o d s p e c i e s , C.  plumchrus  and  201  C. c r i s t a t u s , the  fall  are represented  i n t h e model a s c o h o r t s .  a n d w i n t e r , when t h e s e c o p e p o d s a r e a b s e n t ,  s m a l l e r c o p e p o d s (C. p a c i f i c u s , M e t r i d i a p a c i f i c a , minutus, in  O i t h o n a ) becomes i m p o r t a n t .  During  a group of  Pseudocalanus  This group i s represented  t h e m o d e l s i m p l y a s a b i o m a s s , G, i n t h e manner o f s e c t i o n  4.2. 4.3.3  Parameters. The  C. p l u m c h r u s a n d C. c r i s t a t u s n a u p l i i  are recruited i n  weight  o f .3 pq  C (Fulton,1973).  r e c r u i t m e n t parameter e s t i m a t e s  obtained  i n S e c t i o n 4.1  the  s p r i n g a t an i n i t i a l  recruitment nauplii. factor  T y p i c a l e s t i m a t e s o f 300 a n d 75/m  of 2 t o account 2  were i n c r e a s e d by a  3  for naupliar mortality  t o g i v e 1.10 and 5  f o r C. p l u m c h r u s a n d C. c r i s t a t u s  respectively.  mean r e c r u i t m e n t t i m e was s e t back t o 70 d a y s t o a c c o u n t f o r  naupliar  growth.  The  functional  response  p a r a m e t e r s CO a n d D were  (20 a n d 40 pq  g i v e n t h e same v a l u e s  C.l"  of c o u r s e  weight-dependent and t h e v a l u e  i  C. p l u m c h r u s a n d C. c r i s t a t u s c o r r e s p o n d s fraction weight value  o f body w e i g h t  t o 0.3 d a y " i  1  = 1.0 d a y " a s s i g n e d  an ' a v e r a g e '  C. c r i s t a t u s  weight  scale.  a basal metabolic 0.10 d a y "  1  M  Ingestion i s  = 2.0 a s s i g n e d t o  t o a maximum r a t e a s a  f r o m 2.9 d a y "  f o r a l a r g e (400 pq 1  M  ranging  1  f o r the smallest  C) C. c r i s t a t u s .  t o t h e biomass f r a c t i o n  o f 10 ug C / i n d  initially  r e s p e c t i v e l y ) f o r both  1  d o m i n a n t s p e c i e s a n d f o r s m a l l c o p e p o d b i o m a s s , G.  to  represent  i n t o t h e sampled s i z e c l a s s e s , not r e c r u i t m e n t of  2.5.10* i n d / m The  The  The  corresponds  on t h e C. p l u m c h r u s ,  The p a r a m e t e r b was f i x e d a t 0.075, y i e l d i n g  r a t e a s a f r a c t i o n o f body w e i g h t  f o r smallest nauplii  t o 0.01 d a y "  1  ranging  f o r large  from  202  C. c r i s t a t u s .  The c o m b i n e d b a s a l m e t a b o l i c  r a t e and m o r t a l i t y  r a t e f o r t h e b i o m a s s f r a c t i o n , m, was f i x e d a t 0.075 d a y " . 1  B a s e d on t h e a v e r a g e s i z e o f t h e b i o m a s s f r a c t i o n c a l c u l a t e d a b o v e , t h i s c a n be r e g a r d e d metabolism.  as h a l f m o r t a l i t y and h a l f  The m o r t a l i t y r a t e s f o r C. p l u m c h r u s a n d  C. c r i s t a t u s were i n i t i a l l y In the simple  set equal  o v e r an e x t e n d e d p e r i o d .  According  s u r f a c e w a t e r s as stage  to  represent  1  A simple  of Georgia  C. c r i s t a t u s . by F u l t o n  up  realistic  weight.  way  This  weight  C f o r C. p l u m c h r u s a n d 400 ug C f o r  The v a l u e  f o r C. p l u m c h r u s i s l o w e r  (1973) f o r t h e S t r a i t  of Georgia  than  observed  but c o n s i s t e n t  t h e wet w e i g h t v s l e n g t h r e l a t i o n s h i p d e r i v e d in  leave the  i n t h e m o d e l i s t o have day  classes leave at a s p e c i f i e d over-wintering was f i x e d a t 100 pq  loss rate  have b u i l t  and a p p a r e n t l y  over-wintering departure  of  (1973),  V c o p e p o d i d s , once they  stores.  departure  as a c o n s t a n t  to Fulton  C. p l u m c h r u s i n d i v i d u a l s i n t h e S t r a i t  lipid  a t .025 d a y " .  biomass model, the o v e r - w i n t e r i n g  t h e d o m i n a n t c o p e p o d s was r e p r e s e n t e d  sufficient  basal  with  f o r O.S.P. c o p e p o d s  S e c t i o n 4.1.8.  4.3.4 S i m u l a t i o n R e s u l t s a n d D i s c u s s i o n . The stock  p r e d i c t e d time streams of mixed l a y e r C h i a and s t a n d i n g  (g C m " )  o f C. p l u m c h r u s , C. c r i s t a t u s a n d s m a l l  2  (plotted cumulatively) c o n t r o l of p h y t o p l a n k t o n d e s p i t e t h e much l o w e r w i t h 200 mg C m " ) . 2  a r e drawn i n F i g 52.  Note t h a t t h e  i n t h e s p r i n g i s a s good a s i n F i g 4 9 , r e c r u i t e d biomass  (37 mg C m "  2  compared  The l a r g e r f i g u r e u s e d i n t h e b i o m a s s m o d e l  corresponded to recruitment (individuals greater  copepods  i n t o sampled s i z e  t h a n 1 mm),  classes  not n a u p l i a r recruitment.  A  .1...5-  1 .0 0.5  J  F  M. fl M J  J 'R 'S '0  N  D  MONTH Figure 52.  Predicted Chi a and zooplankton biomass (long dashes = C_. plumchrus, short dashes = C_. c r i s t a t u s + C.  plumchrus s o l i d l i n e y  = t o t a l biomass) f o r 1976, using weight thresholds f o r departure i n cohort model.  204  recruitment result  o f 37 mg C m "  i n the simple  2  i n some d e t e r i o r a t i o n i n p h y t o p l a n k t o n  spring.  T h i s does not occur  The  behaviour  departure  in  by s m a l l  i n summer a n d f a l l  o f C. p l u m c h r u s s t a r t s  proceeds very q u i c k l y .  nauplii.  i s , however, not r e a l i s t i c . r a t h e r e a r l y , i n May, a n d  The d e c r e a s e i n b i o m a s s s t o p s  f o r a while  J u n e a s C. c r i s t a t u s b i o m a s s i n c r e a s e s , b u t t h e n t h i s  also reaches over-wintering bloom o c c u r s  weight and l e a v e s .  i n June and J u l y as a r e s u l t  s m a l l biomass f r a c t i o n  in July.  A  of t h i s  d i s a p p e a r a n c e of b o t h major g r a z e r s and t h i s the  c o n t r o l i nthe  i n t h e s i z e - s t r u c t u r e model because  of t h e h i g h g r o w t h r a t e s a c h i e v e d  The  biomass model would  species  phytoplankton rapid  t r i g g e r s a peak i n  The p h y t o p l a n k t o n  are  c o n t r o l l e d and t h e system approaches t h e q u a s i - e q u i l i b r i u m c y c l e of t h e b i o m a s s model f o r t h e r e m a i n d e r o f t h e y e a r . There i s a simple over-wintering behaviour.  instability  departure  associated with the threshold  s t r a t e g y which i s r e s p o n s i b l e  While a l l day-classes  a r e below  for this  over-wintering  w e i g h t , t h e system i s near a q u a s i - e q u i l i b r i u m s t a t e , w i t h relatively reach in  constant  over-wintering  phytoplankton  pressure. reach  phytoplankton.  As a r e s u l t ,  the remaining  i s a small  weight and depart  day c l a s s e s grow more q u i c k l y .  relaxation i n grazing pressure,  phytoplankton classes.  weight and l e a v e , t h e r e  increase  a b u n d a n c e due t o t h e r e l a x a t i o n i n g r a z i n g  over-wintering  a greater  As t h e o l d e s t day c l a s s e s  abundance and h i g h e r  faster, This  causes  a larger increase i n  growth r a t e s f o r remaining  day  The e n d r e s u l t c a n be s e e n i n F i g 5 2 , where  C. c r i s t a t u s d a y c l a s s e s , r e c r u i t e d a t f i x e d a p e r i o d exceeding  initial  weight  over  60 d a y s , a l l l e a v e w i t h i n a 20 d a y p e r i o d .  205  It  is clear  concentration wintering  cristatus  as  though  a  Two  was a  down  class  at  by a  departure  weight,  when  ug  C  the  slightly,  but  same  departure  the  simple  until  bloom  the  spent  a  prolonged  as  and  a  the  fixed  f o r C.  then  C.  residence  cristatus  of  time a  in July.  i n weight  classes  C.  that  recruitment  cristatus,  occurred  increase day  and  C.  cristatus  individuals they  pass  and  200  departure with  a  and  leave a  departure  weights  of  this 100  800  and  jjg C  p e r i o d was  prolonged  resulting  followed closely  fixed  strategy ensures  still  achieve in  maximum  maximum  by  i t seems  simulation using  plumchrus,  e s t i m a t i o n of  departure  days  and  a  a  over-  52.  once  reach  instability,  in July  having 1  to  tried  in Fig  day" )  In  the  were  plumchrus  of  they  minimum  cristatus,  parameter  individuals  using  (.05  leave.  f o r C.  C.  plumchrus  i s needed  as  device  weight,  basic  of  C.  p e r i o d of  i s r e q u i r e d and  strategies  the  remainder  of  strategy  parameters  this  by  a  small  copepod  persisted.  In that  copepods  rate  the  phytoplankton bloom,  by. s m a l l  constant  f o r C.  through  replacement  departure  respectively  phytoplankton  smooth  strategy, with  respectively  constant  departure  the  first,  minimum  200  a  then  using  the  departure  remain  different  slowed  day  and  i f .the p r e d i c t e d  different  simulations In  is'to  departure,  C.  this.  that  of  due  time  to  of  100  days  period  day  so  This  as  phytoplankton of  i s made u p  classes,  is  plumchrus  i n numbers  earlier  assumed  in a simulation  f o r C.  reduced)  departure  and  i t was  surface waters.  However,  decrease  day  4.1,  departure  period.  remaining  plumchrus  i n the  the  (somewhat The  section  by  that  classes  C. a  180  bloom  plumchrus large  the of  and  later  206  C. c r i s t a t u s  leave at large weights  respectively). a very rapid the  The  increase  i n s m a l l copepod biomass,  I n t h e s i m u l a t i o n s so f a r , C. c r i s t a t u s has  reached too high a departure weight. where C. c r i s t a t u s was  apply.  The  (1974) m o d e l ,  approximate  residence time, model  as a s i n g l e day  class,  p r e s e n t e d i n C h a p t e r 2,  g o v e r n i n g e q u a t i o n s w o u l d be  - f (C) .Z.W' 0  either  In a h y p o t h e t i c a l  p r e s e n t by i t s e l f  a n a l y s i s of S t e e l e ' s  C = r.C  to  i n F i g 52.  too e a r l y , o r , i n the case of a f i x e d  would  a l t h o u g h not  by  i s n o t t o be f o u n d s i m p l y by c h a n g i n g t h e d e p a r t u r e  strategy.  the  ug C / i n d  seems a s t h o u g h a t i g h t c o n t r o l o f p h y t o p l a n k t o n d u r i n g  departure  left  and 1700  p h y t o p l a n k t o n bloom i s i n f a c t c o n t r o l l e d  peak l e v e l s s e e n It  ( c a 400  . . . .  7  W = (e. f ( C ) - b ) .W0  7  z = -e. z  If  the system  changing  remains  near q u a s i - e q u i l i b r i u m , w i t h C c o n s t a n t or  s l o w l y , the l e v e l  of g r a z i n g  intensity, proportional  Z.W-, must c h a n g e s l o w l y w i t h p r i m a r y p r o d u c t i o n . 0  in  7  Chapter  Z.W  0,7  2, t h e w e i g h t W w i l l  the  r a t e of growth  the  mortality In  C.  biomass. will  that  In such a model,  o f C. c r i s t a t u s c a n be r e d u c e d by d e c r e a s i n g  rate, ©.  the s i m u l a t i o n model,  plumchrus  As d i s c u s s e d  i n c r e a s e a s Z d e c r e a s e s so  i s m a i n t a i n e d a t an a p p r o p r i a t e l e v e l .  to  which  C. c r i s t a t u s  with  i s i n i t i a l l y d o m i n a n t n u m e r i c a l l y and by  Decreasing the m o r t a l i t y  initially  i s present  result  r a t e f o r C. c r i s t a t u s  i n a replacement  o f C. p l u m c h r u s  alone  biomass by  207  C.  cristatus  b i o m a s s b e f o r e any  not d e s i r e d .  There w i l l  be  departure  little  takes p l a c e , which i s  c h a n g e i n C.  rate until  i t becomes t h e d o m i n a n t g r a z e r .  r a t e of C.  cristatus  feeding e f f i c i e n c y required.  i n the  reduce the  growth  p r e s e n t , w h i c h i s not  cristatus likely  alone would r e s u l t  b i o m a s s w h i l e C.  to lead to better  plumchrus d e p a r t s .  to over-winter  plumchrus i s  phytoplankton  i t will  C.  p l u m c h r u s and  especially  W-  metabolic  day  increases like  0  7  W" °-  3  C.  the percent  as W d e c r e a s e s .  s m a l l e r , more numerous C. quickly  cristatus.  cristatus  As  of  remaining  According  to  the  i n c r e a s e i n weight  per  T h u s , a p o p u l a t i o n of  should  respond  to small increases i n phytoplankton  plumchrus departs, l e a d i n g to t i g h t e r  be  grazing-  i n c r e a s e i n weight  by C.  law u s e d h e r e ,  in zooplankton  plumchrus  now  composed o f a l a r g e r number o f s m a l l e r i n d i v i d u a l s . plumchrus d e p a r t s , the decrease  changes  c r i s t a t u s a s C.  need not change, but  p r e s s u r e must be made up by an  in a  However, t h e two  t o g e t h e r mean t h a t t h e b i o m a s s of C.  C.  in i t s  r e l a t i v e t o t h a t of C . p l u m c h r u s i s a l s o  r e l a t i v e d e c l i n e i n C.  starts  growth  s i m u l a t i o n model, a decrease  Note t h a t t h i s decrease  c o n t r o l when C.  To  cristatus  much more  carbon  as  phytoplankton  regulat ion. All  t h r e e s i m u l a t i o n s d e s c r i b e d a b o v e were t h e r e f o r e  repeated, 0.01  day"  o f C. C. all  w i t h the m o r t a l i t y 1  and  C.  plumchrus.  cristatus  cristatus  i n g e s t i o n reduced  three departure some C.  cristatus  cristatus  although  f o r the  never reached  times  to that  from  t o s m a l l c o p e p o d s was  strategies,  reduced  t o 0.8  A smooth t r a n s i t i o n o f c o n t r o l  p l u m c h r u s t o C.  strategies,  r a t e f o r C.  obtained  with  weight-dependent  over-wintering  208  weight,  owing  The biomass shown  to  predicted of  C.  in Fig  53.  is  in  individuals The  C.  cristatus  0-7  C/ind,  I t can  fact  departure  of  Chi  be  seen  the  slight  the  that  departure  a  i n the  range  as  grazing  simulation are  of  C.  layer  copepods  and  August.  plumchrus  ca  pq  earlier  C/ind  larger  weights  to  and and  simulation  departure  a  Total  proportional  f o r C. 240  in  triggers  i t c o n s i s t s of  pressure  and  are  increase  plumchrus  in July  peak  weights  than  of  biomass  in this  The  mixed  small  departure  strategy.  i n the  and  maintain  reasonable  a  cristatus  larger  in this  rate.  C.  cristatus  must  maximum  more  by  i n C.  and  W .  mortality  timestreams  caused  increase  biomass  low  plumchrus,  phytoplankton rapid  the  510  jug  using  this  is also  much  smaller. While parameter the  the  importance  of  raises  changes  of  estimation  the  size  of  of  C.  sensitivity  to  small  Provided  using  in  the  growth the  not  about  changes be  of  changes argument  of  by  instability  C.  been  sensitivity  For  example, suggest year  removed  to  of  the a  this  associated  cristatus  the  from  using  and  during the  the this  results  in mortality rates  to  parameter  large variation  year.  with  The  model's  and  growth  in  suspect. in  recruitment  for  abundance  occurs  face  avoided  really  s e c t i o n 4.1  phytoplankton  individual pressure  of  been  number  cristatus  also  effects  predicted  has  parameters.  recruitment  must  has  intrinsic  and  questions  results  parameters  the  copepods  in other  The  bloom  combination,  departure  period  summer  so  as  to  a  single  remains  be  qualitatively  day-class  given  approximately  maintain  mortality losses.  can  a  constant  Increasing  above.  constant, grazing the  209  Figure  53.  As  f o r F i g 52, w i t h f i x e d  lower  g r o w t h and  residence  times  m o r t a l i t y r a t e s f o r C.  and cristatus.  210  recruitment l e v e l produces a l a r g e r must be  s m a l l e r a t any  departure fixed  particular  number o f  time.  strategy, over-wintering w i l l  residence time, departure  weights  Doubling  the s p r i n g r e c r u i t m e n t of both  thousand  individuals/m  fixed 100  r e s i d e n c e time  and  no C.  300  just  For  the  departure  f o r C.  plumchrus  t o 370  ug C / i n d ) .  weights (100  and  5  C.  cristatus  in  biomass f o r the h i g h e r  to  the W  1.10  c o p e p o d b i o m a s s , G.  latter  presence  of r e l a t i v e l y  s m a l l C.  respectively w i t h the  fixed  f r a c t i o n , G, C.  levels  C.  i n even  i n g e s t i o n and  4  and  The  The  smaller  fall small this  when  the  prolonged  C.  cristatus  delays  left.  1.25.10  4  ind/m  2  weights  s m a l l copepod  more t o t o t a l low  again  the  that  c r i s t a t u s has  strategy.  ( F i g 55).  lower  and  ( F i g 5 4 ) , due  in larger departure  contributes relatively  plumchrus departs  the  ug C / i n d )  p l u m c h r u s and  ,as e x p e c t e d ,  r e s i d e n c e time  the  i n t h e g r o w t h of t h e  r e c r u i t m e n t t o 5.10  results  results  s i m u l a t i o n , the  increase in G u n t i l  Halving total  Doubling  the dominant g r a z e r s except  In t h i s  For  threshold departure,  There i s a l s o a s l i g h t  large.  a significant  2  50  average about  parameters chosen are such  i s o u t c o m p e t e d by  are very  weights  and  There i s a marked r e d u c t i o n  and  to a delay  The  smaller.  effect.  (23 t o 100  dependence of m e t a b o l i s m  bloom due  w h i l e f o r the  s p e c i e s t o 200  ind/m  5  recruitment  mean s i z e o f c o p e p o d s p r e s e n t .  fraction  be  reach o v e r - w i n t e r i n g weight. t o 4.10  phytoplankton  will  which  weight-dependent  be d e l a y e d ,  s i n g l e weight  recruitment again  07  the  the expected  strategy, departure  ug C / i n d .  cristatus  has  2  For  individuals  biomass  recruitment  as  results  in a  generally higher c h l o r o p h y l l a l e v e l  i n t h e s p r i n g and  a  C.  J u n e , due  higher  p l u m c h r u s b i o m a s s peak i n May  and  both  to  larger  1 .5 cn X  1.0-  CD  CE IE CJ  0.5  0.0  i  i  i  5M - 5 H >K  CJ CD  1 .OH  cn LU  r^j  0.5  CE LYL  CD  0.0  J  F  M  fl  M  J  J  fl  S  0  N  MONTH F i g u r e 54.  E f f e c t of q u a d r u p l i n g s p r i n g r e c r u i t m e n t i n Fig  53.  D  212  primary  p r o d u c t i o n and  This inverse effect  the l a r g e r  s i z e of  of r e c r u i t m e n t l e v e l  individuals on  while c o n s i s t e n t with a simple q u a l i t a t i v e model, i s r a t h e r According mortality  original and  leaves C.  f o r C.  recruitment  the p r i n c i p a l  increase  p l u m c h r u s and  Departure  C.  weight,  T h i s was  and  5  weights  pg C / i n d  r a t e s s h o u l d be  2.5.10  4  The  ind/m  2  f o r C.  and  500  plumchrus pg  C/ind  other  standing stock.  p a r a m e t e r s D and  CO  For  of  parameters  example, d e c r e a s i n g  should decrease  c o n c e n t r a t i o n , determined  roughly  plumchrus  and  recruitment  zooplankton  an  respectively.  the p r i n c i p a l d e t e r m i n a n t s  The  were u s e d  i n c r e a s e i n C h i a and  t h e q u a s i - e q u i l i b r i u m l e v e l s of p h y t o p l a n k t o n  zooplankton  found 0.015  cristatus.  t o the q u a l i t a t i v e a n a l y s i s ,  and/or t i m i n g .  determine  recruitment.  l e v e l s o f 1.10  c r i s t a t u s were c a 300  weight  total  on d e p a r t u r e  i n t h e c o n t r i b u t i o n o f s m a l l c o p e p o d s a f t e r C.  According  the  i n c r e a s i n g the  effect  e f f e c t s were a s l i g h t  ( F i g 56).  mortality  of  when v a l u e s of © were i n c r e a s e d t o 0.03,  respectively  1  biomass,  understanding  to the q u a l i t a t i v e a n a l y s i s ,  and/or t i m i n g , as d e c r e a s i n g  day"  zooplankton  counter-intuitive.  r a t e s h o u l d have a s i m i l a r  t o be t h e c a s e  present.  and departure will and  the  the e q u i l i b r i u m p h y t o p l a n k t o n  i n t h e s i n g l e day  class  model  by  C = CO  + D. (b+e.W - )/( i . e - ( b + e . W - ) )  In f a c t ,  0  3  0  t h e v a l u e s of b and  6 chosen i n p r e v i o u s s i m u l a t i o n s  p r o d u c e v a l u e s of C c o r r e s p o n d i n g T h i s can  4.3a  3  M  be  reduced  t o .5 mg  to the v i c i n i t y  C h i a.m"  3  of t h e o b s e r v e d  or  higher.  average  (.4  213  Figure  55.  E f f e c t of h a l v i n g spring recruitment  i n F i g 53.  cn x x CD  CE  CJ  £  1 .0 0.5 0.0  i  1 .51  X CJ CD  cn cn LU  1 .0 0.5  CE  cn CD  0.0  JFMR'MJJ.R  S 0 N D  MONTH Figure  56.  E f f e c t of  increasing mortality rates  Calanus i n F i g  53.  for  i  215  mg C h i a . m " ) , by r e d u c i n g D a n d CO t o 30 a n d 15 jug C . l 3  respectively.  1  The r e s u l t i n g C h i a a n d g r a z e r b i o m a s s l e v e l s a r e  shown i n F i g 57; b o t h p h y t o p l a n k t o n a n d z o o p l a n k t o n stock  -  standing  i s r e d u c e d , but d e p a r t u r e w e i g h t s and p h y t o p l a n k t o n  control  are not a f f e c t e d . The q u a s i - e q u i l i b r i u m z o o p l a n k t o n b i o m a s s i n a s i n g l e d a y c l a s s model  i s d e t e r m i n e d by  Z.W - = r.C.e/(b+8.W - ) 0  7  0  4.3b  3  I n c r e a s i n g b and m f r o m .075 t o 0.10, k e e p i n g D a n d CO a t 30 a n d 15 jug C . l  - 1  especially  ,  results  i n an i n c r e a s e d ' p h y t o p l a n k t o n  i n the f a l l  a s C. c r i s t a t u s  level,  leaves (Fig 58).  There i s  a d e l a y i n t h e r e s p o n s e o f s m a l l c o p e p o d s , G, t o t h e d e p a r t u r e o f C. c r i s t a t u s .  The s m a l l c o p e p o d s h a v e s u f f e r e d c o m p e t i t i v e l y by  an e q u a l i n c r e a s e i n b a n d m, a s t h e m o r t a l i t y c o p e p o d s were n o t c h a n g e d . z o o p l a n k t o n biomass carbon  There i s l i t t l e  r a t e s of the l a r g e  change  in total  (cf F i g 57); the increase i n phytoplankton  ( e q u a t i o n 4.3a) b a l a n c e s t h e d i r e c t e f f e c t  i n equation  4.3b. In  most o f t h e s e s i m u l a t i o n s , p h y t o p l a n k t o n h a s s t a y e d  r e l a t i v e l y c o n s t a n t a n d t h e r e s u l t s have been c o n s i s t e n t w i t h t h e q u a l i t a t i v e , quasi-equilibrium theory.  The p a r a m e t e r c h a n g e s  h a v e a l l been a p p l i e d e q u a l l y t o b o t h m a j o r c o p e p o d s . three s i m u l a t i o n s with equal m o r t a l i t y  The  first  r a t e s showed t h a t t h e  r e g u l a t i o n of phytoplankton d u r i n g o v e r - w i n t e r i n g departure i s very s e n s i t i v e copepods.  t o changes d i f f e r e n t i a l l y  affecting  On two o c c a s i o n s p a r a m e t e r c h a n g e s w h i c h  the major slightly  216  Figure  57.  Effect in  Fig  of  53.  decreasing  grazing parameters  CO  and  D  Figure 58.  E f f e c t of i n c r e a s i n g metabolic rates i n F i g 57.  218  a f f e c t e d the r e l a t i v e  s u c c e s s of t h e s m a l l c o p e p o d g r o u p and  l a r g e c o p e p o d s have l e d t o s m a l l p h y t o p l a n k t o n fall  CFig So  f a r , o n l y the p r e d i c t e d timestreams  t h e p e r i o d 1964  t o 1976.  Phytoplankton  were c o n t i n u e d f r o m  including  In the f i r s t  r e c r u i t m e n t t o t a l s o f 1.10  G,  and  5  2.5.10  b i o m a s s and  little  recruitment  4  ind/m  1976,  performed  fixed  from year t o y e a r .  the q u a l i t a t i v e  the  residence  constant  were assumed i n  the s m a l l copepod  biomass,  The p r e d i c t e d ( F i g 33) Given  showed  that  spring  slow-variable analysis  the p r e c e d i n g s i m u l a t i o n r e s u l t s suggest of  2  each year t o the n e x t .  variation  i s fixed,  the f i x e d  simulation,  primary production c y c l e with c h l o r o p h y l l remarkably  year,  These s i m u l a t i o n s have used  a s F i g 53,  time departure s t r a t e g y .  years.  f o r one  L o n g - t e r m s i m u l a t i o n s h a v e a l s o been  same s e t o f p a r a m e t e r s  all  e x c u r s i o n s i n the  54,58).  have been shown. for  the  and  t h a t the p r e d i c t e d c y c l e  z o o p l a n k t o n b i o m a s s s h o u l d a l s o show l i t t l e  year  to year  variation. T h i s d o e s n o t t u r n o u t t o be t h e c a s e , a s shown i n t h e of  m i x e d l a y e r C h i a and  total  zooplankton carbon  S h o r t - l i v e d p h y t o p l a n k t o n b l o o m s and peaks occur  in late  e x t e n t , 1971.  summer i n 1967,  Closer examination  these zooplankton peaks c o n s i s t departure weights  (up t o 1000  ( F i g 59).  unusally high 1972,  1974  zooplankton  and,  to a  of the model o u t p u t  o f C.  plot  shows t h a t  c r i s t a t u s which  jug C / i n d ) a s a r e s u l t  lesser  reach  of  high  the  blooms. The the years  cause  of t h e s e b l o o m s was  involving  somewhat u n e x p e c t e d .  Each of  l a r g e peaks f o l l o w s a severe w i n t e r , i n which  the s m a l l copepod biomass,  G,  falls  t o u n u s u a l l y low  levels,  an  .J T 00 X  YEAR Figure 59.  Predicted mixed layer Chi a and t o t a l zooplankton parameters and fixed recruitment levels of F i g 53.  carbon f o r 1964-76, using the  220  order  of magnitude  result,  there  lower  i s a  than  l a g i n the response  departure  of the large  control.  It i s interesting  minimum  annual  magnitude, maximum tendency  for G  predicted case  co-existence  balance  transition groups.  O.S.P.  peak  i s a very  whereas  phytoplankton  couple  the long-term.  copepods  again  the spring  each  year  through  model  control  to achieve  among  pointed  out that  i n zooplankton  the  a  life  a s an e x p l a n a t i o n  lacking.  stock at Of t h e  years  a t O.S.P. b u t  occur  i n June.  peaks  smooth zooplankton  standing  The p r e d i c t e d peaks  zooplankton  a  the three  i n August, Finally,  a t O.S.P.  from  the previous  long-term recruitment  s i m u l a t i o n , an attempt o f C. p l u m c h r u s  there  follow  than  year.  This  practical  success  was made t o  a n d C. c r i s t a t u s  t o the p r e d i c t e d o v e r - w i n t e r i n g biomass  over-wintering  is clearly  blooms.  the second  theoretical  The  emphasize the  1967 a n d 1972 a r e h i g h  high  the p r e d i c t e d  i s no  i n 1967 a n d 1972 o c c u r  that  grazing  of 4 and there  simulation i s sadly  low y e a r .  the peaks  no e v i d e n c e  In  years,  escape  copepods.  in this  variation  (Fig 60), this  predicted  is  annual  to the  the p r e d i c t e d  by t i m e ,  simulation results  a t once.be  copepods  As a  of the winter,  and l a r g e  of the resource  years.  by 2 o r d e r s o f  or i n c r e a s e over  of the major  I t should  while  varies  factor  of small  required  of small  that,  biomass  a  of phytoplankton  observed  1974  to decrease  long-term  delicate  t o note  than  i n other  and phytoplankton  on t h e s e v e r i t y  by l e s s  strategies  The  copepods  copepod  of p a r t i t i o n i n g  history  of  small  depending  varies  t h e minimum  of each  in  species  e x e r c i s e h a s r a t h e r more  interest,  are poorly  as the f a c t o r s  affecting  known, a n d h o r i z o n t a l  advection  500-i 400300-  Figure 60.  Observed zooplankton wet weights (10 day means) at O.S.P. 1956-1978.  222  is  significant  relating biomass  recruitment  in rather  choosing  a  levels  recruitments  used  efficiencies, efficiency,  were  fashion,  which  would  account  l o w (.04  corresponding  zooplankton  s i m u l a t i o n , and total  over-wintering  of m o r t a l i t y and  f o r C. p l u m c h r u s  -biomass  predicted  reproductive  a n d .005 f o r  a r e shown  timestreams  i n F i g 61.  constant  recruitment,  slightly ind/m ) 2  higher than  average  recruitment  expected.  There  of poorest  recruitment  o f C. p l u m c h r u s  i s very  contributes this  year, 2000  variation  to the control both  species  pg C / i n d  coexistence  of the major  the  partitioning  can  grow  a t lower  food  there  levels  than  while  2  which  departs.  time.  C. c r i s t a t u s  that  In  w e i g h t s (900  the annual  i s no l o n g - t e r m  over  in  undoubtedly  departure  i s predicted,  resource  variation  s  1967, t h e 5  despite  peaks  ( c a 1.5.10  2.1.10 /m , 2  with  i s due t o  annual  a s C. p l u m c h r u s  Again,  copepods  of the food  high,  unreasonable  levels,  This  control,  4  respectively).  i n recruitment  higher.  low, 1.7.10 /m ,  problem  reach  and zooplankton  i s considerable  and i n the year  i s unusually  t o those  o f C. p l u m c h r u s  recruitment  C. c r i s t a t u s  t h e change i n  similar  but the phytoplankton  1 9 6 7 , 1 9 7 1 , 1972 a n d 1974 a r e e v e n  of C h i a and t o t a l  (Note  The p r e d i c t i o n s a r e q u a l i t a t i v e l y  and  at the average  the 'fixed'  scale).  of  was  cristatus). The  in  The  to nauplii  long-term  The r e s u l t i n g  take  over-wintering  by l o o k i n g  provide  so t h a t  undertaking.  converted  i n the previous  there.  which  i s a dubious  biomass  arbitrary  fraction  of over-wintering,  a t O.S.P. t o t h e p r e d i c t e d  of over-wintering  over-wintering  C.  scale  a t t h e same l o c a t i o n  fraction decided  on t h e t i m e  trend and  presumably  due t o  (C. plumchrus ,but t h e l a t t e r  oo X X SI  YEAR F i g u r e 61.  P r e d i c t e d mixed l a y e r C h i a and zooplankton carbon f o r 1964-76 u s i n g c o u p l e d r e c r u i t m e n t .  u>  224  has  the resource The  C.  long-term  plumchrus  changes  6:1  t o about  6.7.10*/m  of  efficiency  2  1.3:1, w i t h  be e x p e c t e d ,  recruitment  4.4  annual  control  i s much  ratio  over  and 5.10 /m 4  several  i fthe  t o .01, t h e  years  from  fluctuating  during  and the peaks  robust,  to  about  about  f o r C. c r i s t a t u s .  2  The b e h a v i o u r  estimates  sensitive  i s doubled  recruitments  improved  i s more  recruitment of  F o r example,  of phytoplankton  disappear.  the recruitment  As  the departure  i n 1 9 6 7 , 1972 a n d  of t h e model  but the r a t i o  with  i s not  this  consistent  o f S e c t i o n 4.1.  Conclusions.  here  According  t o one p h i l o s o p h y ,  should  regarded  be  explicitly,  which  observations. when in  between  i s particularly  shifts  f o r C. p l u m c h r u s  virtually  with  balance  o f C. c r i s t a t u s  recruitment  C. p l u m c h r u s  1974  equilibrium  fall).  i n the over-wintering e f f i c i e n c i e s .  of spring  might  i n the  a n d C. c r i s t a t u s  over-wintering ratio  to itself  i t fails  While  are controlled  history distinct occur  seasonal  a t O.S.P. sound  cycle'  that  nature such  copepods or  reasonable  of the  as  alone,  to  'phytoplankton  error  this  at  'the l i f e  i s the cause  i n zooplankton  in maintaining they  an  useful  hypothesis-model.  grazing' or  'thresholds  against  i s most  revealing  limitations  a t O.S.P.  and a r e a component  stated  a model  thereby  a r e , however,  by z o o p l a n k t o n  of the dominant  constancy'  be c l a i m e d  v e r b a l hypotheses  presented  to rigorous testing  observations,  There  of the type  hypothesis,  subject  due t o t h e complex  simple  O.S.P.  even  t o match  the hypothesis.  usefulness,  as a complex  i s thereby  I t may  a model  of the  grazing  phytoplankton  c a n be a d d r e s s e d  v i athe  225  model  only  within a  phytoplankton Within a  life  i n zooplankton  history  either  accepted, these  model's  i s important predictions  small  above  errors  and  to  corresponds  including  and t h i s  produce seasonal  to  t h r e s h o l d s and  i s not possible i f  assumptions  as evidence  to consider fail  of t h e model a r e  f o r the importance  carefully  t o match  of d e t a i l  or structure  questions field  failures  cannot  studies.  i n t h e model  Much  of  observations  balance  required  among a  control  here  to  prevent  to  the unstable  of growth  the three  small  be d u e the  conclusions.  further  causes  experimental  of the model's  will  hopefully  of the species, o f how  assist  as large  rate,  copepods in  mortality  zooplankton  groups  bloom.  a s s o c i a t e d with  size-structure  to achieve  sensitivity.  phytoplankton  feedback  may  i n the model's these  i s n o t s o much  as i n the model's  delicate  These  do n o t a f f e c t  examination  the problem  of phytoplankton The problem  i n which the  studies.  of the s i m u l a t i o n study  feeding.  flaws  without  The f o l l o w i n g  around  which  invalidate  be a n s w e r e d  of these  has c e n t r e d  transfer  which  the areas  observations.  and s p e c u l a t i o n as t o t h e i r  the d i r e c t i o n  model  by  c o n c l u s i o n s , or t o fundamental  assumptions Such  i t i s possible  roughly  i n the model,  c a n be t a k e n  mortality, etc.  c o n c e n t r a t i o n and a  which  If the other  involving  p h e n o m e n a a t O.S.P. It  to  chlorophyll  biomass  strategy  this  zooplankton  and magnitude,  i s removed.  hypotheses  constructed here,  constant  i n form  of other  dynamics,  t h e framework  observation  in  growth  (reasonably)  cycle  framework  a  smooth  cease  matching A  suspiciously  and recruitment i s i n t h e model  This  has been  i n order  attributed  the over-wintering  226  strategies  used  Another it  conversion  exceeds  least  high,  when  day  AW,  or  allowing  an  a  this,  by  in  occurs,  as  pressure departure  the  is  This  in  grazing  criterion.  hovering a  around  prolonged  of  a  the  departure  series  of  the  orderly with  criterion  not  each  the  the  biological  is  still  phytoplankton departure by  up  grazing for  departure  followed  live  departure  required  level,  is  first.  did  behaviour  will  is  maintain  smooth,  bursts,  AW/W  to  ©,  be  growth.  required  and  individuals first,  through  no  than  to  abundance  ©,  0 is greater  strained  Instead  larger  A  over-  decreasing  leave  that  AW  strategy,  Departure  than  than  day  leaves  appears  low,  strategy  for  optimal  since  should  is less level  is  a  for  increase  or  also  pressure  this  i s higher  If  becomes  disappointing. copepods,  AO  an  problem.  is that,  r e s u l t s using  staying  phytoplankton  copepods  3  7  biomass  stays  and  an  required  strategy  when  cease  phytoplankton  interpretation  following  It  W~ °- , l a r g e r  is  i n d i v i d u a l which  abundance,  (Z.W - ) c o n s t a n t 0  biomass  instability  If  that  weight  our  expectations.  1  expected  phytoplankton  simulation  these  i t follows  day" )  its daily  basis.  increase  A©  whenever minimum  If  plumchrus  amount  of  model.  by  an  will  C.  the  increased  criteria  day  an  in  is  additional a t t r a c t i v e feature to  (by  terms  spring,  layer  above  pressure.  The  in  the  solution, to  proportional  to  the a  potential  grazing  An  to  above  rate  tried  depths,  (Fulton,1973),  Presumably  but  on  in  was  over-wintering  measured  surface  W.A©.  according  occur  eggs  strategy  mortality  fitness,  the  wintering,  a  lower  to  in  at  respiration  individual's  longer  that  a  negligible  far.  over-wintering  i s accepted  experiences  at  so  of abundance occurs  phytoplankton  227  blooms  which  departure to  the  slowly  decline  s t r a t e g y succeeds  instability  Besides  the  problem  strategies  departure  times  rates  too  are  or  over-wintering study  of  the  of  C.  depth  concerning  may  reveal  strategy  a  can  copepodids  The  be  the  the  to  a  weight  leads  of  a  way  to  a l l of  or  mortality seems  failure  recruitments.  timing  or  answer  weight  a  weights  several from  manipulation  uncover  careful  over-  over-wintering  s t u d i e s over  of  A  including  p e r i o d would  of  the  unreasonable  complete  copepods,  Laboratory  quicker  to  aspect  departure.  t h r e s h o l d , which  to  Field  new •  oscillatory  control,  lead  for high  departure  an  The  i f recruitment  variability  inferred. be  can  departure  departure.  level.  weight-dependent  weights  cristatus  pattern  may  low.  during  of  in adding  here  distribution  questions  timing  departure  phytoplankton  biologically,  depths,  the  of  suggested  or  wintering  and  only  departure  high  reasonable  the  associated with  departure  more  to  true  of  years which  stage  a  V  departure  strategy. Aside  from  discrepancy the  fall.  which  reach  metabolic  large  law,  C.  plumchrus  individual  the  annual  problems, observed are  high  i n August  observed  Predicted  for this  and,  biomass  at  C.  O.S.P.  discrepancy  biomass  in  cristatus  phytoplankton due  peak  zooplankton  in a l l years  explanations  is a  zooplankton  ( F i g 34)  zooplankton the  there  r e p l a c e d by  weights.  is still  However,  sharply  possible  sensitivity  p r e d i c t e d and  i n August  ( F i g 53).  declines Three  between Departing  production  time  these  to  the  occurs  at  W0  7  this  biomass (Fig will  62a).  be  discussed. The  first  depends  on  the  o b s e r v a t i o n of  Marlowe  and  Miller  228  200 n  5x3 CD  Lu  150 00 50  J  F i g u r e 62a.  F  M  R  M'-J  . J n ' S ' . Q N l  Average seasonal cycle i n observed zooplankton wet w e i g h t (10 day means,  1956-1978).  229  Figure  62b.  Average s e a s o n a l c y c l e i n i n g e s t i o n v a r i a b l e VI (10 day means, 1969-78).  230  Figure 62c.  Average seasonal cycle i n ingestion v a r i a b l e V2 (10 day means, 1969-78).  231  (1975) t h a t  l a t e - s t a g e C. c r i s t a t u s u n d e r g o e x t e n s i v e  v e r t i c a l migration.  The 150m d a y l i g h t h a u l s  a t O.S.P. may  have u n d e r - e s t i m a t e d z o o p l a n k t o n biomass b a d l y (The but  term zooplankton biomass i s being as p r e d i c t e d  phytoplankton explanation  The  i n the surface  second p o s s i b i l i t y  l o o s e l y here  According  to this  This  c a n be e a s i l y t e s t e d by  o f z o o p l a n k t o n s a m p l e s a t O.S.P.  i s that  t h e m o d e l i s wrong i n i t s  A change i n growth and m o r t a l i t y  C. p l u m c h r u s l e a v e s .  small  According  phytoplankton c o n t r o l .  parameters  copepod biomass w i l l t o t h e W0  law, t h i s  7  i n a lower zooplankton biomass being  dominate as should  necessary t o maintain  I n c i d e n t a l l y , as s m a l l copepods have  higher  maximum g r o w t h r a t e s , t h e i r d o m i n a n c e a t t h i s t i m e  result  i n t i g h t e r , more r o b u s t  possibility  on  t h a t C. c r i s t a t u s t a k e s o v e r f r o m C. p l u m c h r u s a s t h e  be a r r a n g e d s o t h a t  result  waters).  a r e not comparable.  dominant g r a z e r . could  used r a t h e r  time.  by t h e m o d e l a p p l i e s t o z o o p l a n k t o n f e e d i n g  t h e d e p t h and t i m i n g  prediction  at this  simply  t h e n , t h e model may be c o r r e c t b u t p r e d i c t i o n s a n d  observations altering  diurnal  phytoplankton c o n t r o l .  should  This  c a n be r e j e c t e d on t h e b a s i s o f t h e s i z e - s t r u c t u r e d  d a t a c o l l e c t e d s i n c e 1969.  Two d e r i v e d  v a r i a b l e s were c a l c u l a t e d  f o r e a c h s a m p l i n g d a t e f r o m t h e s e t o f d e n s i t i e s N-, o f c o p e p o d s of  l e n g t h L; :  VI  =  Z  V2 = I  Nj . L  2  N; ,L;  i If W  i n d i v i d u a l weight i s p r o p o r t i o n a l t o L 067  , Vl(t) i s proportional  to ingestion  3  and i n g e s t i o n r a t e t o  (for f i x e d phytoplankton  232  abundance). observed should  vary as  to  decline  make V I  In  section  as  L ' 2  L - , 1  According  .  than  An  use  made,  and  Frost,1977).  V2  decline  the  i t was  biomass  increase in small  suggested  ingestion  rate  weight  this  power  L,  that  of  ingestion  Under  this  an  is  copepods  on  of  W  even  than  large  In  to  L 1  vary  than  extreme  V2(t)  VI.  assumption (eg  0  Steele  is proportional fact,  biomass  copepods  then  copepods  more  density.  weights  would  07  small  assumption,  i n August  predominance  individual  is proportional  phytoplankton  sharply  that  v a r y i n g as  relative  fixed  more  reflecting  an  proposal, while  more  namely  at  above  i n August,  4.1,  was  ingestion  the  constant.  placing  7  Rather  4 5  to  (Fig at  both  VI  to  and  62),  O.S.P.  at  this  time. These samples  variables  and  occurs  i n August.  Raw  from  phenomenon.  These  especially  were  150m  not  well-known  However, show A  the a  third  continued  to  to  data  a  bloom  1977  were  consisted  and  were  used  of  individual  estimate  marked  The  decline  primary  350  even  jum  mesh  smaller  collected  to  look  for  the  higher  from  copepods,  densities  than  of  sizes  were  not  recorded,  rates  for  microzooplankton.  i n abundance  i n August  (Table V I I ) .  i s that  production  the  model  these  in  autecology  cycles  of  the  and  smaller  many  forms  primarily  (foraminifera and  net  this  species counts,  much  ingestion  seasonal  possibility  at  of data  Protozoans  found  hauls.  average  high  that  r e s u l t s .of  n u m e r i c a l l y dominant  vertical  difficult  also  1966  Oithona,  the  on  micozooplankton  n e a r - s u r f a c e samples.  radiolaria)  is  based  i t is possible  weatherships  from  are  so  f o r most  i s wrong  in  in August-September.  taxa  is  that  i t  groups  predicting These  are  Table VII. Average r e l a t i v e seasonal abundance of microzooplank (Raw data supplied by J . Fulton and 0. Kennedy.)  Taxa  J  F  M  A  M  J  J  A  S  0  N  D  Foraminifera  0.6 0.8 1.6 2.1 1.7 0.5 0.8 0.2 0.4 2.8 1.6 2.3  Radiolaria  5.2 1.3 0.5 0.8 1.2 0.8 0.5 0.4 0.6 0.9 2.7 4.4  Tintinnids  1.5 0.8 1.2 2.7 1.7 0.7 2.1 0.3 0.4  1.9 5.5  Oithona nauplii  0.9 0.6 0.8 1.2 1.7 0.9 2.6 0.7 0.6 1.9 1.5 0.8  copepodids  0.6 0.5 0.7 1.1 1.0 1.6 1.9 2.1 1.2 4.0 0.5 0.5  ad tilts  0.8 0.5 1.1 1.0 1.0 1.4 1.7 1.0 1.2 4.3 1.0 0.5  234  t h e months o f g r e a t e s t w a t e r c o l u m n s t a b i l i t y a r e n o t l i m i t i n g , one w o u l d e x p e c t  and, i f n u t r i e n t s  high phytoplankton  production.  However, p a r a m e t e r s f o r t h e P v s I c u r v e c o r r e s p o n d i n g low g r o w t h r a t e s were f o u n d  t o very  f o r t h e s e months i n C h a p t e r  3 (Fig  24,27) b u t t h e s e were i g n o r e d , a s t h e u n d e r l y i n g a s s u m p t i o n s o f the e s t i m a t i o n procedure  were n o t met, a n d v e r y  contributed t o the estimates. in August, a c a r e f u l phytoplankton  field  few  1 4  C  profiles  In view of the zooplankton  and e x p e r i m e n t a l  decline  study of  g r o w t h p a r a m e t e r s a t O.S.P. a t t h i s t i m e  seems  warranted. A d i s a p p o i n t i n g aspect  o f t h e m o d e l h a s been i t s f a i l u r e t o  p r o v i d e any c o n v i n c i n g e x p l a n a t i o n f o r t h e year in zooplankton  b i o m a s s a t O.S.P.  The y e a r  w h i c h i s p r e d i c t e d by t h e m o d e l o c c u r s b i o m a s s peak a s a r e s u l t  of v a r i a t i o n  t o year  t o year  variation  variation  i n t h e l a t e C.  cristatus  i n over-wintering  survival  of s m a l l copepods and i s not c o n s i s t e n t w i t h t h e o b s e r v e d variation.  E l i m i n a t i o n o f t h e s e p e a k s , by a l t e r i n g  r a t i o s or changing zooplankton varies  effects direct  biomass which,  little  The  parameters,  from year  will  like  result  recruitment  i n a predicted  the primary  p r o d u c t i o n of F i g 33,  t o year.  simplest e x p l a n a t i o n of t h i s f a i l u r e i s that advective are neglected  effect  i n t h e model.  T h e r e i s good e v i d e n c e  o f a d v e c t i o n on z o o p l a n k t o n  for a  biomass a t p a r t i c u l a r  t i m e s , s u c h a s l a t e 1973 a n d t h e s p r i n g o f 1 9 7 4 , when t h e a p p e a r a n c e o f ' t r a n s i t i o n water'' a t O.S.P. i s a s s o c i a t e d w i t h l o w zooplankton has  biomass  (Fulton,1978).  Such an e f f e c t  t o be r e c o n c i l e d w i t h t h e r e q u i r e m e n t  phytoplankton.  In t h e case  of advection  f o r g r a z i n g c o n t r o l of  of t r a n s i t i o n water,  t h e r e appear t o  235  be  two  rates  possible explanations. are  (Nitrate second  lower  in this  water  concentrations  interesting  quite  different  fact,  recognised  gelatinous recently  been  to  possibility  by  water  the  be  these  phytoplankton  some  unknown  adequate  mass.  appearance  for  for  i s that  (Fulton,1978).  reported  i s that  mass,  appear  in this  forms  One  the  The of  for  A  population  transition and  reason.  growth).  grazer  salps  growth  water  is  i s , in  similar  High  feeding  efficiencies  taxa  (Madin,1974;  have  Harbison  and  McAlister,1979). These in  explanations  zooplankton  biomass  appearance  of  variations  i n the  are  not  variations.  These  advection  to  This  argument  estimation  high  biomass  increase  due of  to  the  unknown  of  high  in  slower  Section  However,  the  model's  recruitment individual  in  possibility of  the  some  could  4.1  extent where  these  the  model  rates  be  by  the  due  the  are  are  either  to success.  parameter large with  themselves  recruitment  the  which  biomass  associated  results  and  annual  over-wintering  results  r e a c t i o n to  the  i s that  for  despite  estimates  variations  l a r g e copepods  responsible  affecting  growth  large  One  changes  factors  the  associated with  m o d e l ..are  to  for  not  recruitment  recruitment  years. with  are  recruitment  results  inconsistent An  spring  suffice  water.  i s supported  uncertainties,  not  which  transition  p r e d i c t e d by  or  do  in  changes.  lower  biomass  weight-dependence  ingestion. If  that  the  primary then  the  the  estimation  phytoplankton production  results  are  control  requirement  does  size-dependence  not of  vary  ignored,  and must  significantly  grazing  per  unit  i t is  accepted  apply, from  biomass  and  year  that  to  is left  year, as  a  236  possible using the as  t h e same  much  from  t o year  An  in mortality  zooplankton  used  increasing 4.3a),  change  here,  thereby  increasing  increasing  effect  stock  wintering  strategies  mortality  rate  weight would  probable secondary argument model,  loss  production against  i n years  increase  biomass None  efficiency  estimates  zooplankton  composition  the simplest  changes i n  in mortality  explanation.  According  phytoplankton standing  stock  production  i n terms  and  to the  production  will  (and t h e r e f o r e  The r e s u l t  secondary  rate  of  additional  e x p l a n a t i o n which  net  the over-  an  or parameters  of  counteracts  The e s t i m a t e s  should  be a  production.  are high  o f low biomass  explanations  to the  (equation  This  small  provide  decrease.  of secondary  C  4.3b) a n d l i t t l e  changes  and h i g h e r  and low i n y e a r s  of these  rate  mortality,  will  expected  o v e r - w i n t e r i n g t i m i n g or  control.  and zooplankton  metabolism)  trophic  However,  of high  be  over-wintering behaviour  i n S e c t i o n 4.1  a mortality  slightly  zooplankton higher  of phytoplankton  vary  has the e f f e c t  carbon  relatively  unrealistic  V I a n d V2  according  Further, with  Large  looking at  size-dependence  rate  production.  either  by  intuitively  In f a c t ,  (equation  here,  variation  i n biomass.  phytoplankton  copepods.  in very  :  might  results.  used  ( F i g 63)  the mortality  primary  decline both  stock.  severely affect  of the major result  rate  of m o r t a l i t y  i n standing  f o r annual  Again,  variations  standing  the equilibrium  direct  fact,  year  the annual  increase  tested  VI and V2.  not e x p l a i n  models  high  c a n be  as f o r the August  variables  as biomass  lower  the  This  technique  'ingestion'  will  to  explanation.  i n years of  (Table VI)  of v a r i a t i o n  i s really  in  satisfactory.  i s consistent  with the  In  25 n  20 -  15-  Figure  63a.  Annual v a r i a t i o n i n i n g e s t i o n  variable  VI  (10  day  means).  ho  239  grazing control hypothesis, estimation results production,  t h e model and t h e parameter  i s that annual v a r i a t i o n s i n primary  n o t p r e d i c t e d by t h e model o f C h a p t e r 3, a r e  responsible.  The p h y t o p l a n k t o n  growth model i s d r i v e n  with  observed s o l a r r a d i a t i o n , mixed l a y e r depth and temperature. i s p o s s i b l e t h a t t h e model i s n o t s u f f i c i e n t l y  It  s e n s i t i v e t o these  d r i v i n g v a r i a b l e s a n d t h a t l a r g e r a n n u a l d i f f e r e n c e s may be predicted  fordifferent  phytoplankton  increasing the r e s p i r a t i o n mixed l a y e r depth  rate w i l l  F o r example,  sensitivity to  also result  i n w i n t e r , as d i s c u s s e d  i n a greater  earlier.  i n Chapter 3 that secchi depth could a f f e c t  production  composition  I t was  primary  but the time s e r i e s i s not adequate t o look  r e l a t i o n s h i p with zooplankton  biomass.  f o ra  Changes i n s p e c i e s  a n d g r o w t h p a r a m e t e r s c o u l d a l s o be r e s p o n s i b l e f o r  changes i n primary data  increase  : however, i t w i l l  l o s s of phytoplankton noted  parameters.  production  : the lack of species  a n d t h e p r o b l e m s i n v o l v i n g *C m e a s u r e m e n t s have 1  been d i s c u s s e d  i n C h a p t e r 3.  The p o s s i b i l i t y  l i m i t i n g m i c r o - n u t r i e n t whose s u p p l y must be c o n s i d e r e d ,  although  composition already  that there  v a r i e s from year  t o year  t h i s a n a l y s i s of the  O.S.P. e c o s y s t e m r e s t s on t h e a s s u m p t i o n t h a t n u t r i e n t does not o c c u r .  As d i s c u s s e d  i n C h a p t e r 1, t h e  a p p e a r t o be a b u n d a n t a t a l l t i m e s , against micro-nutrient  limitation  experiment of M c A l l i s t e r for  phytoplankton  of p h y t o p l a n k t o n  i s some  limitation  macro-nutrients  but the p r i n c i p a l  evidence  i s the s i n g l e culture  e t a l (1960).  Again,  there  i s a need  g r o w t h s t u d i e s a t O.S.P. i n c l u d i n g t h e c u l t u r e i n flasks with grazers  which r u l e out the inadvertent  supply  excluded  under c o n d i t i o n s  of m i c r o - n u t r i e n t s .  240  A correlation  between z o o p l a n k t o n  a b u n d a n c e a t O.S.P.  and  c o m p u t e d Ekman t r a n s p o r t c o m p o n e n t s a t 50 N,160°W some months 9  p r e v i o u s l y was f o u n d by W i c k e t t ( 1 9 6 7 ) f o r t h e y e a r s As  i n a l l such s t a t i s t i c a l  a causal  link  their  advection  difficult  c o r r e l a t i o n s , i t i s not c l e a r whether  i s responsible.  of m a c r o - n u t r i e n t s  Wickett  through upwelling  to the v i c i n i t y  suggested that the supply  i n the Alaskan  o f O.S.P.  It is  with the apparent  i n a l l years.  d i s c u s s i o n s o f a r h a s c e n t r e d on t h e a r e a s o f  d i s a g r e e m e n t between p r e d i c t i o n and o b s e r v a t i o n implications Various  Gyre, and  was i n v o l v e d .  t o r e c o n c i l e s u c h an e x p l a n a t i o n  o v e r - a b u n d a n c e o f n u t r i e n t s a t O.S.P. The  1957 t o 1 9 6 4 .  f o r t h e model and f o r f u t u r e f i e l d  possible a l t e r a t i o n s or extensions  s u g g e s t e d a n d t h e r e a r e many o t h e r s  and t h e i r reserach.  t o t h e m o d e l h a v e been  w h i c h c o u l d be d i s c u s s e d .  T h r e e t o p i c s w h i c h have r e c e n t l y r e c e i v e d c o n s i d e r a b l e a t t e n t i o n in  the modelling  o f m a r i n e s y s t e m s , a n d have n o t been i n c l u d e d i n  t h i s model f o r l a c k of i n f o r m a t i o n , d e s e r v e a t l e a s t a b r i e f mention. The by  size  s t r u c t u r e of phytoplankton  S t e e l e and F r o s t  zooplankton series  Although  there  i n f o r m a t i o n on p h y t o p l a n k t o n  phytoplankton might normally by  (1977) and found t o s i g n i f i c a n t l y  dynamics.  a t O.S.P., v a r i o u s  was i n c l u d e d i n a m o d e l  i s no q u a n t i t a t i v e t i m e  s i z e or species  isolated observations  there are predominantly  affect  suggest  small  composition  that  flagellates.  be a t t r i b u t e d t o s e l e c t i v e g r a z i n g on l a r g e  t h e dominant l a r g e c a l a n o i d copepods, b u t , a c c o r d i n g  (1978),  This  C. p l u m c h r u s h a s an u n u s u a l l y  f a c t , S t e e l e and F r o s t  fine  filtering  cells  to Frost  mesh.  (1977) a t t r i b u t e d t h e dominance of  In  241  C. p l u m c h r u s o v e r  s m a l l e r c o p e p o d s a t O.S.P. t o i t s a b i l i t y t o  feed e f f i c i e n t l y  on s m a l l c e l l s .  microzooplankton  samples taken  The raw d a t a  from  d i v e r s e assemblage of l a r g e diatoms concentrations  (<500 c e l l s . I " ) .  from  the weathership  reveal a  and d i n o f l a g e l l a t e s a t low  The c o n d i t i o n s a t O.S.P. o f a  1  d e e p m i x e d l a y e r a n d a b u n d a n t n u t r i e n t s m i g h t n o r m a l l y be expected  t o favour faster-growing diatoms.  interesting  s e t of problems i n v o l v i n g phytoplankton c o m p e t i t i o n  a t O.S.P. w h i c h may be a d d r e s s e d model a f t e r growth  by a s i z e o r s p e c i e s - s t r u c t u r e d  shipboard experiments  and g r a z i n g .  on s i z e o r  such a model.  species-dependent  I t i s n o t c l e a r how t h e c o n c l u s i o n s r e a c h e d  here c o n c e r n i n g t h e p h y t o p l a n k t o n - g r a z e r for  T h e r e i s c l e a r l y an  i n t e r a c t i o n would change  As a p o s s i b l e e x a m p l e , w i t h a s i n g l e  p h y t o p l a n k t o n v a r i a b l e , a d e l i c a t e c o m p e t i t i v e b a l a n c e h a s been r e q u i r e d amongst t h e t h r e e z o o p l a n k t o n g r o u p s t o e n s u r e transition  of g r a z i n g c o n t r o l .  Some f o r m o f r e s o u r c e  partitioning  by s i z e amongst t h e s e g r o u p s c o u l d r e s u l t  r o b u s t model  behaviour.  The  importance  carnivores  f r o m a b o v e h a s been e m p h a s i z e d r e c e n t l y  and Greve,1977;  rate  i n t h e model i s c e r t a i n l y  The i n f o r m a t i o n on p r i m a r y c a r n i v o r e s a t O.S.P. i s  of v a r i a b l e q u a l i t y . time s e r i e s  (Sonntag  dynamics  R e p r e s e n t a t i o n o f p r e d a t i o n on h e r b i v o r e s  as a c o n s t a n t m o r t a l i t y unrealistic.  i n more  of the s i z e and/or s p e c i e s c o m p o s i t i o n of  i n c o n t r o l l i n g h e r b i v o r e and p h y t o p l a n k t o n  H a r r i s e t a l ,1980).  a smooth  The 150m v e r t i c a l h a u l s p r o v i d e  reasonable  f o r t h e p l a n k t o n i c forms, a l t h o u g h t h e l a r g e r  may be u n d e r - s a m p l e d due t o n e t a v o i d a n c e p l a n k t o n i c c a r n i v o r e s , dominated  (Heath,1977).  by t h e medusae, A g l a n t h a  forms These  242  digital,  the chaetognath S a g i t t a e l e g a n s ,  Thysanoessa r a s c h i i principally  on  the  and  poorly  the amphipod P a r a t h e m i s t o ,  Other nektonic  s q u i d , whose a b u n d a n c e and  known, may  copepods.  euphausiid  s m a l l e r copepods or e a r l y l i f e  of l a r g e r c o p e p o d s . f i s h and  the  be more i m p o r t a n t  F u n c t i o n a l response data  l a c k i n g , w i t h the p a r t i a l  exception  may  feed  history  c a r n i v o r e s such as  stages  myctophid  p o p u l a t i o n dynamic's a r e predators  on  the l a r g e r  for a l l these of t h e  taxa i s  chaetognaths  (Sullivan,1980). I n t r o d u c t i o n o f more r e a l i s t i c  c a r n i v o r e d y n a m i c s , when  possible, w i l l  almost c e r t a i n l y a f f e c t  reached  However, the a p p a r e n t t i g h t c o u p l i n g  here.  phytoplankton hypothesis and  and  grazers  r e q u i r e d by  some of t h e  For  of  the g r a z i n g c o n t r o l  w o u l d seem t o r e q u i r e some d e c o u p l i n g  carnivores.  conclusions  example, the assumption  between  grazers  i n C h a p t e r 1 of  a  h e r b i v o r e m o r t a l i t y r a t e which i n c r e a s e d w i t h biomass r e s u l t e d i n very u n r e a l i s t i c  behaviour  of p h y t o p l a n k t o n  l i m i t e d c o u p l i n g b e t w e e n h e r b i v o r e s and  standing  stock.  c a r n i v o r e s may  desirable.  A c o r r e l a t i o n between r e c r u i t m e n t  a r i s i n g out  of a c a r n i v o r e r e s p o n s e e i t h e r  and  departure  w i n t e r , would tend times  with a constant  or weights  t o numbers o f  mortality rate.  f e e d i n g phenomena on the  sensitivity  to reduce the  to recruitment  again  of occurs  or  threshold  decrease  model d u r i n g the d e p a r t u r e  by m a i n t a i n i n g minimum s t o c k s o f s m a l l The  l e v e l s which  Switching behaviour  small  i n the  sensitivity  t h e p a r t o f c a r n i v o r e s may  of the p r e s e n t  be  mortality rate,  copepods or t o the o v e r - w i n t e r i n g copepod p o p u l a t i o n s preceeding  Some  period  copepods.  i m p l i c a t i o n s of h o r i z o n t a l v a r i a t i o n or p a t c h i n e s s  in  243  plankton also  populations  received  questions (a)  How  much  have  for theories  attention  generally  i s patchiness  of p o p u l a t i o n  i n the last  been  addressed  i n plankton  i n t e r a c t i o n s has  decade.  Two , t y p e s  by t h e s e  populations  of  studies:  created  and  maintained? (b)  How  does  properties have  been  the existence  of p o p u l a t i o n based  of patchiness  affect  i n t e r a c t i o n s , such  on m o d e l s  which  ignore  conclusions  as s t a b i l i t y ,  horizontal  about  which  spatial  effects? Plankton  populations  to  turbulent  mixing  as  smoothing  out s p a t i a l  natural linear  one.  diffusive analysis below This  length  If  of  diffusive scale  has been  processes  grazing linear  reducing  increasing considered,  L . c  of a c r i t i c a l  from  the patch  using  rate  patch  at smaller  depending  length  is a a  simple  This  dimension,  outweigh  evidence  of  in a  1953).  t o be d f o r d e r  i s empirical  thought  L , c  growth. a few  on  growth  that  spatial  scales  i s determined  reduction  i n net growth  (Piatt,1972). of simply  phytoplankton at small  I f growth  a non-linear  of such  fixed  and Slobodkin,  estimated  There  i s thought  patchiness  at a  subject  question  theoretically  of kilometres,  i n phytoplankton  in a  analysis  losses  up t o h u n d r e d s  physical  rate  to the idea  (Steele,1976).  variation by  rise  so the f i r s t  growing  (Kierstead  a r e by d e f i n i t i o n  are traditionally  addressed  of phytoplankton  gave  which  which  variation,  first  environment  kilometres rates  processes  I t was  model  i n the ocean  a system  as a  model,  i t should  length  and d i f f u s i o n  scales;  have  that i s ,  of zooplankton  r e a c t i o n - d i f f u s i o n system i s much  the effect  more d i f f i c u l t .  are  results. For  small  The  244  perturbations approximate applied.  about  linear  For  a  spatially-uniform steady-state,  system  equal  can  be  diffusion  derived  rates  a  Fourier  analysis  phytoplankton  diffusion  again  has  steady-state  is stable  under  spatially-uniform perturbations,  stable  is  unstable  critical For  under  under  length  unequal  models, stable  perturbations  the  which  diffusion  rates  and  linear  theory  uniform of  spatial  aroused  great  deal  of  (Turing,1952;  Segel  Okubo,1974).  Unequal  migrate  truly  variations transfer  linear been  of  terms,  argued  variations  on  state  nature  of  known  small which  under  horizontal variation  (Steele,1976).  If  with a  more  other or  in  i t  of  out.  reaction can  be  diffusionhas  fields  Kopell  Howard,1973;  and arise  shear  in  the  length  spatial the  via  constant  It  has  a  uniform  perturbations.  vicinity such  non-  large-amplitude  destabilize  uniform  possibility  the  turbulence.  correlated,  could  and  .  scales  of  ocean  dispersion  is obviously  regions  less  damped  large-amplitude  the  If a  instability,  could  highly  i t  v a r i e t y of  cascade  scales  is stable  compared  of  the  under  This  ( E v a n s , 1978)  across  that  length  unstable  by  There  energy  (Steele,1974)  steady  well  the  rates  analysis  difficult.  in  in a  Jackson,1972;  variability as  but  i s dominated  are  steady-state  diffusive  interest  vertically  non-linear  i s very  or  i s again  certain class the  If  wavelengths.  there  wavelength.  diffusion  h o r i z o n t a l mixing  A  of  and  a  that  perturbations  variation,  effect.  perturbations  for  shows  intermediate  generated  zooplankton  perturbations,  below  perturbations  where  a l l finite  scale  under  a  uniform  at  smoothing  and  zooplankton,  is  a  for  and  an  as flow  of the  O.S.P. North  past  The is  not  Sea  O.S.P.  of  245  about  3 km.day"  about  spatial variability  time-series.  i s accepted  1  (see Chapter  1 ) , some i n f o r m a t i o n  c a n be d e d u c e d f r o m t h e w e a t h e r s h i p  The h i g h - f r e q u e n c y s a m p l i n g o f s u r f a c e c h l o r o p h y l l  f r o m 1964 t o 1968 ( F i g 13) r e p r e s e n t s s a m p l e s a b o u t The  generally  low v a r i a b i l i t y  there i s l i t t l e Chapter be  6 km a p a r t .  i n these samples suggests  horizontal variation  i n C h i a.  3 t h a t t h e o c c a s i o n a l s h o r t groups  that  I t was n o t e d i n  of h i g h v a l u e s c o u l d  i n t e r p r e t e d a s p a t c h e s o f o r d e r 20 t o 50 km a c r o s s .  typical  l e n g t h s c a l e a t which b i o l o g i c a l  expected  (Steele,1976) .  to explain  This i s a  p a t c h i n e s s might  be  With t h e models used h e r e , i t i s e a s i e r  these h i g h c h l o r o p h y l l o b s e r v a t i o n s as patches  than s h o r t - l i v e d , wide-spread  blooms.  as t h e s e a r e not c l e a r .  There  p h y s i c a l patches  length scale  on t h i s  rather  Causes of outbreaks  i s evidence  such  f o r c h e m i c a l and  (McAllister  e t a l ,1960;  Miyake,1979) According to the l i n e a r  t h e o r i e s , p e r t u r b a t i o n s on l a r g e  enough l e n g t h s c a l e s b e h a v e a s a s p a t i a l l y p r e d i c t a n d i t h a s been a r g u e d stability  similar  t o those reached h e r e , based  i s taken i n t o account in a spatially  dimension  exceeded  which variation  A possible exception could  region i f the c r i t i c a l  patch  t h e s i z e of the r e g i o n (Steele,1974) but t h i s  d o e s n o t seem l i k e l y  f o r a system  P r o p e r t i e s of u n s t a b l e systems, markedly  on m o d e l s  r e l e v a n t when s p a t i a l  (Steele,1974) . restricted  would  that conclusions concerning  ignore s p a t i a l e f f e c t s , are s t i l l  occur  uniform theory  with the introduction  on t h e s c a l e o f t h e S u b a r c t i c .  such as p e r s i s t e n c e , can change of s p a t i a l  variation  ( H i l b o r n , 1 9 7 5 ) , b u t s u c h phenomena d e p e n d i n t r i n s i c a l l y amplitude short-term f l u c t u a t i o n s  on  large-  i n space and time which a r e not  246  observed  f o r O.S.P.  Summary. This  study  has taken  a  phytoplankton-zooplankton explore based  them  on  summarised (a)  What  have  has been  been  rates.  raised  Parslow  copepods, range  learnt  of t h i s  questions  about  regarding  and attempted of a  to  model  and i n f o r m a t i o n from t h e study  c a n be  :  t h e O.S.P.  questions  during  the  cycle  data  ecosystem  as a  dynamics again  data  inconsistencies 3.  has y i e l d e d standing  ecosystem  information stock and  by t h e t y p e s i n some  The d i r e c t  the parameter  and secondary  t h e O.S.P.  study?  imposed  i n Chapter  using  about  of phytoplankton  et. a l ( 1 9 7 9 ) h a s p r o v i d e d  population  o f raw d a t a  related  The l i m i t a t i o n s  at length  zooplankton  the l i t e r a t u r e  of phytoplankton  and apparent  discussed  two  a t O.S.P.  t h e framework  The r e s u l t s  a r e t h e open  analysis  measured  from  hypothesis  study?  the seasonal  growth  within  analysis  series.  by a n s w e r i n g  of the  The  data  of  variables  have  analysis  been of  estimation technique of  i n f o r m a t i o n about  the  p r o d u c t i o n of the major  subject to the limitations  imposed  by t h e d e p t h  sampled. As  should of  time  What  (b)  about  rigorously  by t h e o r i g i n a l  weathership  which  interactions  information taken  obtained  result  more  s e t of v e r b a l  the data perhaps  this  study  information ignored,  used  here  be s t r e s s e d  a r e f o r t h e most at this  and of the author  i n the weathership  either  as i r r e l e v a n t  point  part  that  h a s meant zooplankton  unpublished, i t  the modelling  that time  much  of the  series  t o the q u e s t i o n s posed  bias  has been  here,  or as  247  unusable species  until  further  concerned  exhaustive  is available.  a n a l y s i s of  information  on  zooplankton  species  of  other  data  series  prove for  herbivore and  and  will  i s obtained  and  carnivore  modelling  concerning  grazing  thresholds  and  qualitative potential  Nor  an  to  -Summaries  abundance  of  shortly  has  the  exhausted  feeding  species,  depths  component control  life  a n a l y s i s and of  seasonal  in affecting  and  more  a  of  the  potential  in  basic  number  of  application to  of  : as.further  growth  the  large  and  complex  these  the  time  laboratory  various  models  of  discussed.  Perhaps  more  both  model The  of  and  Details winter emerged  data  questions  interest of  the  as  may  spring  the  be  built  a  the  dominant  of  the  guide  to  future  uncertain  growth and  August  number  by  and  of  of  model  important.  importance  emphasized rates and  secchi  has  data  growth  i n August  research  and  Attention  effect has  where  should at  O.S.P.  particularly  September,  needs  been  puzzles.  analysis  cycle,  The  which  September  interesting  and  in  destabilising  and  the  production.  herbivores  of  The  herbivores  biological  seasonal rates  the  questions  addressed.  in primary  been  months  the  production,  has  raised a  be  recruitment  of  and  respiration  primary  novel  indicate  to  allowed  s t u d i e s have  variations  phytoplankton  production as  of  departure  has  phytoplankton  simulation  annual  a t t e n t i o n on  study  strategies  cycles  effects  focused  this  phytoplankton  stabilizing  over-wintering  of  of  history  importance  determining  be  i s made  the  tested. The  of  of  multivariate statistics,  been  on  autecology  claim  published  as  models  the  series.  annual  be  such  on  No  time  illuminating.  testing  information  this  seasonal  techniques,  may  information  to  be  have paid  248  to  phytoplankton  micronutrient extinction and/or a  limitation.  coefficients  In needed  the to  vertical  subject case  of  outstanding  questions  in this  studies  the  interpretive  ignored  poor  series  time  may  species'  be  predominate  (McAllister  detritus  play  important  planktonic  likely  seasonal  samples  and  collection help  questions  addressed  to  are  and  resolve  concerning  directly  (1979),  The  by  of  the  grazing grazing  although  microzooplankton  have  i n f o r m a t i o n and  partly  because  of  ignorance  and  in winter  where  and  characteristics. detrital-based  Kennedy,1972).  non-phytoplankton ,1960) and pers.  more  the  comm.),  between  In  particulate  seasonal  attention.  transparency,  constant  growth  the  cycle  food view  organic in this  may  and  also  be  the  material of  growth  secchi  depth  chlorophyll  level.  The  microzooplankton  the  dynamics  i n any  carnivores,even  study  i f their  of  role  in  high  carbon  as  role  may  an  phytoplankton varies  of  group  chains of  been  relatively  This  energetic role  There  detritus  be  diurnal  p o p u l a t i o n dynamics  Frost  to  the  feedback  a  size  of  deserves  despite  appears  stratifed  should  remain.  The  light  because  K.Ishi,  water  study.  in  partly  of  through  be  variation  study,  et. a l  and  interesting  an  in this  (LeBrasseur  concentrations  (C.S.Wong  may  feeding  important  can  the  of  problems.  with  the  d i s c u s s e d by  problems  virtually  together  concerning  study  type  of  deeper  surrounding  Obviously,  thresholds  seasonal  future research  These,  possibility  phytoplankton  naupliar stages,  herbivores.  of  and  the  worthy  zooplankton,  migration. of  also of  resolve problems  identification  these  are  for  r a t e s and  Annual  species composition  productive  major  respiration  seasonally  of  the  phytoplankton  will  249  regulation  proves  Although  i t would  to  the question  be  unrealistic  knowledge,  marginal. be p l e a s a n t  : 'What  i s really  to expect  and models  one.  considered  to consider  information  and of the p o s s i b l e q u e s t i o n s .  this  model  biological theoretical  study,  problems  subset  by y i e l d i n g  considered,  of t h i s  gaps  here  i t would  i n our  are necessarily  fresh  I t i s t o be  insights  and r a i s i n g  interesting  a t O.S.P.?',  answer  of the a v a i l a b l e  and e m p i r i c a l , can c o n t r i b u t e  investigation  a definitive  are large  constrained  that  small  happening  There  of the type a  to provide  new  to a  ecosystem.  into  hoped the  questions, long-term  both  250  CHAPTER 5 PARAMETER ESTIMATION AND  5.1  S T A B I L I T Y FOR  A CEPEX ENCLOSURE.  Introduction. In Chapter  2, a q u a l i t a t i v e a n a l y s i s o f t h e  stability  p r o p e r t i e s o f a s i m u l a t i o n model o f t h e N o r t h Sea L a n d r y , 1 9 7 6 ) was that,  presented.  T h i s a n a l y s i s l e d t o the c o n c l u s i o n  following nutrient depletion,  be e x p e c t e d a nutrient approach  stable cyclic  b e c a u s e of t h e l i m i t a t i o n f l u x due  to mixing.  to t h i s cycle,  d e n s i t i e s was  found  (Steele,1974;  behaviour  of p h y t o p l a n k t o n g r o w t h  However, d u r i n g the  the response  t o be c r i t i c a l .  of z o o p l a n k t o n  As low  indicated  t o low  food d e n s i t i e s ,  response  including a longer-term  a c h i e v e t h e same e f f e c t . responses  m i g h t be d i s t i n g u i s h e d  suggested  food The  to  there that these  i n long-term c u l t u r e  experiments  sufficiently  densities. CEPEX e x p e r i m e n t s  Saanich I n l e t  f r o m 1974  p e r t u r b a t i o n and plastic  response  rate, could  (eg P a f f e n h o f f e r , 1 9 7 0 ) i f t h e s e were c a r r i e d o u t a t low  by  physiological  increase in mortality I t was  used  during this period.  2, o t h e r t y p e s o f z o o p l a n k t o n  or a rapid-enough  food  A behavioural feeding  reasonable behaviour  i n Chapter  by  transitory  t h r e s h o l d c o m b i n e d w i t h a z e r o b a s a l m e t a b o l i c r a t e was Steele(1974) to ensure  could  ( G r i c e e t a l ,1977),  t o 1978,  conducted  i n v o l v e d the c a p t u r e ,  o b s e r v a t i o n of c o l u m n s o f w a t e r  enclosures.  enclosed ecosystem  in large,  By a l l o w i n g t h e m a n i p u l a t i o n and i n a l a r g e volume  s e v e r a l months, these e x p e r i m e n t s  in  (1300  m) 3  study of  o v e r a p e r i o d of  were i n t e n d e d t o b r i d g e t h e  b e t w e e n l a b o r a t o r y s t u d i e s of a few  an  s p e c i e s a t one  o r two  gap  trophic  251  l e v e l s and  field  i n v e s t i g a t i o n s where i t i s d i f f i c u l t  p a r t i c u l a r p l a n k t o n i c community over In  an e x p e r i m e n t  O c t o b e r o f 1976, and  adding  S o n n t a g and behaviour  An  t h e e f f e c t s of u p w e l l i n g  Of  r e s u l t s has  particular  A f t e r an  phytoplankton  dropped to very  relatively  constant  estimated  zooplankton  concentration  initial  Chapter  (CEE5) w h i c h bloom, the  that  A systems-identification p o p u l a t i o n p a r a m e t e r s has  originally  size class  a t low,  on  t h e CEPEX t i m e  this  feeding to  constant  levels.  raised  technique  f o r e s t i m a t i n g copepod  a l r e a d y been d e s c r i b e d i n C h a p t e r  f r o m O.S.P.  The  s e r i e s of  This technique  stage  f r o m a CEPEX  4  and  copepod  was  on o b s e r v a t i o n s  e s t i m a t i o n procedure  of  enclosure  performed p o o r l y  s e r i e s compared w i t h s i m u l a t e d d a t a .  tentatively attributed either  in  seemed j u s t i f i e d .  life-history  ( P a r s l o w e_t a_l , 1 9 7 9 ) .  both  zooplankton-phytoplankton  t e s t e d on s i m u l a t e d d a t a and  c o p e p o d d e n s i t i e s by  As  r e f u g e , were n e c e s s a r y  a p p l i e d t h e r e , w i t h some m o d i f i c a t i o n s , t o t i m e d e n s i t i e s by  biomass  nitrate  i m p l i c a t i o n s f o r the g e n e r a l q u e s t i o n s  in this enclosure  was  remained  the average  Parsons(1979) concluded  2, a c l o s e r a n a l y s i s of t h e  interaction  by  remained very h i g h d u r i n g  the p e r s i s t e n c e of p h y t o p l a n k t o n  v i e w of t h e  were  been g i v e n  l e v e l s and  g r a z i n g r a t e s and  t h r e s h o l d s , o r some s o r t of s p a t i a l  In  bubbling)  f o r t h e d u r a t i o n of t h e e x p e r i m e n t .  i n the e n c l o s u r e  S o n n t a g and  low  and  i n t e r e s t here i s the  o f t h e c o m m u n i t y i n t h e c o n t r o l bag  of p h y t o p l a n k t o n  explain  (through a i r  juveniles) to enclosures  a n a l y s i s of the  Parsons(1979).  not b u b b l e d .  period,  time.  c a r r i e d out d u r i n g A u g u s t , September  carnivores (salmonid  investigated.  to follow a  to higher noise l e v e l s  This  than  was  252  expected  i n the  population times)  data,  parameters  i n the  The  CEPEX  or  to  the  ( mortality  variation  rates  and  over  stage  time  of  residence  CEPEX p o p u l a t i o n .  second  serious  implications  for  parameter, e s t i m a t i o n i n g e n e r a l .  These  higher-level  population  parameters  biological  depend  environment to  vary  laboratory  some  sense,  data  average may  a  a  time.  lead  times  to  reasonable  time  individual's I t was  detailed copepods, might  be  scheme  that  can  of  .  to  by  of  that these  functional  expressed  least-squares fitting  parameters  the  i s not  are  assumed  met,  there  r e p r e s e n t any of  the  to to  is  sort  be  no  of  procedure  negative  parameters  stage  et  parameters  al  are  (1979)  numerical  series  of  the  encountered  are  vary,  which  i t  a  response and  may  determine  constant  that  food  life-history  concerning  which  conditions  conditions  and  time  difficulties  single  the  estimate  in  a  1979).  Parslow  observed  of  systems-identification  environmental  t o assume  expected  i t represent, in  non-linear nature  to  the  regulated expected  i s that  assumption  a_l ,  to d e n s i t i e s  were  i n view  et  Where  the by  be  be  of  The  i n which  highly  response  driven fitted  involves  like  suggested  model  Reservations  best  can  n o n s e n s i c a l e s t i m a t e s , even  would  still  time.  tightly  estimates will  (Parslow  one  and  most  this  The  over  the  If  components  the  time.  model  parameter  constant be  over  above  dynamic  time.  Ideally  The  average  that  in fact  residence  i n a l l but  t i m e - v a r y i n g parameter  an  over  and  temperature,food,predators)  time  over  guarantee  physical  mentioned of  constant  on  situations.  of  technique the  (eg  over  estimate  e x p l a n a t i o n has  over  more of •  predators,  stages.  feasibility with  simpler  of  such  a  253  population In and  this  chapter,  numerical  time  series  These  by  a r e used  (a)  data  obtained  of time  from  series  diatoms,  model  from  CEE5.  the question of  i n CEE5,  both  directly  nutrient-phytoplankton-  in a CEE5  Zooplankton which  carbon  (0-20m  C h l o r o p h y l l a and  (c)  nitrate  (d)  zooplankton  a r e used  average)  diatoms,  Model.  in this  paper  1 4  C  group  :  flagellates,  productivity  c o n c e n t r a t i o n b y 4m  by t a x a n o m i c  dinoflagellates,  and u n i d e n t i f i e d  (b)  Growth  of o b s e r v a t i o n s of  pennate  silicoflagellates  identified  growth  model.  phytoplankton  centric  densities  functional  enclosure  to address  i n a combined  5.2 E s t i m a t i o n o f P a r a m e t e r s  consist  i n t h e CEPEX  t o t h e low f o o d them  to estimate  i n a copepod  densities  estimates  inserting  The  i s made  parameters  of copepod  response  zooplankton  an a t t e m p t  response  parameter  copepod and  models.  b y 4m  layer,  layer,  (numbers/liter) as t o t a l  nauplii,  t o s p e c i e s a n d t o one o f t h e g r o u p s  and  copepodids  CI-III,  CIV-V, o r  CVI . The  aim i n t h i s  concentrations species  history  a model  t h e sum o f s q u a r e s  numbers  of zooplankton  phytoplankton growth  of the parameters  i n t h e model  of d e v i a t i o n s between  of zooplankton  i n t h e above  f o r each  groups  a r e many  predicts  forms  numbers  which  a model  in various  of  of zooplankton  life-history  by  p r e d i c t e d and life-  stages.  There which  to drive  i s t o use the observed  and t o o b t a i n estimates  minimising observed  section  stages  growth can take.  254  A d i s c u s s i o n of parameter e s t i m a t i o n v i a systems techniques  i n some s i m p l e  m o d e l s w h i c h do  food c o n c e n t r a t i o n  i n t o account  When t h e e f f e c t of  food c o n c e n t r a t i o n  explicitly,  as  m o d e l w h i c h has  intended as  a weight or stage  here,  c l a s s and  one  assigns  the disadvantage  the  initial  not  allow  for observed  because of  The class  stages  the  on  Z ; ( t ) , and  the  while  and  the  smear  latter  does (eg  used here, p r i m a r i l y  for this  provided  study.  classes.  f  day  For  each  t > t-  (  i n d i v i d u a l w e i g h t by W j ( t ) .  day  is  These  to  Wi  = f(W,,P(t))  5.1a  Zi  = g(W;,Zi )  5.1b  The  initial  number i n day  in  The  i n growth r a t e  t , t h e d e n s i t y on  change w i t h time a c c o r d i n g  Greve,1977)  i t tends to n u m e r i c a l l y  m o d e l u s e d h e r e . i s b a s e d on day  d e n o t e d by  treated  t h e b a s i s of w e i g h t , as  t o S t e e l e ' s m o d e l w h i c h has  day  (1979).  abundance of a g e - c l a s s e s  distribution,  biological motivation  on  of  t r a n s f e r r a t e s between  (eg S o n n t a g and  l a t t e r m o d e l was  its similarity  i , introduced  g r o w t h i s t o be  individual variation  The  e t a_l  v e r s i o n of S t e e l e ' s m o d e l . that  w e i g h t or s t a g e  Paffenhoffer,1970).  much of  food d e n s i t y  (1976) m u l t i - c o h o r t  f o r m e r has  i n Parslow  c a l c u l a t e s the  them t o l i f e - h i s t o r y  the e f f e c t  s t a t e v a r i a b l e s t h e numbers i n  w h i c h f o l l o w s t h e w e i g h t and  Landry's  on  take  a c h o i c e must be made b e t w e e n a  i t s underlying  c l a s s e s a s a f u n c t i o n of and  i s given  not  identification  class i , Z,(t;),  a s s u m i n g t h a t a d u l t s do  not  increase  c a l c u l a t e d d a i l y weight  increment  i s obtained  i n weight but  into nauplii  by  convert  w i t h an  their  efficiency  255  X.  If  , .. . ,W  4  history  represent the i n i t i a l  weights i n the  s t a g e g r o u p s N I - N V I , C I - C I I I , CIV-CV and  respectively,  the t o t a l  g r o u p s c a n be d e f i n e d  life-  CVI  numbers i n e a c h o f t h e s e r e s p e c t i v e  as  where  The possible.  functions  f and g have been c h o s e n t o be a s s i m p l e  Assimilation  i s assumed t o depend  hyperbolically  f o o d d e n s i t y P and a n o n - z e r o b a s a l m e t a b o l i c allowed.  S t e e l e ( 1 9 7 4 ) assumed a W-  0 7  b a s i s of o b s e r v a t i o n s  rate  On  H a r r i s ( 1 9 7 6 ) f o u n d no s i g n i f i c a n t  law f o r b o t h , p a r t l y  deviation  from a w  i n c r e a s e d up t o NV and were more o r l e s s c o n s t a n t  for variation  10  body  to w  (The same g r o u p s o f l i f e - h i s t o r y  were d i s t i n g u i s h e d by P a f f e n h o f f e r ( 1 9 7 0 ) and P a f f e n h o f f e r  of  weight  10  to but  i n growth p a r a m e t e r s a c r o s s t h e groups of  s t a g e s d e s c r i b e d above.  Harris(1976),  and  law f o r  f r o m NVI  I have c h o s e n h e r e t o make g r o w t h p r o p o r t i o n a l allow  on  i n t h e c a s e of  C a l a n u s h e l g o l a n d i c u s , maximum g r o w t h r a t e s p e r u n i t  to  on t h e  the o t h e r hand, P a f f e n h o f f e r  i n g e s t i o n by P s e u d o c a l a n u s e l o n q a t u s a n d , e v e n  CII.  on  i s also  (eg P a f f e n h o f f e r , 1 9 7 0 ) and p a r t l y  general t h e o r e t i c a l grounds.  as  facilitating  t h e use o f t h e i r  t h e CEPEX d a t a . ) T h u s ,  f(W,P) = (Cn.P/(D+P) - F)  .  W  results  stages and  in analysis  256  where C n , D a n d F c a n d e p e n d on W t h r o u g h  their  stage  dependence.  N o t e t h a t t h e p a r a m e t e r Cn r e p r e s e n t s t h e maximum i n g e s t i o n r a t e C, r e d u c e d metabolic  by b o t h a s s i m i l a t i o n e f f i c i e n c y a n d any component o f rate proportional to ingestion,  maximum e x p o n e n t i a l g r o w t h According  s o t h a t Cn - F i s t h e  rate.  t o Sonntag and P a r s o n s ( 1 9 7 9 ) ,  the mortality  i m p o s e d on c o p e p o d s due t o p r e d a t i o n was i n s i g n i f i c a n t  i n CEE5 s o  t h a t c o p e p o d l o s s e s were p r e s u m a b l y due t o ' n a t u r a l m o r t a l i t y ' , possibly Chapter  exacerbated  by low f o o d d e n s i t i e s .  2, c o m p a r a t i v e l y  mortality  little  As d i s c u s s e d i n  i s known a b o u t t h e d e p e n d e n c e o f  r a t e s on f o o d c o n d i t i o n s i n c o p e p o d s .  capita mortality  A constant per  r a t e , 0 , h a s been assumed h e r e w i t h t h e  p o s ' s i b l i t y t h a t e may v a r y o v e r  stage groups being  retained.  q u e s t i o n o f t h e d e p e n d e n c e o f e on f o o d c o n d i t i o n s w i l l  The  be  returned t o i n the d i s c u s s i o n . The  phytoplankton  model.  data  The o b s e r v a t i o n s  f r o m CEE5 were u s e d t o d r i v e t h e  i n F i g 64 r e p r e s e n t p g  c a r b o n / l i t e r , averaged over from c e l l levels  counts.  ( l e s s than  accuracy  of these  phytoplankton  data  phytoplankton  t h e w h o l e bag (0 - 20m), a s e s t i m a t e d  The a v e r a g e c o n c e n t r a t i o n f a l l s 5 pg C . l " ) 1  observations  a f t e r d a y 30. i s called  from the e n c l o s u r e .  t o very low  Unfortunately, the  i n t o q u e s t i o n by o t h e r The  observed  c o n c e n t r a t i o n s o f C h i a f r o m day 30 on a r e t o o h i g h t o be c o n s i s t e n t w i t h the carbon low  a s 6 p g C.pg C h i a "  a" .  The e s t i m a t e s o f  1  C  productivity  _ 1  .hr  _ 1  .  10 p g C.pg C h i  i n t h e t o p 8m o v e r t h e  same p e r i o d r a n g e a s h i g h a s 10 p g C . l ^ . h r " pg C . l  C:Chl a r a t i o s as  and g e n e r a l l y l e s s than  1  1 4  estimates, yielding  1  and average about 5  T h e s e a r e b a s e d on 4 h r i n c u b a t i o n s a n d , even  257  Figure 64.  Phytoplankton carbon (0-20 m average) i n CEE5 ( s o l i d l i n e estimated from c e l l counts, dashed l i n e represents corrected time stream used to drive estimation models).  258  allowing  f o r an e x p o n e n t i a l i n c r e a s e i n p h y t o p l a n k t o n  carbon  d u r i n g t h e i n c u b a t i o n s , when c o m b i n e d w i t h t h e e s t i m a t e s o f phytoplankton carbon h i g h growth The suggest Apart  1 4  c o u n t s , they y i e l d  r a t e s , of order 1 h r  C  -  1  unreasonably  and h i g h e r .  and C h i a o b s e r v a t i o n s a r e c o n s i s t e n t and both  that the estimated phytoplankton carbon  from  carbon  from c e l l  i s too low.  the usual u n c e r t a i n t i e s a s s o c i a t e d with e s t i m a t i n g  from c e l l  counts  (eg M u l l i n  that phytoplankton carbon t h i s c a s e by t h e f a c t c o n s i s t e d almost  the p o s s i b i l i t y  has been u n d e r e s t i m a t e d  t h a t the phytoplankton over  entirely  p r e s e r v a t i o n problems  e t a l ,1966),  of s m a l l f l a g e l l a t e s ,  i s increased i n this period  so t h a t  ( S m e t a c e k e t a_l ,1980) may h a v e l e d t o an  underestimate. It  will  be assumed h e r e  underestimated results  i n F i g 64 by a f a c t o r o f 5.  growth  rate  1  w i t h an a v e r a g e  that half  or 2 d i v i s i o n s per day, assuming  t o d r i v e the zooplankton growth variation  d a t a , as carbon  guide  1  time s e r i e s of p h y t o p l a n k t o n carbon c a n model, t h e q u e s t i o n of  i n p h y t o p l a n k t o n d e n s i t y must be  only guide t o t h i s v e r t i c a l  average.  The  (1000 t o 1400 h r s ) .  Before the r e s u l t i n g  The  v a l u e o f 37.  t h e d a i l y p r o d u c t i o n o c c u r s d u r i n g t h e p e r i o d of  incubation  vertical  assumption  i n t h e t o p 8m o v e r t h e same p e r i o d h a s been  e s t i m a t e d t o be a b o u t 1.4 d a y "  be u s e d  This  i n c o r r e c t e d C : C h i a r a t i o s a f t e r d a y 30 i n t h e r a n g e  25 t o 60 jug C.ug C h i a " average  that f l a g e l l a t e carbon i s  variation  addressed.  i s the c h l o r o p h y l l  e s t i m a t e s a r e o n l y a v a i l a b l e a s a 0 t o 20m  The c h l o r o p h y l l d a t a h a v e been u s e d h e r e a s a  rough  t o r e l a t i v e c o n c e n t r a t i o n s of phytoplankton carbon a t  259  d i f f e r e n t depths, although  the l i k e l i h o o d  : Chi a r a t i o with depth i s recognised.  of v a r i a t i o n s i n the C The e a r l y p a r t o f t h e  e x p e r i m e n t was c h a r a c t e r i z e d by an i n t e n s e d i a t o m b l o o m a t t h e s u r f a c e w h i c h sank o u t a s a c o n c e n t r a t e d became l o c a l l y phytoplankton average.  depleted.  l a y e r as n u t r i e n t s  T h u s , maximum l o c a l c o n c e n t r a t i o n s o f  o v e r t h i s p e r i o d were much g r e a t e r  However, i t w i l l  be assumed h e r e t h a t z o o p l a n k t o n  the average carbon c o n c e n t r a t i o n p e r i o d a s t h i s was a l r e a d y g r o w t h so t h a t  i n the enclosure  h i g h enough t o a c h i e v e  affect  the conclusions  From d a y 22 o n , C h i a c o n c e n t r a t i o n s o v e r t h e t o p 12m, d e c r e a s i n g  while phytoplankton  below t h i s d e p t h .  almost  reached  I f these  a r e l o w and r e l a t i v e l y  1  and a p p a r e n t l y  light  clearance  r a t e s were  to a very  limited  evenly population  below  t h e s m a l l c h l o r o p h y l l g r a d i e n t s and low m i x i n g  ( S t e e l e and Farmer,1977) o b s e r v e d t h e r e f o r e assumed t h a t , a f t e r  i n the enclosure.  day 22, g r a z i n g  This i s 8m,  rates I ti s  occurs  predominantly  i n t h e t o p 8m a n d t h e c o n c e n t r a t i o n o f  phytoplankton  carbon there, obtained  the e n c l o s u r e  and t h e v a r i a t i o n  using  matched  (Sonntag  high net l o s s rate.  i n c o n s i s t e n t w i t h the p e r s i s t e n c e of p h y t o p l a n k t o n  food.  level,  ( g r e a t e r than 1 d a y " )  over the water column, t h e phytoplankton  b e l o w 8m w o u l d be s u b j e c t  available  here.  r a t e s on t h e p a r t o f t h e z o o p l a n k t o n  Parsons,1979).  given  maximum  These h i g h growth r a t e s a r e a p p a r e n t l y  by h i g h c l e a r a n c e  distributed  this  s l o w l y below t h i s  growth r a t e s a r e h i g h  i n t h e t o p 8m b u t a r e much l o w e r  and  over  saw  i n c r e a s i n g the a v a i l a b l e food d e n s i t y f u r t h e r  would not s i g n i f i c a n t l y  uniform  t h a n t h e 0-20m  the average carbon f o r  of C h i a w i t h depth, i s used as  The t i m e s e r i e s o f . f o o d d e n s i t y o b t a i n e d  in this  260  way Fig  and  now  used t o d r i v e the z o o p l a n k t o n  64. While  the aim  zooplankton  here i s t o e s t i m a t e parameters i n the  model d e s c r i b e d a b o v e by  s e r i e s of o b s e r v a t i o n s p a r a m e t e r s can transition  be e s t i m a t e d  w e i g h t s , Wj,  the e s t i m a t e d  zooplankton  parameters,  must be  the  For example,  those The  the  s p e c i f i e d as a c h a r a c t e r i s t i c  For  the  e a c h of t h e t h r e e  t h e n be  fitted  t o t h e l i t e r a t u r e v a l u e s and  therefore f o r the those  t o the  The  objective fitting  p o s s i b l e h e r e by identification  data  (1979).  of the model t o the d a t a  Briefly,  of t h e n o n - l i n e a r  g u e s s a t t h e p a r a m e t e r v a l u e s and  o f e r r o r s b e t w e e n p r e d i c t i o n s and  non-linear optimisation algorithm. ( M a r q u a r d t , 1 9 6 3 ) has t h i s case  The  been u s e d h e r e and  is relatively  stage  groups;  Parslow  of  observations using a  Marquardt  technique  i t s implementation  in  exception.  thresholds to assign  that i s  e_t  initial  r e d u c i n g t h e sum  s t r a i g h t - f o r w a r d w i t h one  model as d e s c r i b e d above uses weight c l a s s e s to observed  systems  i n v o l v e s m a k i n g an iteratively  i n them.  i s made  d e s c r i b e d i n B e n s o n ( 1 9 7 8 ) and  the procedure  their  confidence  m i g h t l e a d us t o p l a c e  the a p p l i c a t i o n  technique  the degree of  size  principal  w h i c h a r e w e l l - e s t a b l i s h e d and  model w i l l  of  species  i n the e n c l o s u r e , I w i l l  which the e s t i m a t i o n procedure  squares  t h a t not a l l the  of l i t e r a t u r e - e s t a b l i s h e d v a l u e s  indicating  time  r e s u l t i n g p a r a m e t e r e s t i m a t e s d i s c u s s e d i n t e r m s of  relation  al  i t to the  : t h e i r values determine  species present  which are u n c e r t a i n . and  i n t h i s way.  growth r a t e s .  begin w i t h a review  fitting  f r o m CEE5, i t i s c l e a r  the s p e c i e s i n q u e s t i o n and  model i s a l s o g i v e n i n  day  The  261  Yj{t)  =  I Z (t).^ (W (t))  where  the  while  Wj(t)  i  and  Z;(t) are  through  The  derivatives  of  calculated,  and  problem  been  has  Here,  a  about  Wj  is a and A  reasonable  5.3  used by  small W  -  (7 =  convergence for a  0.1 of  given  The discussed  of  the  the  result,  predicted functions  that  to parameters  these  of  the be  circumstances.  This  step-functions  <f>j by  As was  )/(cr.W  J + 1  which  the  from  the  <J a p p r o a c h e s used  here  ))/2.  J + a  controls  iterative  set of  indistinguishable  Estimation  under  a  discontinuous  respect  erf(W-W  parameter  of  the  As  by  ( F i g 65).  J+1  are  replacing  Y'j d e f i n e d  functions  technique requires  with  be  overcome  value  predictions, less  Marquardt  cannot  step-functions.  e q u a t i o n s 5.1,  Yj-(t),  ( e r f (W-Wj)/(a.Wj)  0j-(W) .  or  system  predictions  functions  =  the  are  differentiable  i n stage groups,  parameters.  1^(W)  above,  z  densities  smooth  i  (f> , d e f i n e d  parameters  the  J  and  scheme  spread  0,  found  allow  leading  which  obtained using  ^  approaches  to  while  other parameters, those  1^(W)  of  were  to more  thresholds.  results.  application f o r each  of  of  the  the  estimation  dominant  scheme  copepod  will  now  species  in  number  of  be  CEE5 i n  turn. A  Pseudocalanus This  and  field  minutus  species studies.  has  been  the  (eg P a r s o n s  subject et  al  of  a  laboratory  ,1969; P o u l e t , 1 9 7 3 ;  /il  /: /:  0. 0.1  0.4  W Figure 65.  j&jCW) , ^(W)  1.0  10.  (ug C )  for  pg C, W =2.0 pg C. J+]  (Solid l i n e i s ^j(W), dashed l i n e i s (J = 0.2, dotted l i n e i s . •  (W) with  ^ (W) with  (7=  0.1.)  263  Ikeda,1976). over  However, most o f t h e s e h a v e d e a l t w i t h  s h o r t time p e r i o d s or o n l y a t l a t e  behaviour  life-history  stages.  I  h a v e r e l i e d h e a v i l y h e r e on t h e r e s u l t s o f P a f f e n h o f f e r a n d Harris(1976)  f o r another  on l o n g - t e r m  laboratory cultures.  two s p e c i e s a r e q u i t e The t r a n s i t i o n  species, Pseudocalanus elongatus,  weights  W ,...,W h a v e been t a k e n 1  respectively.  4  The f i r s t  and l a s t  than  day" ). 1  between  a r e i n good  f o r Pseudocalanus  Maximum e x p o n e n t i a l g r o w t h r a t e s (Cn-F i n t h e model  u s e d h e r e ) were f o u n d CIII  from  a s 0.08, 0.4, 2.0 a n d 8.0 ug  a g r e e m e n t w i t h v a l u e s g i v e n by F r o s t ( 1 9 7 9 ) minutus.  t o Frost(1979), the  similar.  P a f f e n h o f f e r and H a r r i s ( 1 9 7 6 ) C/individual  According  based  f o r other  t o be s i g n i f i c a n t l y  stages  (about  H o w e v e r , no s i g n i f i c a n t  higher  0.3 d a y " overall  i n g e s t i o n p e r u n i t body w e i g h t  1  compared w i t h  s i z e d diatom  and weight  at 4 different  (Thalassiosira  While  c o u l d be o b t a i n e d  f o r any p a r t i c u l a r copepod weight  r o t u l a , 20-  e s t i m a t e s of a h a l f - s a t u r a t i o n c o n s t a n t  graphs of c l e a r a n c e r a t e or i n g e s t i o n r a t e versus each food d e n s i t y , t h e s c a t t e r i n e s t i m a t e s o f D.  a r e b a s e d on t h e same d a t a  in their  data  from  D  their  body w e i g h t f o r  results  i n high  F o r e x a m p l e , t h e g r a p h s i n F i g 66  s e t b u t one u s e s P a f f e n h o f f e r a n d  H a r r i s ' s r e g r e s s i o n l i n e s through  t h e i r clearance rate vs  data while the other  i s b a s e d on t h e i r  p e r u n i t body w e i g h t  fordifferent  first  f o r D l e s s than  y i e l d s a value  found  itself.  22 pm d i a m e t e r ) .  uncertainty  0.18  r e l a t i o n s h i p was  P a f f e n h o f f e r a n d H a r r i s c u l t i v a t e d P. e l o n g a t u s d e n s i t i e s of a m o d e r a t e l y  f o r s t a g e s CI t o  weight  average i n g e s t i o n r a t e s  food c o n c e n t r a t i o n s . 15 pg C . l "  1  The  ( f o r an  i n d i v i d u a l o f 10 pg d r y w e i g h t ) w h i l e t h e s e c o n d g i v e s a v a l u e  264  J  0  50  100  Food  cone.  I  L  150  200  ( p g C . I" ) 1  150 r  i-  1  0  50  Food ure  66.  (a) ( C l e a r a n c e dry  1  100  150  cone. rate)  (ugC.I  * vs food  weight Pseudocalanus  1976). (b)  1  Line  (Food vs  200 - 1  )  concentration  ( P ) f o r 10 p g  ( f r o m F i g 3, P a f f e n h o f f e r  and H a r r i s ,  : Ingestion  = 0.69.P/(12.6+P)  (pg C . d a y " ) .  concentration  P ) / ( % body weight  ingested  food  Line:  _i  concentration  % body w e i g h t  1  ( P a f f e n h o f f e r and  ingested  p e r day =  p e r day)  Harris,1976). 181.P/(41.+P).  265  greater An  than  1  from the d a t a  a decrease  of P a f f e n h o f f e r and  Yet, t h e i r  o x y g e n c o n s u m p t i o n by  c o n c e n t r a t i o n of 115  over  elongatus the  first  production  On  r a t e s as h i g h a s  Harris(1976) pg C . l " , 1  f o r as  t h a t , at a  an  food  food day"  1  food  naupliar production could with average female  r a t e r e q u i r e s an e f f i c i e n c y X o f 0.2  continue  production  p e r day  (from a  f o r females  or  ratio.  o b s e r v a t i o n s o f d e n s i t i e s of P. m i n u t u s i n CEE5 a r e An  a n o m a l o u s f e a t u r e i s t h e a p p e a r a n c e of l a r g e  numbers o f a d u l t s on day or CIV-V b e f o r e  19 w i t h no  t h i s day.  i m p l y t h e p a s s a g e of t h e s e  s i g n o f any  cohort  in CI-III  I f t h e d a t a were a c c u r a t e , t h e y individuals  f r o m NVI  to'CVI  would  in less  3 days which i s q u i t e i m p o s s i b l e a c c o r d i n g to the r e s u l t s  P a f f e n h o f f e r and H a r r i s ( 1 9 7 6 ) will  t o 0.11  Under t h e a s s u m p t i o n s of our m o d e l , s u c h  shown i n F i g 67.  than  0.04  found  l o n g a s 35 d a y s ,  f o r a 50:50 m a l e : f e m a l e The  i f anything,  t h e o t h e r h a n d , m e a s u r e m e n t s of  15 d a y s b e i n g a b o u t 4 p e r  their F i g 6).  0.1  r e s u l t s suggest,  food  Ikeda,1977).  P a f f e n h o f f e r and  P.  with decreasing  s m a l l copepods i n the absence of  basal metabolic  (Marshall,1973;  in  H a r r i s by l o o k i n g  i n g r o s s g r o w t h e f f i c i e n c y down t o t h e l o w e s t  c o n c e n t r a t i o n they used.  suggest  r a t e i n Pseudocalanus might  i n gross growth e f f i c i e n c y  concentrations. increase  C.l" .  e s t i m a t e of b a s a l m e t a b o l i c  be o b t a i n e d for  40 pg  and  references cited  be assumed h e r e t h a t t h e a d u l t s o b s e r v e d  were p r e s e n t a r e due  f r o m day  1 and  t h a t the e a r l i e r  t o the p r o d u c t i o n of n a u p l i i  phytoplankton  bloom.  Only the C I - I I I  therein.  i n the  copepodite  CIV-V time  It  enclosure stages  by a d u l t s d u r i n g t h e and  of  initial  series  will  266  F i g u r e 67.  Observed d e n s i t i e s  o f P s e u d o c a l a n u s i n CEE5.  E r r o r b a r s r e p r e s e n t 95% c o n f i d e n c e b a s e d on P o i s s o n s u b - s a m p l i n g  limits  statistics.  267  be  fitted. The e r r o r b a r s drawn i n F i g 67 a n d i n t h e c o r r e s p o n d i n g  figures  f o rthe other  based s o l e l y  species represent  on t h e s u b - s a m p l i n g  p r o p o r t i o n o f t h e sample t a k e n raw  counts  95% c o n f i d e n c e  statistics.  A relatively  on e a c h d a y was c o u n t e d  a r e assumed t o be P o i s s o n d i s t r i b u t e d  f i x e d by t h e t r u e c o n c e n t r a t i o n i n t h e o r i g i n a l does n o t t a k e  i n t o account  intervals small  and t h e  w i t h a mean sample.  This  the degree t o which c o n c e n t r a t i o n s i n  t h e pumped s a m p l e a r e r e p r e s e n t a t i v e o f t h e p o p u l a t i o n s i n t h e e n c l o s u r e ; t h i s q u e s t i o n h a s been a d d r e s s e d  by Lawson a n d G r i c e  (1977). The f o l l o w i n g p a r a m e t e r s were n o t e s t i m a t e d b a s i s of t h e r e f e r e n c e s d i s c u s s e d above. Wj were t a k e n  The t r a n s i t i o n  a s 0.08, 0.4, 2.0 a n d 8.0 ug C / i n d .  r e p r o d u c t i v e e f f i c i e n c y was t a k e n  recruitment.  The e s t i m a t i o n r e s u l t s  be d i v i d e d i n t o two c a t e g o r i e s .  The  The f i r s t  t h i s d i d not f i x the f o r P. m i n u t u s c a n  were o b t a i n e d by  a s s u m i n g t h a t g r o w t h a n d m o r t a l i t y p a r a m e t e r s were across stages.  weights  a s 0.1 b u t , a s t h e i n i t i a l  number o f r e p r o d u c t i v e a d u l t s was e s t i m a t e d , naupliar  b u t f i x e d on t h e  Some j u s t i f i c a t i o n  constant  f o r t h i s c a n be f o u n d i n  P a f f e n h o f f e r and H a r r i s ' s i n g e s t i o n d a t a . The i n i t i a l slowly.  trial  with a l l parameters f r e e converged  T h i s s l o w c o n v e r g e n c e was due p r i m a r i l y  of t h e model t o d i s t i n g u i s h between d i f f e r e n t metabolic  r a t e F on t h e b a s i s o f t h e s e  f i x e d a t each of three d i f f e r e n t  very  t o the i n a b i l i t y  v a l u e s of t h e b a s a l  observations.  When F was  v a l u e s , 0.0, 0.03, 0.10,  c o n v e r g e n c e was r a p i d a n d d i r e c t a n d a u n i q u e optimum w i t h a l o w SSQ was o b t a i n e d  i n each case.  The t h r e e r e s u l t i n g  parameter  268  s e t s and t h e c o r r e s p o n d i n g Table V I I I ; given  t h e p r e d i c t i o n s and o b s e r v a t i o n s f o r T r i a i  i n F i g 68a.  SSQ i s o b t a i n e d 0.03  small.  As F i s i n c r e a s e d , t h e t e c h n i q u e  N o t e t h a t t h e maximum e x p o n e n t i a l g r o w t h r a t e , Cn-  i s roughly constant  somewhat l o w e r The  f o r F = 0.0 b u t t h a t t h e i n c r e a s e i n SSQ f o r F =  a good f i t t o t h e o b s e r v a t i o n s by i n c r e a s i n g Cn a n d  d e c r e a s i n g D. F,  1 are  I t c a n be s e e n f r o m T a b l e V I I I t h a t t h e l o w e s t  o r F = 0.1 i s v e r y  maintains  SSQ a r e g i v e n a s T r i a l s 1, 2 a n d 3 i n  principal  than  as- F c h a n g e s  t h a t found  ( 0 . 1 3 t o 0.14 d a y " ) , b u t 1  by P a f f e n h o f f e r a n d H a r r i s ( 1 9 7 6 ) .  e f f e c t of i n c r e a s i n g F i s t o a l t e r D but t h e v a l u e s  of D a r e a l l w i t h i n t h e range c a l c u l a t e d above from P a f f e n h o f f e r and  H a r r i s ' s data. The  s e c o n d s e t o f e s t i m a t e s was o b t a i n e d by m u l t i p l y i n g t h e  growth r a t e f o r C I - I I I  by a f a c t o r  1.7, f o l l o w i n g P a f f e n h o f f e r  and  H a r r i s ' s o b s e r v a t i o n s of higher growth r a t e s i n these  The  p a r a m e t e r s Cn, D, 9, a n d t h e i n i t i a l  again estimated  number o f a d u l t s were  f o r F = 0.0, 0.03 a n d 0.10 d a y " . 1  c o n v e r g e n c e was r a p i d a n d d i r e c t  i n a l l cases  As b e f o r e ,  and t h e r e s u l t i n g  p a r a m e t e r e s t i m a t e s a n d minimum SSQ a r e g i v e n a s t r i a l s 6 i n Table V I I I . set  The SSQ a r e s l i g h t l y  (Table V I I I , F i g  little  lower  68b) a n d now d e c r e a s e  the change w i t h F i s a g a i n  small.  stages.  than  4, 5 a n d  f o r the f i r s t  as F i n c r e a s e s  although  The p a r a m e t e r Cn c h a n g e s  w i t h t h e i n t r o d u c t i o n o f t h e 1.7 g r o w t h f a c t o r a n d t h e  major e f f e c t understand  i s a l a r g e i n c r e a s e i n D.  as the growth of n a u p l i i  This behaviour  i n the enclosure  p r i m a r i l y a t high food c o n c e n t r a t i o n s , f i x i n g r a t e , w h i l e the growth of C I - I I I concentrations,  fixing  i s easy t o  takes  t h e maximum g r o w t h  t a k e s p l a c e a t low f o o d  t h e v a l u e o f D.  place  Table VIII. F i n a l parameter estimates and corresponding SSQ errors f o r Pseudocalanus.  ( Subscripts 1 . . . 4 r e f e r to stage groupings NI-VI,  CI-III,CIV-V,CVI respectively. A " " means that the parameter value i s f i x e d at the value given f o r the stage group above; an a s t e r i s k means that the parameter value i s f i x e d , n o t estimated.) Parameters  Cn  1  Cn  2  C n  3  Cn  D  2  D  3  D  4  F  l 2  F  F  3  F  4  9  1  V 9  3  e  4  1  2  3  4  5  6  .14  .16  .23  .16  .18  .24  II  ti  II  .28  .30  .41  II  II  it  .16  .18  .24  11  II  II  II  II  II  34.  20.  9.9  72.  34.  16.  II  II  II  II  II  it  it  II  II  II  ii  II  II  II  4  l  D  Trials  Lt. adults SSQ  *  *  *  it  *  .00  .03  .10  II  II  it  .00 n  II  II  II  ti  II  II  II  II  it  II  * .03 .05  * *  .03  • io .17  :  :  .10  it  it  .01  .01  .01  .01  .01  .01  II  II  II  II  II  It  II  II  II  II  II  II  II  II  II  II  II  It  1.84  1.85  1.86  1.93  1.94  1.88  74.  75.  77.  66.  63.  60.  20 CIV-V  11  ^  1  -*  0  *¥• 1  V  70  0 U C  o  ] 20 <  CI — II  11  0  •V  ^  Figure  Time 68a.  (days)  Comparison o f p r e d i c t i o n s observations ( t r i a l 1).  70 ( s o l i d symbols) and  (open symbols) f o r P s e u d o c a l a n u s  271  Figure  68b.  Comparison of p r e d i c t i o n s observations (trial  6).  ( c l o s e d symbols)  and  (open symbols) f o r P s e u d o c a l a n u s  272  The  estimated  the c o h o r t  initial  is relatively  b e t w e e n 1.78  and  1.96  number of a d u l t s r e q u i r e d t o p r o d u c e  constant  per  liter.  across a l l t r i a l s , This estimate  i s of  i n v e r s e l y p r o p o r t i o n a l t o t h e assumed e f f i c i e n c y X. to  The  of t h e has  of  low v a l u e o f e i s a l s o w o r t h y of comment.  h a v e been l i t t l e o r no low  been assumed c o n s t a n t  by o b s e r v a t i o n s o f C I - V . r a t e s of n a u p l i i copepodids,  as  found  alone,  a c r o s s s t a g e s and It is entirely  by P a f f e n h o f f e r and but w i t h o u t  course  production, There appears  r a t e as a  result  Note t h a t t h i s m o r t a l i t y is really  than  o b s e r v a t i o n s on  mortality  those  Harris(1976)  rate  determined  possible that  i n t h e e n c l o s u r e were h i g h e r  laboratory cultures, nauplii  increase in mortality  food c o n c e n t r a t i o n s .  lying  of  in  Pseudocalanus  i t i s impossible to estimate a n a u p l i a r m o r t a l i t y  rate. B Calanus pacif icus Calanus p a c i f i c u s  ( h e l g o l a n d i c u s ) i s the  t h r e e d o m i n a n t c a l a n o i d c o p e p o d s i n CEE5.  I t has  s u b j e c t o f a l a r g e number of l a b o r a t o r y and (Paffenhoffer,1970,1971,1976;  M u l l i n and  l a r g e s t of  field  been  the  the  studies  Brooks,1970;  F r o s t , 1 9 7 2 , 1 9 7 5 , 1 9 7 9 ; R u n g e , 1 9 8 0 ) , b e i n g one  of the b e s t - s t u d i e d  of m a r i n e p e l a g i c copepods. The  transition  weights  Paffenhoffer(1971)  as  respectively.  The  fact  g r e a t e r weight  range than  generation  t i m e has  0.1,  Wj- have been t a k e n 1.0,  t h a t C.  10.,  from  70. jug C p e r  individual  p a c i f i c u s grows t h r o u g h  P. m i n u t u s i n r o u g h l y  been commented on by F r o s t  the  a much  same  ( 1 9 7 9 ) and  the  maximum e x p o n e n t i a l g r o w t h r a t e s r e p o r t e d by  Paffenhoffer(1976)  explain  for nauplii,  this,  being  0.41,  0.41,  0.33  and  0.2  CI-III,  273  CIII-V twice  and  CV  t o young a d u l t  t h o s e f o u n d f o r P. This  i n the  maintained  reproductive of  0.14  sex  eggs per  food  concentrations  Under t h e  a s s u m p t i o n s of  efficiency  f o r a 1:1  p a c i f i c u s females  r a t e o f a b o u t 40  t o 50 d a y s a t h i g h  (Paffenhoffer,1976).  almost  to n a u p l i a r weight i s a l s o  f e c u n d i t y of C.  a production  These are  elongatus.  l a r g e r a t i o of a d u l t  reflected  o v e r 40  respectively.  of a b o u t 0.28  ratio  which female per  day  the model, a  f o r females or a v a l u e  i s r e q u i r e d to produce  of  X  this  fecundity. The  r e s u l t s of P a f f e n h o f f e r ( 1 9 7 0 , 1 9 7 1 ) and  emphasize the the  t y p e and  o f f e r e d as  d e p e n d e n c e of t h e particularly  food.  the  s i z e of the p h y t o p l a n k t o n  s t u d i e s were d e s i g n e d t o l o o k d e n s i t y on  food  (50 and  type  100  t i m e s as h i g h  ingestion  using adult ranging  f r o m a b o u t 50  the  two  ( M u l l i n ejt a l ,1975) and could here.  be  suggesting  (1972) g r a z i n g  ug  C.l" .  d i s t i n g u i s h e d on  be  b a s i s of  I have t h e r e f o r e c o n t i n u e d  consistency.)  of  the  t o use  Runge(1980) found h i g h e r  one  borealis) about for  experiments  values  of  D  were  fitted  f u n c t i o n a l response  distinguished  i t i s exceedingly the  rather  that D  ( F r o s t ' s data  1  r a t h e r than a c u r v i l i n e a r  models c o u l d not  type  u n i t body w e i g h t were o n l y  Frost's  t o 150  food  the'diatom Lauderia  lower d e n s i t y ,  large.  on  cells  concentrations  females a l s o suggested rather high  to a r e c t i l i n e a r but  e f f e c t of  when two  of  1  r a t e s per  at the  i s rather  at the  ug C . l "  D,  (1970,1971) l o n g - t e r m c u l t u r e  growth, but  were u s e d , c l e a r a n c e 1.2  h a l f - s a t u r a t i o n constant,  Paffenhoffer's  than food  Frost(1972)  statistically  u n l i k e l y that time s e r i e s  the  curvilinear  clearance  rates  they  fitted form f o r and  274  consequently  lower  v a l u e s f o r D i n C.  r e p o r t e d , a l t h o u g h he pretreatment 15 ug C . l "  i n d i c a t e s t h a t t h e s e may  effects.  H i s r e s u l t s suggest  f o r the l a r g e s t c e l l  1  p a c i f i c u s than F r o s t  s i z e used.  d a t a o f M u l l i n and  the r e s u l t s  are c a l c u l a t e d  t h a t D f o r g r o w t h l i e s w e l l b e l o w 19 ug C . l " 1  for Thalassiosira.  estimate basal metabolic  1  increase with increasing  Brooks,1970).  respiration  markedly highest  r a n g e of C.  Mortality  d e p e n d e n t on for nauplii.  rather  1  and  stages  of  low  (Marshall,1973;  r a t e s i n c u l t u r e were f o u n d  to  be  f o o d d e n s i t y ( P a f f e n h o f f e r , 1 9 7 0 ) and  to  be  There i s a p o s s i b l e c o m p l i c a t i o n i n the  CEPEX e n c l o s u r e s a s t h e l a t e  s t a g e s of C.  v e r t i c a l m i g r a t o r s i n the w i l d not  late  p a c i f i c u s suggest day"  and  impossible to  food c o n c e n t r a t i o n ( M u l l i n  r a t e s of o r d e r 0.03  Ikeda,1977).  suggest  the l a b o r a t o r y c u l t u r e  E s t i m a t e s of o x y g e n c o n s u m p t i o n by  copepods i n the s i z e  the  f o r Gymnodinium  r e s u l t s as g r o s s g r o w t h e f f i c i e n c y a p p e a r s t o d e c r e a s e than  from  (Table IX)  I t i s again  r a t e s from  as  I f average  t o CIV  b e l o w 49 ug C . l "  to  v a l u e s of D a s low  e x p o n e n t i a l g r o w t h r a t e s o v e r NI Brooks(1970),  have been due  p a c i f i c u s are  ( M u l l i n and  strong  B r o o k s , 1 9 7 0 ) and  be a b l e t o s u r v i v e i n t h e e n c l o s u r e s a t any  may  food  concentration. The  time  s e r i e s of o b s e r v e d  a r e shown i n F i g 69. attempting weights  pacificus densities  A d u l t s were n o t o b s e r v e d  t o f i t the model t o t h i s d a t a  were s e t a t 0.1,  e f f i c i e n c y X a t 0.14. was  C.  e s t i m a t e d and  1.,  Again  10. and the  a f t e r day  s e t , the  1.  In  transitional  70. jug C / i n d . and  initial  i n CEE5  the  number of a d u l t s p r e s e n t  a d u l t s were assumed t o p r o d u c e eggs f o r o n l y 4  d a y s a s t h e y were n o t o b s e r v e d  on day  5.  275  Table IX. Exponential growth rates f o r Calanus over NI to CIV at 12°C and d i f f e r e n t concentrations of Gymnodinium and T h a l a s s i o s i r a (.calculated from Table 3 and F i g 2, M u l l i n .and Brooks, 1970) .  Gymnodinium Food concentration (ug C . l " )  19.  23.  54.  78.  271.  318.  Growth rate ( d a y )  .24  .19  .26  .30  .33  .33  Food concentration (ug C . l )  49.  63.  165.  205.  750.  750.  Growth rate ( d a y )  .25  .25  .29  .31  .31  .34  1  -1  Thalassiosira - 1  -1  20  CIV-V  Figure 69.  Observed d e n s i t i e s of Calanus i n CEE5. ( E r r o r bars as i n F i g 67.)  277  In and  the  CVI  initial  growth r a t e s i n NI-VI,  were assumed t o f o l l o w a f i x e d  half-saturation stages.  The  particular,  c o n s t a n t , D,  r e s u l t i n g best  was  o f 100  constant  ug C . l " . 1  the o b s e r v a t i o n  f i t was  47  c o n s i s t e d of s m a l l f l a g e l l a t e s  Frost's  (1972) r e s u l t s ,  the  late  clearance  growth p a r a m e t e r s of s t a g e s r e a c h i n g CVI  would r e s u l t  p r e d i c t i o n s and  value  t h e p a r a m e t e r s e t and SSQ  final  i n F i g 70a  reproduce the and  are  rates.  sharp  than  of  Dry-v  SSQ  > the  given  shows why:  coincident rise  the  same SSQ  Higher  values  slight  decreases  The  the  high on  t h e b a s i s of  p a c i f i c u s might  H o w e v e r , any  be  change i n the  them f r o m between  a l g o r i t h m converged  in t r i a l  in  X.  1, T a b l e  The  the model i s u n a b l e in CI-III earlier  and  rapidly The  is entirely  to  C I V - V on day  increase in CI-III  parameter estimates  o f Cn As  i s obtained,  (trials  3, T a b l e  D  i n the  i n s e n s i t i v e to  2 and  19 and  and  the  r a t e F; when F i s f i x e d a t 0.0  of F a r e c o m p e n s a t e d f o r by  reason  this,  based  l i t e r a t u r e v a l u e s d i s c u s s e d above.  v a l u e of the b a s a l m e t a b o l i c the  in  t h e c o m p a r i s o n of p r e d i c t i o n s and  c a s e o f P s e u d o c a l a n u s , t h e SSQ  0.08,  of C.  i n e q u a l l y good a g r e e m e n t  i n c r e a s e i n CIV-V.  lower  f o r w h i c h , on  stages  c o m p r o m i s e s w i t h a s m a l l e r and  a later  across  p o p u l a t i o n when CIV-V were  CIV-V w h i c h p r e v e n t e d  i s r a t h e r h i g h and  observations  a c t i o n was  the  observations.  W i t h t h e new to  To p r e v e n t  increased to a  T h i s r a t h e r ad hoc  that the phytoplankton  t o h a v e low  on.  D f o r C I V - V was  CIV-V  1:1:0.8:0.5 and  rather u n r e a l i s t i c ;  present  expected  ratio  CI-III,  assumed t o be c o n s t a n t  a d u l t s a p p e a r e d f r o m day  half-saturation level  trial,  or  X).  i n c r e a s e s i n Cn  and  D.  f o r the c o m p a r a t i v e l y  poor f i t here i s t h a t  the  278 Table X. F i n a l parameter estimates and corresponding SSQ e r r o r s f o r Calanus. ( Subscripts 1...4, " " and  as i n Table V I I I . )  Parameters  Trials  Cn  l  1  2  3  4  5  6  7  .26  .24  .32  .18  .20  .26  .25  .24  +  8  +  +  Cn  2  "  "  "  .31  .34  "  .43  .41  Cn  3  .21  .19  .26  .14  .16  .21  .20  .19  .13  .12  .16  .09  .10  .13  .12  .12  9.5  10.6  7.3  0.0  10.0*  8*6  0.0  10.0*  II  II  II  II  II  H  II  Cn. 4  •p.  11  2 D D. 3  100.* 100.* 100.* 100.* 100.* 100.* 100.* 100.* 9.5 10.6 7.3 0.0 10.0* 8.6 0.0 10.0*  4  F  x  F  2  * .000  .020 "  * .080  . . . .  * .020 ..  .015 > 0 2 4  .023 -.  .022  .000 /  > 0 3 7  0 0 0  F„ 3  .016  .000*  .064*  .016* .012  .018  .018  .000  F, 4  .010  .000  .040  .010  .007  .012 . .011  .000  &  .030  .030  .030  .054  .047  .056  .107  .081  it  II  II  II  II  ti  II  II  11  II  II  II  II  II  It  II  II  II  11  II  II  It  II  x  a  .  .  .  2  Q 3  Q  I n i t . adults SSQ  .23  .23  .23  .84  .60  .62  2.3  1.2  195.  195.  195.  130.  152.  124.  75.  86.  +  I n these t r i a l s , observations on day 16 were ignored.  20 CIV-V  4>  0  o  4-  0  70  z CD  c D TJ c  20 Cl-lll  < 4>  4>  0 Figure 70a.  Time  (days)  -4—4  70  Comparison of predictions ( s o l i d symbols) and observations (open symbols) f o r Calanus ( t r i a l  280  observations  a r e i n c o n s i s t e n t w i t h t h e known g r o w t h  c h a r a c t e r i s t i c s o f C. p a c i f i c u s . p e r i o d of a p p a r e n t l y  high  by d a y 19 a t l o w e r  if  suggest t h a t , over a  f o o d d e n s i t y , C. p a c i f i c u s n a u p l i i  more t h a n 16 d a y s t o r e a c h CIV  The d a t a  C I , y e t almost h a l f of these  food c o n c e n t r a t i o n s .  This  growth r a t e s i n NI-VI a r e a s h i g h as those  same f o o d  concentrations.  reached  i s impossible  i n C I - I I I at the  I n f a c t , an i n d i v i d u a l  r e a c h i n g CIV  f r o m NVI i n 3 d a y s w o u l d have t o grow a t an e x p o n e n t i a l  rate  exceeding  from  0.77 d a y " , a l m o s t t w i c e  l a b o r a t o r y c u l t u r e s w i t h abundant If in  t h e maximum r e p o r t e d  1  we a r e p r e p a r e d  C I - I I I than  obtained.  food.  t o concede a higher  i n NI-VI, a considerably  maximum g r o w t h  minimum SSQ i s p o s s i b l e ( t r i a l  f r o m F i g 70b t h a t t h e r e s u l t i n g  rate  b e t t e r f i t c a n be  F o r e x a m p l e , i f t h e f a c t o r 1.7 f o r C I - I I I o v e r  used f o r Pseudocalanus , i s i n t r o d u c e d , in  a considerable  4, T a b l e X ) .  increase  NI-VI,  reduction  I t c a n be s e e n  i n growth r a t e  during  CI-III  has a l l o w e d  CI-III  a n d t o m a t c h t h e peak d e n s i t y on day 19 more c l o s e l y .  model i s s t i l l not  t h e model t o e l i m i n a t e t h e e a r l y i n c r e a s e i n  u n a b l e t o p r o d u c e an i n c r e a s e  s u r p r i s i n g l y , and t h e a l g o r i t h m  effort  t o do s o .  A similar  f o r c e s D towards zero  f i t , w i t h a somewhat h i g h e r  SSQ, c a n  final  o f Cn i m p l i e s a maximum g r o w t h r a t e f o r NI-NVI o f  0.16  (trial  i n an  w i t h D f r o z e n a t 10 ug C . l "  1  d a y " a n d f o r C I - I I I o f 0.27 d a y " , s t i l l 1  1  maximum f o u n d by Unless  rates f o r stages increase  5, T a b l e X ) . The  w e l l below t h e  Paffenhoffer(1976).  we a r e w i l l i n g CI-III,  i n C I - I I I before  The  i n C I V - V by day 1 9 ,  be o b t a i n e d value  took  t o allow unreasonably high  growth  t h e m o d e l must e i t h e r p r e d i c t an day 19 o r a n i n c r e a s e  i n CIV-V  after  281  20 CIV-V  A  o Z  JL.  0  ±  70  cu C  D -a  120  Cl-lll  i 0 Figure 70b.  Time  4\  -kr-  (days)  70  Comparison of predictions (solid.symbols) and observations (open symbols) f o r Calanus ( t r i a l 4)  282  day  19.  Faced with the e n t i r e  ' c h o o s e s ' t o do t h e l a t t e r .  s e t of o b s e r v a t i o n s ,  However, t h e o b s e r v a t i o n  III  on d a y 16 may be more open t o q u e s t i o n  the  bag was l o s t .  omitted  CI-III  an  increase take  i s returned  as a sample o f p a r t o f  6, t h e o b s e r v a t i o n s  to.  The m o d e l i s s t i l l  i n CIV-V by d a y 19 ( F i g 7 0 c ) . a t l e a s t 11 d a y s t o r e a c h  subsequently  t o reach  on day 16 a r e  the factor  day  f o r the f i r s t  16 a l l o w s  unable t o achieve  To do s o , n a u p l i i must  CI-III  CIV-V, a t lower  Reintroducing  (Fig  and l e s s than 8 days  food  concentrations.  1.7 a n d n e g l e c t i n g t h e o b s e r v a t i o n s t i m e an i n c r e a s e  70d) a n d a s i g n i f i c a n t r e d u c t i o n  C.l" ,  a similar f i t with a s l i g h t l y  i n SSQ ( t r i a l  (trial  8, T a b l e X ) .  III  from t r i a l  former  higher  7, T a b l e X ) .  1  respectively.  by  higher value  u s i n g much h i g h e r  To make s e n s e o f t h e d a t a ,  t h e low o b s e r v a t i o n  observation  The with  conclusions.  It is  f o r t h e growth model t o r e p r o d u c e t h e o b s e r v a t i o n s of  C. p a c i f i c u s w i t h o u t  either  obtained  Paffenhoffer(1976).  We a r e t h e r e f o r e l e d t o t h e f o l l o w i n g  reasonable.  SSQ c a n be  low but t h e second a g r e e s r a t h e r w e l l  t h e maximum r a t e s r e p o r t e d  impossible  D a t 10 ug  The maximum g r o w t h r a t e s f o r N I - V I a n d C I -  8 a r e 0.24 a n d 0.41 d a y "  i s , of course,  on  i n CIV-V on day 19  A g a i n t h e a l g o r i t h m t e n d s t o make D z e r o ; on f r e e z i n g 1  of low C I -  a n d t h e 1:1 r a t i o b e t w e e n maximum g r o w t h r a t e s i n N I - V I  and  still  In t r i a l  the algorithm  of C I - I I I  growth r a t e s than a r e i t must be assumed  on day 16 o r t h e h i g h  o f CIV-V on d a y 19 i s i n c o r r e c t .  maximum g r o w t h r a t e i n C I - I I I of D i s necessary  that  In e i t h e r case,  a  than i n NI-VI and a low  t o f i t the remaining  observations  closely. Especially  i n the l a t e r t r i a l s  where t h e e a r l y  r i s e i n CIV-V  283  20 CIV-V  A o Z cu u c D ~D c ZD  <  0  70  20 C1 — III  4  A  0 F i g u r e 70c.  l I Time  I  !  i  70  (days)  Comparison of p r e d i c t i o n s observations  ( s o l i d symbols) and  (open symbols) f o r Calanus ( t r i a l  (* t h i s o b s e r v a t i o n not  fitted.)  6)  284  20  CIV-V  A o Z  0  70  cu  U  c o ~a c D  20 Cl-I  <  -i—1—i—i—  0  Figure  Time  70d.  Comparison  of p r e d i c t i o n s ( s o l i d symbols)  observations (*  this  70  (days)  (open symbols)  o b s e r v a t i o n was  not  f o r Calanus fitted.)  and  (trial  7).  285  is  reproduced,  estimated.  a very high m o r t a l i t y  rate  (0.08 d a y " ) i s 1  This i s r e q u i r e d to explain the rapid decrease i n  copepodite densities after  day 1 9 .  The m o r t a l i t y  assumed c o n s t a n t a c r o s s s t a g e s b u t , a s n o t e d it  i s impossible to estimate mortality  r a t e h a s been  f o r Pseudocalanus,  rates during the naupliar  s t a g e s w i t h o u t o b s e r v a t i o n s o f C. pac i f i c u s n a u p l i i . t h e r e may be r e a s o n s nauplii.  t o suspect  The c o p e p o d i d s ,  lower m o r t a l i t y  In f a c t ,  rates for  e s p e c i a l l y CIV-V, were e x p o s e d t o low  d e n s i t i e s o f s m a l l p h y t o p l a n k t o n c e l l s a n d P a f f e n h o f f e r ' s (1970) results  suggest  t h a t t h i s may h a v e l e d t o i n c r e a s e d m o r t a l i t y  rates.  Attempts  to migrate v e r t i c a l l y  i n a shallow enclosure  c o u l d a l s o have c o n t r i b u t e d t o h i g h e r m o r t a l i t y copepodids.  Finally,  t h e number o f i n i t i a l  produce the cohort i n t r i a l s observed  i n CEE5 on day 1 : t h i s number c o u l d be r e d u c e d the f i t t o the copepodids  mortality  rate.  C Paracalanus The t h i r d  parvus species here, Paracalanus parvus,  growth dynamics.  i s known o f i t s f e e d i n g a n d  i s a r e c e n t l a b o r a t o r y and f i e l d  s t u d y o f f e e d i n g a n d egg  of Paracalanus  (Checkley,1980).  o f a d u l t f e m a l e s a n d e g g s h a v e been t a k e n  a s 3.0 a n d 0.02 pg C / i n d r e s p e c t i v e l y . for  i s the smallest  The m a j o r s o u r c e o f i n f o r m a t i o n known t o t h e  p r o d u c t i o n by a d u l t f e m a l e s weights  without  by d e c r e a s i n g t h e n a u p l i a r  t h e t h r e e and c o m p a r a t i v e l y l i t t l e  author  adults required to  7 and 8 i s s e v e r a l t i m e s t h a t  affecting  of  r a t e s i n the  NVI t o CI a n d C I I I  from t h i s  The t r a n s i t i o n  The study  weights  t o C I V a r e n o t known a n d h a v e been  guessed  to  be 0.1 a n d 0.6 pg C / i n d , r o u g h l y i n k e e p i n g w i t h t h e p a t t e r n  of  g r o w t h i n t h e o t h e r two s p e c i e s .  Paracalanus  i s intermediate  286  between P s e u d o c a l a n u s and C a l a n u s  i n terms of t h e r a t i o of a d u l t  f e m a l e w e i g h t t o w e i g h t o f e g g , b u t i n maximum e x p o n e n t i a l rate,  i t appears t o match or exceed C a l a n u s p a c i f i c u s ,  e x c e e d i n g l y short g e n e r a t i o n time of as l i t t l e (Sonntag and P a r s l o w , 1 9 8 0 ) .  growth  h a v i n g an  a s 14 d a y s  T h i s c o r r e s p o n d s t o a maximum  a v e r a g e e x p o n e n t i a l g r o w t h r a t e o f 0.36 d a y " . 1  The p a t t e r n o f  g r o w t h r a t e o v e r s t a g e s i s unknown. The  f e c u n d i t y of P a r a c a l a n u s females i n response t o food  s u p p l y h a s been w e l l  s t u d i e d by C h e c k l e y ( 1 9 8 0 ) , who f o u n d maximum  r a t e s o f p r o d u c t i o n o f o r d e r 50 e g g s / f e m a l e  day, again matching  t h o s e o f C. p a c i f i c u s , a l t h o u g h t h e d u r a t i o n o f b r e e d i n g i n P a r a c a l a n u s i s unknown.  The d a i l y  e g g p r o d u c t i o n r e p r e s e n t s more  t h a n 0.3 o f t h e f e m a l e body w e i g h t p e r day a s c o m p a r e d w i t h t h a n 0.1 f o r P s e u d o c a l a n u s a n d C a l a n u s .  An e f f i c i e n c y X e q u a l t o  1.0 f o r f e m a l e s o r 0.5 f o r a 1:1 s e x r a t i o i n t h e model t o r e p r o d u c e t h i s h i g h Checkley  obtained  4 jjgN.l"  f o r a diatom  be d i f f i c u l t  i s required  fecundity.  1  o r about  20 t o 40 ug C . l "  on t h e C:N r a t i o o f t h e p h y t o p l a n k t o n .  flagellates  i n CEE5 o r t o s t a g e s o t h e r t h a n a d u l t to justify.  1  f o r t h i s parameter  zooplankton, Ikeda(1977) to drop t o about  females  might  b a s e d on t h e r e l a t i o n s h i p  b e t w e e n i n g e s t i o n a n d egg p r o d u c t i o n .  1  i t to the  He a l s o r e p o r t e d v e r y l o w b a s a l  0.01 d a y " ,  v a l u e s up t o 0.12 d a y " .  1  T h i s was  ( T h a l a s s i o s i r a ) and e x t r a p o l a t i n g  m e t a b o l i c r a t e s , about  interval  in adults  f o u n d a c o n s i s t e n t h a l f - s a t u r a t i o n c o n s t a n t f o r egg  p r o d u c t i o n of about depending  less  H i s 95% c o n f i d e n c e  was r a t h e r w i d e , h o w e v e r ,  allowing  In a study of r e s p i r a t i o n i n found oxygen  0.02 u l . h r "  1  after  consumption  by P a r a c a l a n u s  3 days of s t a r v a t i o n ,  giving  287  a  ' b a s a l ' m e t a b o l i c r a t e o f a b o u t 0.08  ammonia e x c r e t i o n and l e s s t h a n 0.03 While  dry weight  response  'natural'  mortality  lower v a l u e s ,  r a t e s f o r P a r a c a l a n u s have  emphasized the l a c k of l i p i d found  f o r an a v e r a g e  t o low  l o s s suggested  1  s m a l l c o p e p o d and  starvation  However, b o t h "  1  day" .  been r e p o r t e d , C h e c k l e y this  day" .  reserves in  t h a t females  could survive  of 5 days o n l y .  Some s o r t of  f o o d d e n s i t i e s m i g h t t h e r e f o r e be  not  mortality  reasonably  expected. The  time- s e r i e s o f o b s e r v e d  copepodids nauplii  i n CEE5 a r e shown i n F i g 71.  were s e t a t 0.02,0.1,0.6 and  the r e p r o d u c t i v e e f f i c i e n c y  of  total  f i x e d a t 0.5  3.0  d e n s i t y of  The  transition  pg C r e s p e c t i v e l y  and  as d i s c u s s e d above.  A  f e a t u r e o f t h e P a r a c a l a n u s d a t a i s t h a t c o n s i d e r a b l e numbers a d u l t s are present l a t e r  p r o d u c t i o n of n a u p l i i than those observed, controlling after for  The  Paracalanus  i n a l l s p e c i e s i s a l s o g i v e n i n F i g 72.  w e i g h t s Wj  new  d e n s i t i e s of  i n the experiment;  to prevent  the  by t h e s e a d u l t s i n numbers much g r e a t e r found necessary  t o set the  parameters  a d u l t f e c u n d i t y so t h a t n a u p l i i  production  ceased  a b o u t day  30.  i t was  T h i s was  a d u l t s e q u a l t o 0.4  respectively.  The  accomplished  d a y " , 80 pg C . l " 1  f i t to the data  1  and  0.08  i s more o r l e s s  t h e v a l u e s t a k e n by t h e s e p a r a m e t e r s , n a u p l i a r p r o d u c t i o n , and  by s e t t i n g Cn, D and day"  1  insensitive  p r o v i d e d they prevent  t h e y h a v e been f i x e d  the i n i t i a l  s e r i e s of t r i a l s ,  r a t h e r than  been assumed t o be due produce n a u p l i i  t o an  initial  b e g i n n i n g on day  1.  the observed  cohort  has  p o p u l a t i o n of a d u l t s which In t r i a l  1, a l l g r o w t h  to  late  estimated. In  F  and  288  Figure  71.  Observed d e n s i t i e s o f P a r a c a l a n u s i n CEE5. ( E r r o r b a r s as i n F i g  67.)  160  Time Figure 72.  (days)  Observed densities of t o t a l n a u p l i i i n CEE5. (Error bars as i n F i g 67.)  290  mortality in  the  p a r a m e t e r s were assumed  case  of  Pseudocalanus,  t o be  the  i m p r o v e d g r e a t l y when t h e  frozen.  The  Table the  XI.  dip  The  SSQ  in CI-III  early  increases The  the  of  i n CIV-VI  the  C.  i s reduced  the  f i t to remaining although  While  the  in  the  has  observed  increasing  t o grow and  F  considerable XI)  with  entirely  (trial  latter  CVI  after  by  Table  day  of  XI),  still  As  but  (Fig ignored.  day  54  does  increase  when t h e  an  total  excess that  i n t h e model a t  in .  density to  1  and  was  f i t to CI-III  nauplii  of C I - I I I ' s i s nauplii  On  a l l o w i n g Cn  achieved  are  food  enclosure.  for copepodids,  i n minimum SSQ  the  predicted  although  i n the  Cn  that  this.  33,  day"  The  the  the model, w h i c h has  2 i s the  CI-III  t o 0.12  independently  this  i t seems l i k e l y  this  model  surprising  are  73a;  model.  during  2,  in  peak.  the  i n d i c a t e an  after  day  on,  a much b e t t e r  the  i s a l s o much i m p r o v e d  reproduce  which prevent  reduction  by  was  in F i g  c u t s o f f the  observations  the  reach  for nauplii  to vary  and  ignored  in t r i a l  33  seen  reproduced  to remain h i g h .  p r e d i c t e d from day  concentrations  Table  nauplii  be  As  very  given  i n CIV-V d e n s i t i e s . a f t e r  observations,  of  can  I t i s not  dropped to near-zero;  disappearance  continuing  be  data.  e a r l y CIV-VI  NI-CV.  r a t e , F,  are  were q u e s t i o n e d  observations  Another discrepancy  nauplii  16  are  copepods, cannot  numbers a r e  reason  ignored  numbers of CIV-V and  'conserve'  also  day  p r e d i c t e d decrease  CI-III  estimates  in CI-III are  when t h e s e  agree with  combined  on  the  cannot  pacificus  SSQ  73b),  16  rise  observations  analysis  not  day  basal metabolic  parameter  i s h i g h and on  w h i c h smooths out  best  over  r a t e of c o n v e r g e n c e was  slow and  resulting  constant  for  a (trial  in particular  3, (Fig  Table XI. F i n a l parameter estimates and corresponding SSQ errors f o r Paracalanus. ( Subscripts 1...4, " " , '*' and ' + ' as i n Table X.) ;  Parameters  Trials 1  2  +  3  4  Cn  1  .22  .24  .36  .52  Cn  2  "  "  .34  .47  Cn  3  "  "  "  .36  &  Cn.  <fe  &  <fc  .40  .40  .40  .40  4 D  1  20.  23.  61.  41.  D  2  "  "  "  41.  D  3  '*  "  "  50.  D,  80.*  80.*  80.*  80.*  F  x  .03  .03  .12  .14  F  0  "  "  II  II  •TH  *  *  *  .03* II  *  .03* II  3 F,  4  9  X  0  2  _  .08*  .08*  .08*  .039  .051  .037  11  "  "  II  II  "  "  II  .08* .020 .070 it  3  e  .020  4 I n i t adults  SSQ  .80  1.27  1.95  2660.  1137.  760.  .36 +4.60 juveniles. 1584.  50  0  "CVI  •—  ^—^-V  70  50  CIV-V  * •, f CD O  ro "D  0  t 70  C  _o < 100  CI-III ^7  t  t Iff  y  0  Figure 73a.  Time  (days)  70  Comparison of predictions ( s o l i d symbols) and observations (open symbols) f o r Paracalanus (trial 1).  293  50  CVI  t —'—v70  o 21  50  CIV-V  CD O  t  c  CD "O  V—  c  v  * t  t 70  jQ <  100  CI  t  V  ^r  • JL  0  Time Figure 73b.  (days)  -9  70  Comparison of predictions ( s o l i d symbols) and observations (open symbols) f o r Paracalanus ( t r i a l 2).  294  73c).  The  r e s u l t i n g s e t of p a r a m e t e r e s t i m a t e s  is characterized  by much h i g h e r  maximum g r o w t h r a t e s and  for  estimated  maximum g r o w t h r a t e f o r n a u p l i i i s  still  As  c a s e o f P s e u d o c a l a n u s and  the  CI-V.  low,  The  however.  minimum SSQ for  CI-V  i n the  i s more o r  half-saturation  l e s s ' i n d e p e n d e n t of  although high  values  the  n e c e s s a r y here t o e x p l a i n the  early  increases  observations, could  not  differs  on  be  observations  on  day  allow  are  by an  19.  However, t h e  c o n s i s t e n t : an  increase  This  could  be  on  16  2 cohorts,  for  s t a r t i n g on  a short  period  c o p e p o d i d s on r i s e t o the  The  (0.1  the  c l a s s e s , one the  fact that  model was  other  day  are  one  early  an  day  11  increase  c l a s s i s at  is  in  day  to a d u l t s which  1 and  the  the  to  low  therefore  t o 0.8  day  19.  due  to  This  number o f a d u l t s ind/1  changed to a l l o w  by two  day  that  reproduce  reproduce i n time to  initial  increases  other  CVI  fault.  prepared to accept the  due  1,  Paracalanus  the  day. 16 and  1 w h i c h m a t u r e and  i n d / 1 ) and  s t a r t i n g on  i n C I - I I I on  l a r g e C I - I I I peak o b s e r v e d on  s u p p o r t e d by low  day  later  as g e n u i n e , i t a l s o seems p l a u s i b l e  the data represent  in  increase  predict  a c a s e where t h e m o d e l ' s i n a b i l i t y  CI-III  day  chosen  are  f i t the  for  v a r i a t i o n i n g r o w t h r a t e w i t h i n a day  density  CVI  model t o  still  data set  i n CIV-V on  e s p e c i a l l y i f we  and  and  for Pseudocalanus in that  Alternatively,  is  CVI  the  b a s i s of a s i n g l e c o h o r t  found.  from t h a t  followed  i n CIV-V and  the  of F  observed naupliar d e n s i t i e s .  A s e t .of p a r a m e t e r s w h i c h w o u l d a l l o w the  Calanus,  value  o f F f o r N I - V I and  constants  give  is observed  11.  initial  day  c o n s i s t i n g of a d u l t s w h i c h p r o d u c e e g g s f o r 5 d a y s c o n s i s t i n g of  both i n i t i a l  day  juveniles.  c l a s s e s and  The  number of i n d i v i d u a l s  the w e i g h t of  individuals in  the  50  CVI  M7  t  t  t •  JJL_  70  50  CIV-V o  •V +  * ^  70  CD O  c cd  "D  c  100  CI-III  Z3  <  t  t  Time Figure 73c.  (days)  70  Comparison of predictions ( s o l i d symbols) and cbservations (open symbols) f o r Paracalanus ( t r i a l 3).  296  j u v e n i l e day c l a s s were e s t i m a t e d a l o n g w i t h g r o w t h and parameters.  The  parameters  same a s t h o s e u s e d a b o v e .  controlling Convergence  poor, however; the parameter given  i n T a b l e XI  (trial  s e r i e s c o m p a r e d i n F i g 70d. low and  the second  observed. CI-III  Because  increase  t h e l o w e s t SSQ  rather  found i s  t h e p r e d i c t e d and o b s e r v e d  The  SSQ  o b t a i n e d i s not  in CI-III  time  particularly  i s not as s t e e p as  of c o n t i n u e d n a u p l i a r g r o w t h ,  50.  The  problem  here  p h y t o p l a n k t o n c a r b o n a f t e r day p e r m i t a s e t of growth naupliar  the decrease i n  19  parameters  after  The  16 and  day  ( F i g 64)  i s n o t g r e a t enough t o  which w i l l  a l l o w the  rapid  increase  i n CI-  19, w h i l e p r e v e n t i n g n a u p l i a r g r o w t h  into  30.  parameter  variation  too  i s that the observed decrease i n  growth necessary t o reproduce the l a r g e  b e t w e e n day  CI-III  e s t i m a t e s o b t a i n e d here i n v o l v e c o n s i d e r a b l e  across stages; this  t h a t a l l s t a g e s a r e now due  f o r t h i s c a s e was  i s a l s o t o o s l o w and p r e d i c t e d n a u p l i a r d e n s i t i e s a r e  low by day  III  r e p r o d u c t i o n were t h e  set giving  4) and  mortality  is partly  justified  by t h e  p r e s e n t over a range of food  t o t h e two c o h o r t a s s u m p t i o n .  The  maximum g r o w t h  fact  densities rates  (0.38,0.44,0.33) are h i g h e r than those found f o r a s i n g l e c o h o r t and  i n g e n e r a l agreement w i t h the average  the g e n e r a t i o n time. CV and a t t h e u p p e r f e e d i n g on d i a t o m s are required CVI.  There  v a l u e s of D a r e r a t h e r  end o f t h e r a n g e by C h e c k l e y .  to explain  suggested  High m o r t a l i t y  t h e low s u r v i v a l  day"  1  from  similar  for N l -  for adult  females  rates  from C I - I I I  i s no e v i d e n c e t h a t t h e m o r t a l i t y  a s low a s 0.02 initial  The  rate calculated  rate  i n CI-VI t o CIV-V t o  for nauplii  e x c e p t t h a t t h e e s t i m a t e d s i z e s of the  day c l a s s e s a r e a l r e a d y r a t h e r h i g h c o m p a r e d t o  is  50  CVI  J3L  0  70  50 CIV-V o  i CD O  c  cj)  *  0 4> 70  o  CO "O cz ZD  < 100 CI-III 0>  ©  4>  cb  0 F i g u r e 73d.  Time  (days)  70  Comparison o f p r e d i c t i o n s ( s o l i d symbols) and o b s e r v a t i o n s (open symbols) f o r P a r a c a l a n u s (trial  4).  298  o b s e r v a t i o n s and would need t o be e v e n h i g h e r mortality  r a t e s were i n c r e a s e d .  5.4 S t a b i l i t y  of the P h y t o p l a n k t o n - z o o p l a n k t o n  In t h e p r e c e d i n g s e c t i o n , ignored,  zooplankton  observed  phytoplankton  interaction  phytoplankton parameters. all  a s k what  phytoplankton  ingested  daily  efficiency  For are  E f which  o f 0.7 y i e l d e d  agreement  average  efficiency  made t o c o n v e r t  1 4  C  carbon  rate proportional to  was t a k e n  a s 0.7,  following  v e r s i o n o f S t e e l e ' s model, Landry  with  metabolic  the average  i s 24.9 pg C . l " . d a y "  1  compared w i t h t h e 1  given  observations to daily  densities  calculated  p r o d u c t i o n o f 25.7 pg C . l " .  fortuitous,  rate.  t o 0.5.  day 30 on, when p h y t o p l a n k t o n  1  used  the a s s i m i l a t i o n  a value of Ef equal  constant,  daily  i s probably  to allow  the a s s i m i l a t i o n  of metabolic  f o r the l a t t e r  ingestion  feeding  T h i s r e q u i r e d the e v a l u a t i o n of  of b a s a l and i n g e s t ion-dependent  low and r e l a t i v e l y  first  had on  phytoplankton  i n c o r p o r a t e d both  t h e p e r i o d from  calculated  As a  model was t h e r e f o r e a l t e r e d  and t h e t o t a l  In a l a t e r  h i s value  zooplankton  the zooplankton  was c a l c u l a t e d .  Steele(1974) .  efficiency  w o r t h y o f study.-  s p e c i e s t o f e e d s i m u l t a n e o u s l y on t h e  Assimilation  a combination  i s also  impact  and any component  ingestion.  Combining  the phytoplankton-zooplankton  The z o o p l a n k t o n  three zooplankton  efficiency  However, a s n o t e d by  concentrations according to the estimated  observed  an  d y n a m i c s were  b e i n g e s t i m a t e d on t h e b a s i s o f  concentrations.  i n the enclosure  we m i g h t  Interaction.  the phytoplankton  parameters  S o n n t a g and P a r s o n s ( 1 9 7 9 ) ,  step,  i fnaupliar  the crude  This  close  assumptions  p r o d u c t i o n and t o  299  convert  zooplankton  growth to i n g e s t i o n ,  the g r a z i n g parameter  but  i t does s u g g e s t  estimates are at l e a s t  i n the  right  that  ball-  park. The  q u e s t i o n of the s t a b i l i t y  zooplankton  interaction  (May,1974) i n d i c a t e growing  which  Holling,1965) oscillations rate  functional i n the  stable  shown  or type-3  remains  is  The  1  occurring  appears  t o be  behaviour  16m  *C  Given  and  a  2,  to unstable the h i g h t u r n o v e r  would e x p e c t  the  system  levels  that  to  of t h i s  of s p a t i a l  of t h e p h y t o p l a n k t o n  indicate and  limited.  suggests  nitrate  There  with g r e a t e r than  light  a response  a  a Sonntag type  on  refuge for  r e s p o n s i b l e f o r the p e r s i s t e n c e at  30 on.  i n t h e t o p 8m  that  (Holling,1965).  o r some s o r t  h i g h , the n i t r a t e  data  observed  response  the average  day  indicate  i f the p r e d a t o r p o s s e s s e s  of t h e CEPEX d a t a  from  between 8 and  lead  s i m p l e models a l s o  c o u l d be  While  t h e t o p 8m  depth.  1,  have s u g g e s t e d  examination  enclosure  will  i n CEE5, we  functional  constant  explanation.  between a p r e y w h i c h i s  of r e s o u r c e l i m i t a t i o n  populations.  of t h e z o o p l a n k t o n ,  relatively closer  response  is possible  the phytoplankton,  in  two  i n Chapter  Parsons(1979)  the part  p r e d a t o r - p r e y models  quickly.  equilibrium  threshold  interaction  phytoplankton-  i s f e e d i n g a c c o r d i n g t o a h y p e r b o l i c (Type  down v e r y As  an  Simple  i n the a b s e n c e  of p h y t o p l a n k t o n  break  and  that  exponentially  predator  remains.  of t h e  that  an  A  alternative  c o n c e n t r a t i o n i n the  profiles  show v e r y low  i s a steep 16  i n CEE5.  levels  nutricline  ug a t N O ' . l "  1  below  this  most o f t h e p r i m a r y p r o d u c t i o n  p r o d u c t i o n below t h i s I t seems p o s s i b l e  that  depth the  i n t h e e n c l o s u r e c o u l d be a n a l o g o u s  to  that  300  described  i n chapter  depletion,  with the n u t r i e n t  phytoplankton zooplankton The  into  following  t h e upper  stabilizing  nutrient  layer  limiting  the phytoplankton-  interaction. daily  respiration  by z o o p l a n k t o n  r u n d i s c u s s e d above was 12.5 jug C . l " . i s recycled  of approximately  as n u t r i e n t  i n the  Assuming  1  0.4 o f t h i s influx  flux  g r o w t h and t h e r e b y  calculated  combined  2 f o r S t e e l e ' s model  (Steele,1974), a net n u t r i e n t  20 pq C ( e q u i v a l e n t ) . 1 " . d a y " 1  1  i n t o the  t o p 8m i s r e q u i r e d t o meet t h e demands o f p h y t o p l a n k t o n The or  nutrient  g r a d i e n t from 8 t o 16m i s a p p r o x i m a t e l y  200 mg C ( e q ) . m "  rate  of 1 m .day" 2  C(eq).m" .day" 2  (ie  1  1  ( a s s u m i n g a C:N r a t i o  4  would m a i n t a i n  into  t h e upper  25 jug C ( e q . ) . l " ) o v e r  Farmer(1977) r e p o r t mixing 0.26 c m . s e c " 2  observed  nutrient  compatible To  with  test  zooplankton were a d d e d  A diffusion  C(eq).m"  3  S t e e l e and  2  1  so that the  rates are c e r t a i n l y  demand.  of the proposed n u t r i e n t - p h y t o p l a n k t o n  t o the zooplankton  and p h y t o p l a n k t o n  model,  following  and p h y t o p l a n k t o n  components  the f o r m u l a t i o n of equations a r e :  R = -A.R.P/(B+R) + V.(RO-R) + U . r e s ( P ) P = A.R.P/(B+R) - i n g ( P )  where r e s ( P ) and i n g ( P ) r e p r e s e n t t o t a l and  ingestion  4  o f 200 mg  0.4 t o 2.2 m . d a y " ,  nutrient  The n u t r i e n t  at.m"  r a t e s f o r t h e e n c l o s u r e s of 0.05 t o  the c a l c u l a t e d  interaction,  growth.  2 mg  p r o v i d i n g 25 mg  g r a d i e n t and d i f f u s i o n  the s t a b i l i t y  Steele(1974).  layer,  or approximately  1  a flux  of 7:1).  t h e 8m d e p t h p e r d a y .  1  that  r a t e s as c a l c u l a t e d  zooplankton  by t h e c o m b i n e d  respiration  zooplankton  301  model u s i n g t h e ' e s t i m a t e d phytoplankton to  f e e d i n g parameters  c o n c e n t r a t i o n P.  model t h e si-nking o u t o f t h e d i a t o m  species composition  from  diatoms  flagellates.  T h i s was a v o i d e d  with observed  phytoplankton  phytoplankton  and n u t r i e n t  the observed pg  In f a c t ,  The  in-situ about  value  as i n i t i a l used  determined  phytoplankton  1.4 d a y " , 1  limitation,  densities  the zooplankton  up t o day 30.  d y n a m i c s were t h e n  to  this  to 2 day" , 1  was c h o s e n  o r about  i n t h e t o p 8m.  temperature.  s t a b l e manner a f t e r  from out  from  lower  the i n i t i a l  of Chapter  phytoplankton  and a p p r o x i m a t e l y  1  been  to explain  concentrations  1  2.  as a  o r 0.05 pg  v a l u e s have  o f 20 p g C ( e q ) . 1 " . d a y "  F i g 74 t h a t  A  The h a l f - s a t u r a t i o n  I t i s needed  into  1  so as  the top  Steele(1974) . the system  day 30 a s e x p e c t e d  analysis  was t a k e n  1  as 5 pg C ( e q ) . ! "  a s 0.4 f o l l o w i n g  c a n be seen  t o be  The v a l u e s o f V a n d RO were c h o s e n  flux  As  t o be h i g h e r .  3 div.day"  i s r a t h e r low a l t h o u g h  8m a n d U was t a k e n  equations  under c o n d i t i o n s o f n u t r i e n t  (Goldman a n d M c C a r t h y , 1 9 7 8 ) .  qualitative  on w i t h  g r o w t h r a t e s have been c a l c u l a t e d  supposedly  provide a nutrient  It  switched  by t h e o b s e r v a t i o n s d i s c u s s e d a b o v e .  g r o w t h r a t e s up t o 0.75.A a t t h e low n u t r i e n t observed  model  The  i n the phytoplankton-nutrient  f o r g r o w t h has been t a k e n 1  reported  by d r i v i n g  conditions.  r e a s o n a b l e maximum a t t h i s  at.NOg.l" ;  bloom o r t h e change i n  to s i l i c o f l a g e l l a t e s to  t h e maximum g r o w t h r a t e  of A equal  constant  h a s been made  1  parameters  were l a r g e l y  no a t t e m p t  v a l u e o f P (17.87 jug C . l " ) a n d a low v a l u e o f R (20  C(eq).!" ) 1  and t h e p r e d i c t e d  behaves  ina  on t h e b a s i s o f t h e  The o s c i l l a t i o n  and n u t r i e n t  20 pg p h y t o p l a n k t o n  resulting  c o n c e n t r a t i o n s damps  carbon.I"  1  are  30  o  CD  =3.  cz o  CD  O  c ex. 0  30  Figure 74.  Time Predicted phytoplankton a f t e r day  30.  (days) concentration  7  i n CEE5  0  303  maintained rate also  f o r the  stays r e l a t i v e l y  In view of  this,  zooplankton  carbon  days.  hold  parameter  on  were v e r y  zooplankton  a t about  surprising  based  that  25  ug  ingestion  C.1" .day" . 1  1  the p r e d i c t e d  p r e d i c t e d and  on  observed  similar.  its  estimation technique  c o n s i d e r a b l e promise  zooplankton  performance  data, w i l l  on  this  data  of p a r a m e t e r  observation  I t i s hoped  to the  factor  error.  As  affecting noted  of  individuals  sample  the  that  samples,  counting  approximately  same t o t a l  errors  over  counted  time.  f o r any  o n l y on  relative  results  i n an  and  on  that  distribution  s t a g e s and also  on  placed  the number of  the s t r a t e g y one  individuals  t h e number o f  g r o u p of  data  of e s t i m a t e s i s  the a c t u a l  number of  i t s a b s o l u t e abundance but  to other  the  a commonly p r a c t i s e d  irregular  T h i s i s because  particular  the  are based  i s , on  in counting these  sample,  i n t e r m s of  parameters.  It i s rather interesting  the  of  reliability  72  to  of  a discussion  i t s d e p e n d e n c e on  such  employed  each  that  above, the c o n f i d e n c e l i m i t s  splitting;  counted.  appears  t y p e s of e x p e r i m e n t s  o b s e r v a t i o n s i n F i g 67,69,71 and  statistics  here  investigation  set, p a r t i c u l a r l y  i n t h e e s t i m a t i o n of  obvious  presented  f o r the  e s t i m a t e s and  s e r v e as a g u i d e  of most v a l u e An  as a t o o l  growth dynamics.  reliability  not  The  Conclusions. The  on  series  40  constant  i t i s not  time  phytoplankton  5.5  remaining  of from  of o b s e r v a t i o n individuals  s p e c i e s depends i t s abundance  s t a g e s or s p e c i e s .  These c o n f i d e n c e t o w h i c h t h e pumped  limits  do  samples a r e  not  take  i n t o account  r e p r e s e n t a t i v e of t h e  the  degree  enclosure  304  and, and  in  this  Calanus  series large  the  local  anomalous are  or  data  several an  confidence  could  in  of  squares  and  I have  Where  smaller-scale  Even  a  values  single  would  time  numbers.  listed  in  experimental In was  not  rather even  designs  a l l cases, possible high  lower  to  ad  very hoc  ambiguity  or  The  data,  they  types for  establish  of  will data  Paracalanus any  rates.  d e n s i t i e s but  i t seems  perhaps  be  justified.  ambiguities  in  a l l , represent  ambiguity more  and  are  informative  collected.  nauplii  preference This  indication  phytoplankton  of  be  introduced  rigorous  may  suggest to  a  understood,  after  sources  estimates  observation  some  certain  of  used  certain  more  minimized,  the  approach  give  uncertainty  obvious  metabolic  a  observations  that  such to  be  minimising  better  experiments,  of  by  of  uncertainties  here  are  can  parameter  variations in  errors  Some  for  make a n y  parameter  except  to  basal food  SSQ  the  distribution  content  remain.  hope  for  allowed  data  been  series  the  algorithm  e r r o r s were  zooplankton here  the  the  to  Marquardt  e r r o r s , the  laboratory  approach  errors  limits  be  sampling  i f observation  parameter only  of  observation  irregular  of  features' in  sensitivity  statistical  i s an  the  in  above  parameters.  This  assuming  evaluated  discussed  the  observations.  or  time  i s known,  erros  of  ignoring  These  of  the  questionable  troubling.  distribution  While  anomalous  Pseudocalanus  of  (Benson,1978).  by  by  the  analysis.  the  sum  except  of  u n s a t i s f a c t o r y degree  approximate  weighted  features  particularly  explained  introduced  Jacobian  give  be  i n one  and  the  series  not  errors  Where  to  time  could  procedure into  regard,  between  might  likely  and  be that  adults, i t zero  and  accomplished direct  at  305  information stage  densities The  depends  through  or  of course  fixed  i n order  been  reduced  production  (Mullin  constant is  coincided so  that  types  with  could  diatoms  flagellates.  been  were  size  paid  weight  This to a  generations. the  size  found f o r  would  to species  and q u a l i t y  the respect  estimating  a  have  and egg  In CEE5,  from  high  constants  estimated.  determined  and should  therefore  The  here  half-saturation  composition  to  i s varying  in species flagellates  concentrations for different  t o low, food  half-saturation  primarily apply  zooplankton  Frost,1972;  t h e change  carbon  on  i t deserves  single  phytoplankton  half-saturation  n o t have  densities  with  between  confusion  to silico-flagellates  the change  obtained  This  Paffenhoffer,1970;  In g e n e r a l ,  separate  constants food  has not been  from  t o assume  i s exposed several  identified  particle  t o be m i s l e a d i n g .  composition  value.  for confusion  had been  f o r i n g e s t i o n when  likely  require  et al,1979).  and Brooks,1970;  Poulet,1973).  would here  parameters  stage  This  of the cohorts  necessary  absolute  i f each  time.  estimated.  of food  rates  i t was  over  and m o r t a l i t y r a t e s , as has been  i f nauplii  effect  ingestion  which  (Parslow  had been  life-history  f o r growth  density  of growth  their  only  tendency  i n food  so t h a t  to estimate  some  as  constants  the progression  of v a r i a t i o n  recruitment  models  The  stages,  densities  was  initial  simpler  with  c a n be a v o i d e d  There  as w e l l  necessary.  of h a l f - s a t u r a t i o n  history  of food  of i n d i v i d u a l s  on a v a r i a t i o n  pattern  assumption  of  be  c o i n c i d e d here  life  stage  range  may  estimation  variation  some  on t h e w e i g h t s  by b e h a v i o u r  t o ingestion of  a t low  306  Both  i n t e r m s of  manipulating laboratory  and both  food" d e n s i t y and  experiments  zooplankton involves  reducing observation errors  may  parameters.  be  (Droop  of more immediate  equilibrium  phytoplankton approach  of  and  and  this  experimental  Scott,1979)  transient  nutrients type,  timestreams,  those  as d a t a  manipulations  Features  (Paffenhoffer,1970),  and  included  and  quantity  and  i n an  tested  i n the  field.  reproduced. agree,  field,  the  field  see  i f the  I f t h e o b s e r v a t i o n s and  i t may  the  l a b o r a t o r y and  be  concluded  although data  one  has  represent.  that no  the  and  The  be  s u c h as same day  also  with  be  field  class life  food  usefully  of t h i s  type.  from l a b o r a t o r y  traditional  insert  the  Brooks,1970;  ratio  study  predictions  approach  parameter  values  observations are judged  to  laboratory values apply  in  indication  (Perhaps  the  at p a r t i c u l a r  i n sex  interactive  been t o c o n s t r u c t a s i m u l a t i o n model, i n the  i n the  of w e i g h t s  changes  for  system could  here,  status (Mullin  and  an  used  studying  the e x p e r i m e n t a l  however, t h e c o n c l u s i o n s drawn  have t o be  obtained  By  ( P a f f e n h o f f e r , 1 9 7 0 ) , might  parameterised  Ultimately, studies  Harris,1976)  c o u l d be  between p a r a m e t e r s  the v a r i a t i o n  In  from p r e d i c t e d  individuals  stages with n u t r i t i o n a l  quality  has  across  of  been done f o r  Meyer,1972).  n e g l e c t e d i n t h e model u s e d  of g r o w t h r a t e  phytoplankton  investigation  analysis.  of  in estimating prospect  of  as has  to e s t i m a t e parameters  chosen.  Paffenhoffer  the  (eg C a p e r o n and  greatest discrimination  history  and  behaviour  allowing  variation  cultures  the e s t i m a t i o n t e c h n i q u e  d e s i g n as w e l l  technique's a b i l i t y  use  P e r h a p s t h e most e x c i t i n g  and  i n t e r m s of  composition, s m a l l e r - s c a l e  the maintenance of c o n t i n u o u s  zooplankton  and  as  are  t o how  the p r e d i c t i o n s  good a are  test  307  completely  insensitive  to  changes  combinations  of  disagree  i t i s suspected  be  and  parameters).  responsible, this  the  Again  parameter  sets  technique  offers  where may  placed  estimates freezing it  has  been  levels  on  do  as  fresh errors  to on  been  refugia.  A  model  and  of  the  need  as  a  just  that  they  are  not  the  I chose  the  to  questions estimate  It  argued  with  the  that  phytoplankton-zooplankton the  phytoplankton  that  stability  at  of  I  zooplankton  should  grazing to  model  uncertainties carbon  to  concerning  discussed  earlier.  as  constant  a  or  spatial  thresholds  the model  this  do data  zooplankton does  raise  estimation.  parameters  the  used,  between  explain  parameter  fitted  of  nutrient  phytoplankton  have  field  be  low,  p r o p e r t i e s of  concerning  observed  can  thresholds  necessary  limits  technique  nutrient-phytoplankton-zooplankton  model  be  of  result  best,  possible.  i n CEE5  for grazing prove  At  confidence  the  is  is  other  l a b o r a t o r y and  data  may  'tuning'  agreement  discrimination  phytoplankton here  to  problems.  worst,  whether  values  estimation  estimating others  not  comparison  of  and  and  At  does  interesting  view  estimates  this  zooplankton  in  understood,  well  these  or  predictions  whether  non-linear  are  basis  any  to  to  Again  exist;  set.  The  as  until  approach  explained  without  alternative  indication  indicate the  no  and  i n parameter  parameters  well.  parameters  here,  changes  statistically.  p e r s i s t e n c e of  has  could  i s no  parameters  observations  leaves  altering  parameter  values  limitation,  Here,  a  certain  The  some  might  compared  parameter  not  there  observation  be  by  the  that  approach  s i m u l a t i o n model  obtained.  If  in certain  by  driving  the  concentrations.  nutrient-  full  data  set,  the  true  level  especially of  308  Apart which all  from t h e n u m e r i c a l p r a c t i c a l i t y  would r e q u i r e the simultaneous  o f s u c h a scheme,  e s t i m a t i o n of parameters f o r  s p e c i e s o f z o o p l a n k t o n , i t s r e s u l t s c o u l d be d i f f e r e n t  following qualitative  sense.  d r i v e n by t h e o b s e r v e d  When t h e z o o p l a n k t o n m o d e l i s  phytoplankton d e n s i t y , p r e d i c t e d weights  d e p e n d e x p o n e n t i a l l y on g r a z i n g p a r a m e t e r s . concentrations, corresponding increase In  A t low food  i n c r e a s i n g Cn o r d e c r e a s i n g D w i l l  i n c r e a s e i n e x p o n e n t i a l growth  in individual  i n the  weight  after  lead to a  r a t e , and a l a r g e  sufficient  time has e l a p s e d .  t h e combined n u t r i e n t - p h y t o p l a n k t o n - z o o p l a n k t o n model,  p r o v i d e d t h e e q u i l i b r i u m a t low n u t r i e n t c o n c e n t r a t i o n i s m a i n t a i n e d , p h y t o p l a n k t o n p r o d u c t i o n and t o t a l i n g e s t i o n a r e f i x e d by t h e n u t r i e n t species w i l l will  flux.  zooplankton  Changes i n D a c r o s s  n o t change p r e d i c t e d i n g e s t i o n o r growth  r a t e s but  change t h e e q u i l i b r i u m v a l u e of p h y t o p l a n k t o n c a r b o n .  nutrient  flux  fixed,  w o u l d be t h r o u g h metabolic  rates.  t h e most d i r e c t  t h e growth  way t o a f f e c t g r o w t h  gross e f f i c i e n c y ;  A very d i f f e r e n t  With  p a t t e r n of  rates  that i s , through parameter  d i s c r i m i n a t i o n m i g h t be e x p e c t e d . There i s another n u t r i e n t based  model.  l i m i t e d by n u t r i e n t phytoplankton rate.  growth  i n t e r e s t i n g aspect t o the behaviour An e q u i l i b r i u m w i t h p h y t o p l a n k t o n  levels  r a t e , A, e x c e e d s t h e z o o p l a n k t o n c l e a r a n c e 1  (Eppley,1972). on o b s e r v e d relatively  growth  i s o n l y p o s s i b l e p r o v i d e d t h e maximum  The v a l u e o f A u s e d h e r e , 2 d a y " ,  t h e maximum p o s s i b l e  of t h e  i s probably close to  f o r phytoplankton at t h i s  temperature  The c a l c u l a t e d z o o p l a n k t o n c l e a r a n c e r a t e s ,  both  and p r e d i c t e d p h y t o p l a n k t o n p o p u l a t i o n s , a r e c o n s t a n t a t a b o u t 1.4 d a y "  1  f r o m d a y 30 o n , i n s p i t e  309  of  the  5 to  fact that  10  fold  period.  It  Paracalanus  increase i s only  P a r a c a l a n u s and zooplankton  i n d i v i d u a l s , f o r example,  i n weight  the  high  and  ingestion  mortality  rates  rate  over  estimated  C a l a n u s c o p e p o d i d s w h i c h keep t h e  clearance  rate  low  enough t o a l l o w  undergo a the  same  for  total  the  equilibrium  to  persist. The  source  carnivores  are  possibility plausible,  of not  that  food  t h e y were c a u s e d  results.  since  densities  over-depletion  rates  in culture. of  zooplankton mortality explicit assumed  the  behaviour  ecosystem example of  i n the of  i n the  seem and  food  densities  low  (1970)  mortality  inconsistent,  but  seems and rates  were  i s in  fact  (1976) f o u n d no  speculated  during  response  to  mortality  nutrient  might  low  may  be  food  rate the  b a s e d model  CEE5 e n c l o s u r e  i n Chapter  a transient  well  marked low  2  that  approach to  prevented  densities.  on  If  the  i n Pseudocalanus e l o n g a t u s at I t was  t h i s phenomenon.  known.  Parsons,1979),  Harris  models used h e r e ,  the  low  i s not  Paffenhoffer's  equilibrium  d e p e n d e n c e of  was  by  f a c t that  phytoplankton  stable,nutrient-limited  no  The  Paffenhoffer  in mortality  rates  ( S o n n t a g and  f o r P s e u d o c a l a n u s may  encouraging, increase  important  mortality  e s p e c i a l l y i n v i e w of  Checkley's(1980) estimated  these high  by  a  Although  n u t r i t i o n a l status  estimation suggest  results  that  the  have c o n s t i t u t e d  an  and  a  310  Chapter D I F F U S I O N , SINKING AND  6.1  GROWTH OF  continue  s t u d i e s of w e l l - m i x e d p h y t o p l a n k t o n  to y i e l d  g r o w t h r a t e on  more d e t a i l e d k n o w l e d g e o f  conditions  ( D r o o p , 1 9 7 4 ; Goldman and  recognise  that  the  McCarthy,1978).  population  as  i m p o r t a n t as  in-situ  Moreover, recent  more s u b t l e the  simple  certain highly  success.  An  task,  by  no  these study  example i s S v e r d r u p ' s c r i t i c a l  means  /(l-exp(-k.z  C R  assumed t o  be the  r e s p i r a t i o n i s i n d e p e n d e n t of d e p t h  and  to l i g h t  growth z  M  )) = I / ( k . I ) e  to  Under  is proportional f o r net  even  d e p t h model  M  the c r i t e r i o n  to  considerable  mixed t h r o u g h o u t a mixed l a y e r of d e p t h z .  f u r t h e r assumptions that  of  past-history effects  have e n j o y e d  (Sverdrup,1953), i n which phytoplankton are  C R  model  m a t t e r ; the  that  sinking  s i m p l i f i e d approaches  phytoplankton populations  z  must  rates.  modelling  production  considering  stud i e s (Marra,1978a,b) p o i n t  l i g h t - d e p e n d e n c e of g r o w t h  uniformly  To  of  history  m i x i n g and  growth r a t e s .  i n t e r a c t i o n s b e t w e e n m i x i n g and  Nevertheless,  cell  o c e a n s , one  turbulent  mixing alone i s a challenging  complete.  dependence  i s d i s t r i b u t e d i n s p a c e and  p h y s i c a l p r o c e s s e s p r e c i s e l y i s no turbulent  and  cultures  However, i n  i n the  p h y s i c a l g r o w t h p r o c e s s e s s u c h as be  the  of n u t r i e n t s , l i g h t  natural phytoplankton populations  on  PHYTOPLANKTON.  Introduction. Laboratory  may  6  c  < z  .  i n t e n s i t y , Sverdrup C R  , where  derived  311  Here  I  i s the average  e  photosynthetically  (P.A.R.) p e n e t r a t i n g t h e intensity. to data found  s u r f a c e and  This criterion  from W e a t h e r s h i p  to explain  M  was  I  c  active  i s the compensation  successfully  i n the A t l a n t i c  other data  sets  radiation  applied and  (eg P a r s o n s  has  by  light  Sverdrup  s i n c e been  and  LeBrasseur,1968). Another of  simple  phytoplankton  who  diffusion varying  d e v i s e d by  Riley,Stommel  steady-state p r o f i l e s  K and  constant  sinking  a s a s t e p - f u n c t i o n of d e p t h .  realistic  profiles  assumptions  growth r a t e , simulation (Riley  to m o d e l l i n g the depth  concerning  diffusion,  has  rule),  sinking  and  become more p o p u l a r  and  depth  have been d e v e l o p e d  et  a l ,1977; The  the a n a l y t i c  results  m o d e l s have t e n d e d  solutions.  of  interest  of S v e r d r u p  towards  complexity  i s not  However t h i s  d e p e n d e n c e of particular  obtained  with depth  of  Numerical  w i t h the advent  possibly  also  rate  solutions  of  with  t h e h e l p of  treating  h e r b i v o r e s as  (eg W i n t e r  computers  f u n c t i o n s of  e_t a l ,1975;  Jamart  Wrobleski,1977).  same d e g r e e  resulting  (eddy)  t e c h n i q u e s under more  a number of n u m e r i c a l m o d e l s and  when a  w w i t h growth  grazing.  numerical  nutrients  time  rate  Bumpus(1949),  result  the v a r i a t i o n  phytoplankton, and  which  distribution  These a u t h o r s  using numerical  e t a_l o b t a i n e d t h e i r  slide  and  population i s subject to a constant  rate  steady-state  a  was  c o n s i d e r e d the  phytoplankton  approach  solutions  has  been  or R i l e y  conducive  the  can  and  Numerical  r e a l i s m ; the  to obtaining a n a l y t i c  a l s o means t h a t  structural  parameter v a l u e s used  shown i n e x t e n d i n g  et a l .  increased detail  complexity on  not  be  assumptions  investigated  the and only  the in a  312  limited  sense, u s i n g s e n s i t i v i t y  approximate  analysis.  Analytic  or s i m p l i f i e d models can p r o v i d e u s e f u l  s o l u t i o n s of insights  into  t h e b e h a v i o u r o f t h e s e more c o m p l i c a t e d n u m e r i c a l m o d e l s ,  as  as e x p l a i n  start  field  observations.  I n t h i s c h a p t e r , a modest  i s made by e x t e n d i n g t h e r e s u l t s o f S v e r d r u p t o n o n - u n i f o r m l a y e r s and n o n - z e r o more r e a l i s t i c  6.2  sinking  dependence of growth  Review of a S i m p l e The  r a t e s , and  and  Model.  dt  s i n k a t r a t e w(z)  the p a r t i a l  dzS  rate  and grow a t r a t e p ( z ) ,  6.1  '  ( R i l e y e t a l ,1949). z = 0 (zero flux  downwards f r o m t h e s u r f a c e  d i f f e r e n t i a l equation:  bz  dz/  function  P h y t o p l a n k t o n c e l l s a r e s u b j e c t t o eddy  diffusivity-K(z), yielding  depth.  c o n c e n t r a t i o n o f p h y t o p l a n k t o n i s t a k e n t o be a  time t .  mixed  those of R i l e y e t a l t o a  r a t e on  P ( z , t ) o f d e p t h z, m e a s u r e d v e r t i c a l l y z=0,  well  The  boundary  c o n d i t i o n s a r e KdP/dz=w.P a t  t h r o u g h t h e s u r f a c e ) and P  0  i s assumed t o be a f u n c t i o n o f d e p t h o n l y ;  nutrient-limitation  as  in particular,  i s n o t c o n s i d e r e d and any e f f e c t  p h y t o p l a n k t o n d e n s i t y on t h e e x t i n c t i o n c o e f f i c i e n t light availability  i s ignored.  Growth  z-*co.  of and  These s i m p l i f i c a t i o n s  hence  are  c o n s i s t e n t w i t h a d i s c u s s i o n of the o n s e t of t h e s p r i n g bloom, in Sverdrup a b u n d a n t and  ( 1 9 5 3 ) , or of a r e g i o n  i n which n u t r i e n t s  are  t h e p h y t o p l a n k t o n c o n c e n t r a t i o n i s k e p t low  g r a z i n g , as appears As m e n t i o n e d  t o be t h e c a s e  above,  by  i n the S u b - A r c t i c P a c i f i c .  R i l e y e t a l (1949) c o n s i d e r e d t h e  case  as  313  of  K,w  constant  u(z)=  and it  looked  p  L  -d  - d  0 < z < z  at this  T = p .t,  6.1, using  the  yr  as  given  bs  2  0 5  s  o f p(z)  .  non-dimensionalized  When  given  proceeding,  these  are substituted  i n 6.2 a n d t h e a s s u m p t i o n  f ( l - S).P  0 < 5 <  <  I-  S > S  S.p  parameters  6.3  3  co, 5g  and  5  are  by.  5  s  boundary  bP/dS  The  to introduce  The n o n - d i m e n s i o n a l  oo = w / ( 2 . ( j u . K ) °- ) ,  The  Before  equation  d!p - 2 . G U . 3 _ P +  obtained.  6.2  g  z. ( j u / K )  t h e form  hP =  is  g  solutions only.  point  S=  s  constant,  function:  z > z  i s convenient  into  step  for steady-state  variables  K,w  r  and u ( z ) a  S = d/p  s  5=  ,  3  z .(p /K)°-  5  3  s  .  conditions are  — 2. co .P a t 5 =0 a n d P -» 0 a s 5 -* co .  steady-state  solution  general  significance  special  case  solutions  of R i l e y  by t r e a t i n g  of a general  o f 6.3 o f t h e  e t a l c a n be g i v e n  the steady-state  eigenvalue  form  problem;  that  a more  solution i s , by  as a seeking  314  X.  P( $ , T ) = e x p (  Then  0  p  satisfies  ).p(S  T  the  the  problem  ODE's w i t h S  >  S  p -» 0  5  as  =  S  $ ~*  0 0  followed problem  here  )  in  the  $  5  > S  3  6.4  5  regions  the  r e q u i r e m e n t .that  is easily it will  conditions  6.4  and  0  p,p'  5  <  p'(0)  =  be  linear <  S$  ,  2.co.p(0),  continuous  at  -(co  shown  > 0  (1-u>  2  and  - ( 8  the  method  more  )=exp(co.5  p(S  ( S +X  +  2  that  a  is  difficult  )h(S  )  s o l u t i o n to  i s given  ))  + X ))  0 5  ,  9  0 < S < 5  ) ) .h  conditions  3  =  in  + A ) ) . h  - ( S  2  ( C.(cos(a.(S - S  =  indirect  and  gives:  continuity  S~ + X )  an  useful  Letting  f ( l - 6 u  be  solved;  prove  later.  in  easily  a  <  (ODE)  second-order  , and  Lc.exp(-b.(S - 5  where  two  boundary  treated  boundary  h(S  solving  coefficients  I  2  S  ) .p  the  as  0 = h"+  1- to -(  to  <  to  problem  substituting  can  +X  0  equation  .  9  This  It  -(8  reduces  constant  , subject  3  differential  f l - ( S + X ) .p  I Thus,  .  ordinary  p"-2.co.p'+  =  )  5  6.5  5  > S  6.5  3  satisfying a l l  is possible  only  if  8 +/\  0,  <  S  by:  -  b.sin(a.(5 - S  3  ))/a)  0  ))  b  >  < 5  5  =  ( co  2  + S + X ) °- . 5  The  non-  3  >5  3  315  dimensional  tan(a. S  growth r a t e  2  =  A g r a p h of the  one  f ( a ) vs a  5  2  (0, ( 1 - cu ) °- ) 2  in Figure  correspond  5  the. v a l u e  of  S  s o l u t i o n or many s o l u t i o n s o f  value  plot  case, of  only  a or  that  the  s o l v i n g the  of  the  figure.  ( F i g 76). For  c o n d i t i o n s and, vertical  by  the  a of  the  necessary  for a steady-state  The  steady-state  reduction  dimensionalisation representation consideration  of of  i n the has  ),  A=  increase  subject  i n the be  of  ) of  by  S  that p(5  s o l u t i o n s of  to  ).  the  the  i t can g  vertical  6.6  non-  graphical  ( F i g 76).  be  ) is  ).  s o l u t i o n s of  the  these  On  =_/Y(co,S  complete 6.6  is  closely .  f r e e p a r a m e t e r s by  simple,  has  population,  interest,  l o s s r a t e of  number of  some a p p r o x i m a t e  contour  possible for  e i g e n s o l u t i o n , p(S  i s given  a  a  here.  j\.{ou,S$ )-S  should  the  smallest  interest  a population  s o l u t i o n and  allowed  relevant  t o the  In  f u n c t i o n J\{ u>, S9  a  s o l u t i o n s are  a non-dimensional  of  no s o l u t i o n ,  numerically,  population  that  be  i n t e r p r e t a t i o n s are  corresponding  values  interval.  i s of  6.6  of a i n  d e p t h of p o s i t i v e g r o w t h  sufficient  i f steady-state  in this  may  in this  as  r a t e of  concluded  profile  Only values  , there  3  8 < _/Y( cu, Sg  growth  after  75.  ~X+S  + S  Two  given  profile  approximated  of  s i n k i n g r a t e and  maximum s u s t a i n a b l e  hand,  6.6  5  to permissable  6.6  eigen-condition  largest solution  been g e n e r a t e d  other  eigen-condition:  s o l u t i o n corresponding  l a r g e s t value  By  non-dimensional  this  the  2  i s shown  D e p e n d i n g on  latter  the  satisfy  f(a)  interval  S +A .  must  = a. ( 6o + ( l - a - ) °- ) / ( a - cu. ( 1 - a ) °- )  )  g  A  The  in certain  316  Figure 75.  An i l l u s t r a t i o n of the e i g e n c o n d i t i o n 6.6 showing graphs of t a n ( a . J ) ( s o l i d l i n e ) and f ( a ) (dashed line).  317  318  limiting  cases  biological of  6.6  A+ Now  =  S  rates.  only  Sg  =  3  and  processes  exist  8  is quite  z  ~  2  f o r 2 . co  2.  depths  rate,  with  characteristic  a  l a y e r .of d e p t h  s  $  For  population  1/S  A +S  *  A+  8  »  -  1  the  a  9  a  expected  sinking  term  77/2 and  2  3  2  to  approximation 3  -  greater,  TT  /S  2  .  2  3  The  vary of  the  by  .  the  mixing  growth  w  markedly  ratio  For below  would  g  growth  of  result  rate  l.cu.S^  obtained  ,  from  the  growth  scale  and  the  with on  depth w,  over  in a  over  depends  z  biomass  loss  time  gross  layer  u n i f o r m l y mixed  for mixing  6.6  high  (Fig 7 7 a ) .  g  for sinking  growth  to  predicted  rate  non-dimensional  (rr/(2.5 )) .  co  and  solutions  exponentially  p o p u l a t i o n were  the  or  z  integrated  total  with  (a. S 3 ) *  than  ) to  compared  co = 0 ( 1 / 5 ^ )  physical  approximately  within  i s just  3  scale  be  given  . (JU . z ) / K s  or  g  time  The  1 -  greater  the  ),  3  uniform  s  correction  1,  can  layer.  cv«  If  »  3  i s long  this  s  the  of  relevant  corresponds  K/(u .z ).  K/(ju .z ), 5  g  i s just  layer  g  If  w.z .u /K,  rate  which  ).  3  1  3  depth  g  R.p(0)/(p .z )  S «  much  ( p .z .p(0) s  are  population declines  5  (  loss  the  . ( A + 8 ) = ju .z  production  balance  5g<< 1,  and  i s almost  sinking  s  <  population  zero  the  For  so  5  over  of  .5g  CU  s  The  p  involved.  . ( j u / K ) °- ,  3  i s mixed  rate,  revealing  6.6.  within If  co=0  (or  320  These  two c a s e s  a>=0, maximum is  p  reduced  + d = JU  so  corrrespond  biomass  only  S  -  diffusive  pronounced  to different  occurs  at the surface  by d i f f u s i v e  7T , K / ( 4 . z 2  2 3  profiles.  For  ( F i g 77b) a n d g r o w t h  losses.  In dimensional  with  For 6O=0(l),  terms,  ),  losses decrease  sub-surface  depth  maximum  K.  (Fig 77c).  there  i s a  The d i m e n s i o n a l  growth  equation i s  p  5  + d = p  There  - w /(4.K)  are several  sinking  loss  uniformly a  determined  population  of these  existence  large  g  i s because  The appearance terms  rates.  maxima.  property used  effect  equation.  as i t would  loss  i t s dual  of changing  K depends  has  rate  K in  role in  Increasing K from  layer  rate i s  of the d i f f u s i o n  losses  The  fora  the sub-surface  emphasizes  growth  reduces  the euphotic on t h e r e l a t i v e  two-losses. of pronounced  -Sg a n d m o d e r a t e  by R i l e y  surface  form  on w / z ,  (K/w) a n d t h e s i n k i n g  and mixing  The o v e r a l l  The  noted  This  )  features of t h i s  not depend  layer.  2 3  l o s s e s but increases mixing  importance  for  2  by w / ( K / w ) .  determining  zone.  does  of order  sinking  sinking  TT .K/(Z  interesting  term  mixed  thickness  both  -  2  s  co  e t a l (1949),  sub-surface  i s rather who  used  I t i s not c l e a r ,  i s the result  to approximate  p{z).  This  in solutions  interesting.  I t was  i t to explain observed  however,  of the rather  maxima  t o what  unrealistic question  extent  sub-  this  step-function  will  be a n s w e r e d i n  321  the  next  mqre  6.3  section,  realistic  A More The  Riley  where  form  e_t a_l t o a l l o w  form  that  mean  intensity  to  making  realistic  this under  discussion, it  replaced  at that  assumption  see Sverdrup(1953)).  of f i n i t e  rate.  extreme  to the step-function  surface  Finally,  coefficient  Sverdrup's  k for light,  = u .exp(-k.z)  bx  £p-2.cu.&j>  bS*  may  b$  quite  The r e s u l t s  o b t a i n e d by  form,  since  saturation  takes  detailed  critical  rate  and  realistic  these  a constant  the form:  ( e x p ( - 2 . S //? ) - S ) .P  depth  the opposite  or i n h i b i t i o n  6.1  using  non-zero  sense  more  assuming  as b e f o r e , equation  +  to the advantages  be  - d  s  Non-dimensionalizing  &P=  p(z)  (1953)  (for a  diffusion  assumption,  near the  intensities  i t r e p r e s e n t s i n some  involving  rates  are several  ( S t e e l e , 1 9 6 2 ) a r e i n t e r m e d i a t e between  Under  u(z)  There  by  while  i s proportional  extension of Sverdrup's  sinking  profiles  growth  The a s s u m p t i o n  light  chosen  solutions  by S v e r d r u p ' s depth  depth.  here.  low s u r f a c e  to the case  production  analytic  p r o d u c t i o n a t each  represent a direct  criterion  f o r u ( z ) was p r e s u m a b l y  f o r simple  i s now  assumption light  form  the general p r o p e r t y of higher  This  i s obtained for a  Model.  step-function  surface.  solution  of p ( z ) .  Realistic  maintaining  an a n a l y t i c  becomes:  at the  two. extinction  322  where  t h e new  non-dimensional  2. ( p / K ) °- /k.  /3 =  exp(/\.T+cx>.S  h"  ,  boundary  5 ^  co .  change  conditions:  The s o l u t i o n  of independent  2  2  6.8  Provided  v  boundary  h'(0) =  h(S )  P =  satisfying  6.7  cy.h(O),  exp(a>.5  c a n be o b t a i n e d  variable.  Letting  ).h(S  by means  first  boundary provided The  2  of a  y = yC?exp(- S//3  ) , h(.5 )  equation  of order  j3 . ( A + 6" + ou  y=  i s p o s i t i v e , the solution  2  + C  6.8  2  2  ) . 0-5  c a n be w r i t t e n a s :  .Y (y).  2  v  conditions  on H ( y )  are:  a s $-»co  7  0.  or y  -cu.H(fi)  H' (/3 ) =  behaves  + ou  0 as  )  yields  ).h(S ) = y " ^ . H ( y ) - » 0  expicu.S  The  o f t h e form  2  t o 6.7  i s Bessel's  X+8  = q.J (y)  The  with  by  + y.H* + ( y - / 3 . ( / \ + S + a p ) ) . H = 0  Equation  H  i s given  + o; )).h=0  = H ( y ) a n d s u b s t i t u t i n g i n 6.7  y .H"  (3  eigensolutions  ).h(S ) a r e s o u g h t ,  (exp(-2.S / / 5 ) - ( X + S  +  and  As b e f o r e ,  5  s  parameter  boundary  like  y"  v  a s y - > 0.  condition )\ + 8 second  >  condition  implies As J  is satisfied  y  that  behaves  provided  C  = 0,  2  since  Y  v  like  y  v >  u>.(3 ; t h a t i s ,  v  a s y - » 0, v  this  0. boundary  condition  produces  the  eigen-condition:  323  j ; ( /3 ) =  -cv.J (/3  )  v  Only values  j3  of  .  6.9  f o r 0 < y < (3  such t h a t J ( y ) > 0 v  positive  and  therefore meaningful population  of  and  J (y)  J (y) v  zero It  of J ( y ) , y  i t is clear  i s a property  obtain  of  (3 > V  condition  i s given  v  a meaningful  condition  f o r the  Equation  6.9  can  be  6.9  considered. neighbourhood  T h i s was  F i g 79  For  (3«  of  the  plot  the  the  origin  must e x c e e d  j  so  also a  that  .  y  the  in order  to  necessary  of  TV  i s given  i f some l i m i t i n g  Taylor  can  be  cases  a  79.  underlying  the  are  in a  v  to obtain  as  in Fig  for J  expansion  used  \+S  to obtain  p h y s i c a l processes  results  1,  first  satisfied  solved numerically  into  sketch  denotes the  > V,  y  A  to  profile.  A contour  and  j  < 1 must be  2  step-function  further insight  condition  + o»  y  t h a t /?  functions that  solution.  A.{uo,f3).  function Again,  A+S  profiles.  If j  from F i g 78  Bessel  or  i n F i g 78.  correspond  the  approximate  solution  X+S  =  This  ( {3/2)*  is identical  function is  those  2.cu  A  explained  the  on  population  a s s o c i a t e d with  growth r a t e . i s  production constant  to  over  /3/2)  to the  growth p r o f i l e  simply  small,  the  -  approximate if  5$  solution  i s equated  p h y s i c a l grounds. i s mixed over change  d e t e r m i n e d by  the  i n t e g r a t e d biomass. d e p t h s of  order  z  g  (3/2.  For  5  As  ratio  As or  to  depths  i n growth.  f o r the  g  step-  The or  agreement  (3  very  l a r g e compared explained  earlier,  of  integrated  biomass  i s more or  1/k,  total  with  production  less is  the  324  Figure  78.  An  illustration  showing  of the e i g e n c o n d i t i o n  graphs of the f u n c t i o n s  6.9,  J „ ( x ) , J^,(x) .  326  same  under An  j 3 »  approximate  1,  law.  t h e two g r o w t h  treatment  i s r a t h e r more  An a s y m p t o t i c  L/^V  i s needed.  laws  complicated  This  descent  function  (Sommerfeld,1949).  that,  )\ + S  The  'mixing  This the  cu  loss'  i s much  term  slower  by  p( S )  = exp( co. 5 ) . J  For  V  zero  over  in  large, an  v  J ( y ) decreases  growth  the  layer  approximately increase  of width  with  by a  factor  a s f3  growth  V,y  for large by a p p l y i n g  this  /  3  with  t h e method o f  expansion,  i t c a n be  .  /3'  like  a s j3  2/3  term  increases.  ( c o n s t . Sg " ) f o r 2  The e i g e n - s o l u t i o n p ( S  ) is  )) .  i n c r e a s e s towards  0(/tf  at a depth  law r e s u l t s  1 - 2 . o r  i n X+S  2  as y  v  exponential  case,  i n (1//9 ),  profile.  p(6* ) t h e r e f o r e o c c u r s  biomass  Using  ( (3 . e x p ( - S /{3  interval  limiting  the corresponding  s t e p - f u n c t i o n growth  /3/2.  S$ =  representation of the Bessel  decreases  than  given  )  y  - constant.^ "  2  or  f o r the exponential  of J ( y  integral  to leading order  = l -  = 1/k,  g  c a n be d e r i v e d  steepest  shown  z  of the other  expansion  t o an  when  l / 3  ).  S  the  first  The s u b s u r f a c e of order  / ?  l  /  maximum  and t h e  3  i n a r e d u c t i o n of growth  rate i n  e x p ( - 2 . 5 /j3 ) w h i c h i s  1-constant.ft  '  2  /  3  .  i n c r e a s e s c a n be s e e n  The  slower  by c o m p a r i n g  F i g 79  F i g 76. An  /3 a n d  example  of the sub-surface  OJ i s g i v e n  in Figure  80.  maxima  which  occur  The s u b - s u r f a c e  for large  maxima  found  by  327  Riley  e t §_1  (1949) a r e  d e p e n d e n c e of pronounced  The  sub-surface  d/u  contour  .A( co, f3 ) continued  to  I /l c  light  = c9  i s not.  the  79.  Provided  _/\_  rate  followed  rate  on  by  appreciated high,  by  the  if  K  of  growth  6.4  i s zero,  Effect  such  realistic vertical  be  used  For  regions  same  fact,  8 =  In  i s assumed t o  given of  i n the  S,  the  be  contour  ( to, (3 ) s p a c e where  is possible  from t h o s e  b o t h p r o p o r t i o n a l t o K"  of  the  water column  a straight  line  the  i s a l w a y s an  result  This  g r o w t h r a t e was considering  two  will  be  through  must  the  origin  increase  commented on  m i x e d more or  of  earlier  of  ou.  inevitably  less  the  K)  moves  i n Fig-  diffusion and  can  If K  is  uniformly  d e p t h and On  ,  i n growth  extreme s i t u a t i o n s .  regardless  population  (decreasing  reversible effect  Sverdrup's c r i t i c a l  0-5  where  positive  other  sink out,  hand,  regardless  rate.  of  A direct not  fairly  are  impossible, the  that  {3  population  be  can  depth c r i t e r i o n .  population  through a depth exceeding growth w i l l  unrealistic  n u t r i e n t d e p l e t i o n and  ( F i g 79)  a decrease.  population  possible for a  intensity.  stability  co > 0,  an  It i s intriguing  s i n c e growth r a t e  e  the  ( to, j3 ) o u t w a r d a l o n g  very  of  co and  Since  of c h o o s i n g  sinking rates.  d i v i d e s those  g r o w t h of  increasing  be  a b s e n c e of  plot  a l s o equals  s  depth.  Sverdrup's c r i t i c a l  proportional  it  i n the  result  maxima a r e  i n m i x i n g and  manner a s  the  g r o w t h r a t e on  growth p r o f i l e variation  not  a Mixed  Layer.  comparison  of  the  p o s s i b l e s i n c e h i s model  uniformly-mixed  l a y e r , not  a  above r e s u l t s  i s b a s e d on  the  with  Sverdrup's i s  assumption  s e m i - i n f i n i t e l a y e r with  of  a  finite  0  5  20  40  Figure 80  C h a r a c t e r i s t i c phytoplankton p r o f i l e f o r co  = 0.7,  (3=  20.  329  diffusion  rate.  It is relatively  simple  mixed l a y e r i n t o the model d i s c u s s e d c o n s t r a i n t s of the analytic  s o l u t i o n s , t h i s can  6  be  0 < z <  z  z >  z  This  that  used w i t h  that  i s , by  the  of  be  solved  z > z  and  flux across  gradient  Recent measurements of m i x i n g r a t e s  so  that  S  values, i s very  of o r d e r small.  1 cm .sec" 2  The  h e r e ; S v e r d r u p ' s r e s u l t s can  limiting  be  the  r e s u l t s obtained  i n t h i s way  the  problem for z < z  M  SP  = K  .  - w.  £P  reduces  section; and  M  z = z. M  using  The  1  of P a t z = i n the or  case  less  w i t h K -> oo  z . M  thermocline  have  (Osborn,1980),  £ = 0 is treated  r e g a r d e d as a l i m i t i n g and  cu=0.  case For  of £"=0,  to:  + (ju . e x p ( - k . z ) - d ) .P s  to  becomes  i n v o l v i n g a d i s c o n t i n u i t y i n the  low  the  i n a s i m i l a r manner  M  flux condition  the  m i x i n g r a t e s below  solutions in 0 < z < z ,  of  for  z :  first  continuity  yielded  a desire  i n the  of c o n c e n t r a t i o n  the  M  growth s t e p - f u n c t i o n  continuity  finite  M  p r o b l e m can  obtaining  a  a c c o m p l i s h e d by m a k i n g  i s s m a l l , c o r r e s p o n d i n g t o low  mixed l a y e r .  Within  eddy d i f f u s i o n a s s u m p t i o n and  =  where  introduce  above.  diffusion coefficient K a step-function  K(z)  to  6.10  330  with  3_P  boundary  =  w.P  at  conditions:  z=0  Eigensolutions can  be  p(S  C  from  They  take  ) = expico.S  where  The  the  0  at  M  to  by  ).(J (y)  non-dimensional  proceeding  +  v  v  z=z .  eigen-condition  in  the  rate  previous  )  C .Y (y)) 3  v  and  cu.Y (/3)).  ))/(Y;(/?)+  v  as  growth  form:  / 5 . e x p ( - § /j3  =  =  6.10  ( J ( / 3 )+ co.3 {(3  =  3  y  ^_P  corresponding  obtained  section.  and  for  v  (A + £ +  y=  2  ) °-  i s now  5  rather  complicated:  ( y . J ( y ) - / ( 3 . 6 o . J ( y ) ) / ( y . Y ( y ) - (3. co . Y ( y M  v  M  v  M  M  ( j ; ( /3 )+ c u . J ( /3 ) )/(Y;  ((3  y  The  constant  parameter as  y  ^  the  mixed  6.11  can  of  , co a n d  It 6.10 mixed  and  layer be  to  solved f3.  layer i s of  k.z  i s the  M  M  )  that  at  interest  the  6.11  with  those  layer.  This  comparison  new  As  to  A+S  the in  the  is  by  at  limit the  interpretation  the.bottom  before, as  previous  compare  predicted  non-dimensional  intensity  obtain  6.11  physical  surface.  recovered to  the  direct  light  r e s u l t s of  here  a  =  ft))  v  has  of  ))  M  to.Y (  +  numerically  be  v  where  and  ratio  The can  M  j3.exp(-fj/2)  =  equals  exp(-k.z )  infinite  M  v  a  ^->  Sverdrup by  equation  function  section  for  a  J\. semi-  oo .  results  facilitated  the  of  for the  implicit the  in  uniformly-  introduction  331  of  a new  of  the  non-dimensional  r e l a t i v e m a g n i t u d e of  zone and  growth r a t e .  obtained  from  a  e f f e c t of  series  of  The  The  changing  contour  the  plots  contour  simple  plot  the  can  Ci, ^  vs  euphotic can  above.  be  be  Of  d i f f u s i o n rate  the K  and  demonstrated  for d i f f e r e n t  extension  for a uniformly-mixed  of  Sverdrup's theory. layer  p .(l-exp(-k.z ))/(k.z ) s  On  defined  d i f f u s i o n rate /\ +5"  from t h e  = jt(.^,0.,p)  d e p e n d s on  of  s  rate  +  J ,Lo,j3)  ft  only  loss X 8  function  function  i n a u n i f o r m l y mixed  A  sinking  = k.w/u , a measure  by  values  (3.  of  a  the  ^,fi,(3,  parameters the  p a r a m e t e r fl=4.cv/j3  M  sinking  d i v i d i n g by  depth z  The  i s , by  i s obtained  a v e r a g e growth simple  by rate  integration,  - d  M  rate  of  layer  w  imposes an  u , s  the  additional  loss  rate  non-dimensional population  equal  to  growth  w/z . M  rate  becomes :  X =  ( l - e x p ( - k . z ) ) / ( k . z ) -d/p M  Substituting  >+S  =  The dual  ^ , fl  and  s  S,  -  w/(p .z ) s  M  t h i s becomes  ( l - e x p ( - J )-£!)/§  Contour p l o t s 1,3,10  for  M  (from  of  \+8  (6.11)) are  d e p e n d e n c e of  r o l e of  vs  6.12  _Q,J shown  X + 8  m i x i n g whenever  on  for  (3=0  in Figure (3 , J~L  sinking  (from  ( 6 . 1 2 ) ) and  (3 =  81. and  ^  again  i s present.  In  reflects the  case  the  Figure  81a.  Contour plot of the function A. for  (3=0.  (il,  0  1.0  0.5  n 3  F i g u r e 81b.  C o n t o u r p l o t o f t h e f u n c t i o n JV ( f l , £ ) for  (3=1.  Figure  81c.  C o n t o u r p l o t o f t h e f u n c t i o n JV. ( f l , for  (3 =3.  Figure  81d.  Contour p l o t of f o r (3=10.  the  function  J\  {fl,  336  p7=0, f o r i l > 0, d e c r e a s i n g i n c r e a s e i n > + <5  an  as mixing  and  then  out  o f t h e mixed l a y e r  effect  81a,b,c,d). to  mixing  converse  In the f i r s t  losses  /3=1, t h e t i m e depth  81b).  from  scale  f o r mixing  (3 i s t o i n c r e a s e  t h e e u p h o t i c zone due  i n the second  over  case,  the  F o r f3 l a r g e ,  > 1 as that  layer  significant  c a n be c o n c l u d e d  'mixed l a y e r '  time  from t h e  t h e o r y occur a t moderate v a l u e s of ^ . In  from  F i g 81 t h a t  u n l e s s the time  i s much g r e a t e r t h a n  the p h y t o p l a n k t o n . .@ ) i s l a r g e .  a good a p p r o x i m a t i o n  on fi  This w i l l  layer  theory  t h e u n i f o r m mixed  scale  of mixing  (]£-*<»).  layer  is a  over the  t h e minimum d o u b l i n g t i m e o f 5M = z . (p. /K) °-  o c c u r when  The s e m i - i n f i n i t e except  i s t h e same f o r  M  layer  for a limited  5  s  t h e o r y then  r e g i o n where  becomes  ^ is  small.  6.5 A G e n e r a l N e c e s s a r y It  C o n d i t i o n f o r Growth.  was n o t e d e a r l i e r  necessary  M  d e p a r t u r e s from t h e  d e r i v e d i n the s e m i - i n f i n i t e  good a p p r o x i m a t i o n  the l i g h t  a t t h e s u r f a c e but d e p a r t u r e s  f o r |S=10, t h e dependence o f A+S  (=2.^  large, the  t h e o r y o c c u r o n l y a t l a r g e v a l u e s o f £j =k.z  layer  uniformly-mixed  It  F o r cJ  ( 1 / k ) i s o f t h e same o r d e r a s t h e d o u b l i n g  the phytoplankton  fact,  decreases  forf l large ( F i g  A+S  case,  first in  r a t e due t o s i n k i n g  or increasing  s m a l l and decrease  u n i f o r m mixed  §  rate  o u t w e i g h l o s s e s due t o s i n k i n g ;  penetration  (Fig  results  i s true.  For  of  as the l o s s  increases (Fig 81a).  of decreasing mixing  M  below t h e e u p h o t i c zone  i n A+S  i n a decrease  forf i  A +8  £ by d e c r e a s i n g z  for a solution  that  the c o n d i t i o n  f o r both  >+S+co  step-function  2  < l  is  and e x p o n e n t i a l  337  versions  of ju(z),  although  eigen-conditions. possible general  from  A  i t was  simpler  a n d more  the d i f f e r e n t i a l  version  of equation  derived  general  equation  6.4  from  rather  different  derivation i s  itself.  Consider  a  :  p" - 2.co.p' + ( q ( 5 ) - A - S ) . p = 0 where  i t i s assumed  dimensionalizing that  q(5 )  h"  (q(S  +  1.  S .  all h'(0)  2  The  > 0.  2  p(S ) =  + > + £ ) ) .h =  zero-flux Since  for  a  for  the f i n i t e  and n e i t h e r  met.  general  Thus,  mixed  layer  population  growth,  i f or a  rate,  ).h(5 ) as before  at  so gives  increases  with  5,  ) . h —>  nor the c o n d i t i o n  h ' ( SM  discussed  X+S  and h" > 0 f o r  S =0 m e a n s  exp(co.S  on p o p u l a t i o n s  in particular,  growth  < 0 for a l l S  condition  the condition  the requirement  condition  sinking;  expico.S  > 0 a n d h'  semi-infinite layer,  i n non-  0  boundary  h'(0)  used  o f t h e maximum  2  a l l S  net  Letting  scale  > 1 , q ( 5 ) - ( X + 5'• + co )  for  be  the time  i s the inverse  ) - ( a>  >+£+cu  If  that  h  2  subject  )=-6o.h(SM  to diffusion s  0 - 5  )  ) >  population,  quite  1,  continued  i s not  Discussion. As  aimed  mentioned  at a better  i n the introduction, understanding  this  chapter  of the processes  of  a  and  possible.  6.6  )  s e c t i o n , can  < 1 represents  co (= w / ( 2 . ( K . j u ) steady-state  increases  0 a s S->co  i n the previous + u>  that  has  been  diffusion,  338  sinking growth  and  growth  rates  explanation relevance remains  and of  of  and  a  this  and  their  interactive  effect  vertical  distribution,  rather  specific  set  The  theory  to  i s addressed  of -data.  real  here  on than  an  question  phytoplankton  by  phytoplankton  considering  of  the  populations three  related  questions: (a)  to  what  involved  in  the  populations invalidate (b) are  or  theory what  the  are  extent  do  the  (when  assumptions  compared  i s assumed  to  i n more  and  what  approximations i s known  complex  of  numerical  real models)  conclusions?  the  parameter  p r e d i c t e d by  the  ranges  theory  for  likely  which  to  be  interesting  encountered  results  in  the  oceans? (c)  do  p r e d i c t i o n s of  qualitatively The a°lready be The  to  discussed  mentioned. i n more  mixing  mixed  layer  occurs.  the  basis  of  the  thermocline.  assumption represented the  recent  that by  surface  A  more  l a y e r s of  affair,  wind-mixing  and  the  by  being  models  discussed  probable mixing eddy  was  very  within  diffusion  model  and  across  justified  will  is  the  mixed  layer  coefficient, a  heating  i n the  no on in  the  K.  much  transient  an  which  rates  error  be  now  by  earlier  diffusion  to  have  realistic.  of  appears  through  here  i s represented  low  c h a r a c t e r i z e d by  stratification  last  general  source  ocean  the  thermocline  assumption of  least  oceans?  column a  at  i n the  most  water  below  the  correspond  underlying  as  the  the  measurements  constant  complicated  Those  latter  turbulent a  observed  detail  bounded  This  theory  underlying  p h y s i c a l s t r u c t u r e of  upper  in  phenomena  assumptions been  the  can  be  Mixing more  periods summer  of and  339  by c o n v e c t i v e m i x i n g a n d w i n d s t o r m s The  i n t h e w i n t e r (Denman,1972).  l a c k o f a b e t t e r a l t e r n a t i v e , and i t s w i d e s p r e a d use i n  n u m e r i c a l m o d e l s ( W i n t e r e t a_l ,1975; J a m a r t considered to j u s t i f y  e t a_l ,1977) a r e  t h e use of t h e d i f f u s i o n a p p r o x i m a t i o n  here. The  a s s u m p t i o n s made r e g a r d i n g g r o w t h  r a t e and i t s  d e p e n d e n c e on d e p t h a r e t h e same a s t h o s e made by S v e r d r u p and much o f t h e d i s c u s s i o n t h e r e i s s t i l l  valid.  number o f phenomena s u c h a s p h o t o i n h i b i t i o n adaptation  (Jorgansen,1969)  (Marra,1978a,b) effects,  However a  (Steele,1962), light  and s h o r t - t e r m v s l o n g - t e r m  a r e now b e t t e r u n d e r s t o o d .  The  complex one.  w i t h some  i n the case of v e r y s m a l l m i x i n g  q u e s t i o n of p h y t o p l a n k t o n s i n k i n g The a s s u m p t i o n  responses  In view of these  t h e p r e d i c t i o n s o f t h e model must be v i e w e d  caution, especially  (1953)  rates  rates.  i s also a  of a c o n s t a n t , p o s i t i v e  sinking  rate  has  t o be s u s p e c t f o r m o t i l e p h y t o p l a n k t e r s s u c h a s f l a g e l l a t e s  and  some d i n o f l a g e l l a t e s .  Even f o r n o n - m o t i l e p h y t o p l a n k t e r s  such as p e l a g i c diatoms, a non-zero of p l e n t i f u l sinking  nutrient  i s not c l e a r l y  sinking  i n the presence  established.  r a t e s have u s u a l l y been o b s e r v e d  been u s e d  rate  Positive  (Smayda,1970), and have  i n p r e v i o u s m o d e l s ( J a m a r t e t a l ,1977; P a r s o n s a n d  T a k a h a s h i , 1 9 7 3 ) , but under  some c o n d i t i o n s z e r o o r e v e n  positive  b u o y a n c y i n d i a t o m s h a s been r e p o r t e d ( S m a y d a , 1 9 7 0 ) . Both t h e b i o l o g i c a l and p h y s i c a l parameters  involved  model c a n r a n g e o v e r a t l e a s t an o r d e r o f m a g n i t u d e dimensional parameters  and t h e non-  a p p e a r i n g i n t h e t h e o r y have a  c o r r e s p o n d i n g l y l a r g e range. maximum g r o w t h  i nthe  If u  s  i s assumed t o be c l o s e t o t h e  r a t e of p h y t o p l a n k t o n , v a l u e s of about  0.5 t o 1  340  day"  a r e a p p r o p r i a t e (Chan,1978).  1  c e l l s may r a n g e f r o m of  l e s s than  0 t o 5m/day (Smayda,1970) w i t h l o w e r  ocean can v a r y from  i n c l e a r ocean water  1  (Jerlov,1968).  (=k.w/ u ) i s 0 t o 5.  water  Extinction coefficients  <0.04 m"  t u r b i d c o a s t a l water j  values  1.0 m/day b e i n g a p p r o p r i a t e f o r s m a l l e r c e l l s a t  high nutrient concentrations.  fi  Sinking rates f o r healthy  s  i nthe  t o >0.5 m"  1  The c o r r e s p o n d i n g  in  range of  T a k i n g w=0.5 m/day a n d r e a s o n a b l y  clear  (k < 0 . 2 5 , s a y ) g i v e s a r a n g e f o r J l . o f 0.025 t o 0.25, t h e  region of i n t e r e s t Mixed  i n F i g u r e 81.  l a y e r depths  in  highly  stratified  in  some s e a s o n s .  c o a s t a l waters  i n t h e ocean can vary  r e g i o n s t o over  Although  where z  M  f r o m a few m e t e r s  100 m e t r e s i n t h e open o c e a n  l a r g e v a l u e s of k t e n d t o occur i n  i s small,  ^ = k.z  c a n r a n g e f r o m <1 t o  M  >10. The  c o r r e c t v a l u e f o r K i s a s u n c e r t a i n i n most c a s e s a s t h e  d i f f u s i o n approximation al  itself.  I n a n u m e r i c a l m o d e l , J a m a r t et.  (1977) u s e v a l u e s o f 10 t o 40 c m . s e c " 2  m .day" 2  1  f o r t h e mixed l a y e r  o r a b o u t 100 t o 400  1  i n t h e open o c e a n o f f O r e g o n .  s m a l l e r v a l u e s may be a p p r o p r i a t e i n s h e l t e r e d , e n c l o s e d and  v a l u e s as low as 1 t o 5 m .day" 2  were r e c o r d e d  1  diameter  experimental enclosures i n Saanich  ,1977).  With  r a n g i n g from The cover  of  the model.  i n 10m ( S t e e l e et. a_l  s  <0.05 t o >20 a r e p o s s i b l e . of parameters  the complete  i n n a t u r e a p p e a r t o be s u f f i c i e n t  range of b e h a v i o u r s  The c h a p t e r  i s concluded  found  i n the a n a l y s i s  by e x a m i n i n g t h e  i m p l i c a t i o n s o f t h e t h e o r y w i t h r e g a r d t o two s p e c i f i c The  first  waters  t h e v a l u e s o f k a n d ] u g i v e n a b o v e , v a l u e s o f (3  ranges  to  Inlet  Much  concerns  points.  R i l e y e t a l 's ( 1 9 4 9 ) d e r i v a t i o n o f a  341  pronounced  s u b - s u r f a c e maximum.  phenomenon  is s t i l l  profile by  (3,  for large  Jamart  et  possible  a l are  not  w a t e r , and  i t follows  that  likely  in oceanic  been  But  o c c u r , even  i f the  i n the  layers.  In  current  al  ,1977).  solely  However, t h e  to a balance  without  nutrient  do  f a c t , observed  nutrient et  K  mixed  clearest  sinking  f o u n d below a mixed  and  v a l u e s of  used  layers,  ocean  p r o n o u n c e d deep c h l o r o p h y l l maxima  c h l o r o p h y l l maxima a r e depleted,  this  production  most o c e a n i c  between d i f f u s i o n and  mixed  shown t h a t  exponential  c h a r a c t e r i s t i c of  fi > 4 w i l l  to a balance  w i t h an  j3 > 10.  say  v a l u e s of  solely  I t has  layer  explanations  existence  of  r e l y on  d i f f u s i o n and  depletion,  is s t i l l  at  waters.  The  second p o i n t  i s concerned  is  this  usually (Jamart  s u b s u r f a c e maxima  sinking,  in sheltered  seem  deep  which  between  possible  not  due  growth,  least theoretically  w i t h the  usefulness  the  of  Sverdrup's  spring  bloom c o n d i t i o n  diffusion  and  sinking.  I t was  noted  that  s i g n i f i c a n t changes  theory  occur  only  when  is  u n i f o r m mixed large.  v a l u e s of  layer  If values S will M  be  f o r K of  100  obtained  only  to  400  to  of  extension  the  include  ,{p /K)  °- )  are  used,  large  M  n^.day"  1  f o r deep mixed  If. t h e s e d i f f u s i o n r a t e s  can  mixed  layers  which are  open P a c i f i c  before  the  criterion region  spring may  of (3  v a l u e s of Sverdrup's  be of  jS  can  bloom  t h e o r y may  s h a l l o w e r mixed  1 and  occur  also  and to  large  ^  be  in Figure  81.  w a t e r s and  the  s i g n i f i c a n t there,  of deep  Atlantic  Sverdrup's  These s i t u a t i o n s c o r r e s p o n d  in sheltered  layers.  i n the  , significant corrections  required. order  used  5  s  M  layers,  100m.  i n the  be  to  S (=z  order  present  due  to  Again,  the larger  correction  despite  the  to  342  CHAPTER 7 MATHEMATICAL  7.1  ANALYSIS OF DEEP CHLOROPHYLL  Introduction. O b s e r v a t i o n s o f deep c h l o r o p h y l l  from  t h e I n d i a n Ocean  of M e x i c o Pacific  Ocean  (Anderson,1972).  and p r e d i c t a b l e  feature  is  t o Jamart  Explanations simple  such as the c o l l e c t i o n  to sophisticated (Jamart  chemical  properties  o f t h e water  while  mixing,  the l a t t e r  sinking  rate  concentration  light include  and  c e l l s at  incorporating  Both  t h e p h y s i c a l and  column and t h e p h y s i o l o g i c a l  variations intensity  in density,  to.  The  temperature,  and n u t r i e n t c o n c e n t r a t i o n  t h e dependence of p h y t o p l a n k t o n c o n t e n t on l i g h t  intensity,  growth, nutrient  mathematical  t r e a t m e n t s , due t o R i l e y ,  ( 1 9 4 9 ) , h a s been d i s c u s s e d and expanded  the previous chapter.  I t was shown t h e r e t h a t  s u b s u r f a c e maxima c a n be g e n e r a t e d limitation  but t h a t  from  temperature.  Stommel and Bumpus  rates,  of s i n k i n g  e_t a l ,1977).  and c h l o r o p h y l l  One o f t h e e a r l i e s t  nutrient  in structure  o f t h e p h y t o p l a n k t o n have been a p p e a l e d  include v e r t i c a l  turbulent  phenomenon t h e r e a d e r  n u m e r i c a l models  hypotheses  former  and low  e_t a_l ( 1 9 7 7 ) .  multiple  attributes  of t h i s  t o be a  at middle  o f t h e phenomenon have r a n g e d  hypotheses  pycnoclines  maxima a p p e a r  of oceans  F o r an e x t e n s i v e r e v i e w  Ocean and G u l f  S t e e l e , 1 9 6 4 ) , and the  Thse  latitudes. referred  maxima have been r e p o r t e d  ( S a i j o , 1 9 7 3 ) , the A t l a n t i c  (Hobson and L o r e n z e n , 1 9 7 2 ;  widespread  in  MAXIMA.  this  and v e r t i c a l required  i n the absence  variation  pronounced of both  in sinking  eddy d i f f u s i o n  rates  upon  or mixing  lower  than  343  those  usually  subsurface two  c o n d i t i o n s shared  levels  Yentsch  layer  a r e very  with a pronounced T h i s has r e s u l t e d  a n d S t e p h e n s , 1969;  trap'  V e n r i c k e_t  by t h e s i m u l a t i o n r e s u l t s o f  d i s c u s s i o n h a s been prompted  o f t h e maximum t o t h e c o m p e n s a t i o n (1960) p o i n t e d o u t t h a t t h e n a t u r e  equation  f o r phytoplankton  concentration sinking  must o c c u r  traditionally  to estimate  compensation  (Winter  e t a l ,1975) have s u g g e s t e d of compensation  depth  c o n s e n s u s t h a t deep c h l o r o p h y l l shade-adapted c e l l s this  maxima b a s e d given.  section,  Although  depth  with  provided depth. level  (Parsons  t o low l i g h t  t h a t t h e 1% l i g h t  ejt a l  intensity level  and t h e r e now a p p e a r s  maxima c o n s i s t  i s an  t o be a  of a c t i v e l y -  ( E p p l e y e t a_l ,1973).  a theoretical  on c o u p l e d  depth,  below t h e 1% l i g h t  recent d i s c u s s i o n s of a d a p t a t i o n  growing,  S t e e l e and  of the c o n s e r v a t i o n  rate are constant  ,1977),  underestimate  depth.  above t h e c o m p e n s a t i o n  r a t e and d i f f u s i o n  used  by t h e  r e q u i r e s t h a t t h e maximum  However, w h i l e maxima a r e o f t e n f o u n d  is  Steele,1964)  maximum a s a ' n u t r i e n t  i s supported  (often  e t a l (1977) .  relation  In  layer  (Anderson,1972;  i n t h e mixed  Anderson, Parsons  Some t h e o r e t i c a l  that  below t h e m i x e d  (Venrick,McGowan a n d M a n t y l e , 1 9 7 3 ) .  ,1973), a view w h i c h  Jamart  o b s e r v a t i o n s of t h i s  often coinciding  a v i e w o f t h e deep c h l o r o p h y l l  In f a c t , the  e t a_l were a s s o c i a t e d , w i t h  thermocline),  when n u t r i e n t  ( A n d e r s o n , 1969; al  t o be f o u n d  t h e maximum i t s e l f  nutricliiie in  They t e n d  to occur  low,  mixed l a y e r s .  by most o t h e r  o r below a s e a s o n a l  and  to oceanic  maxima e x p l a i n e d by R i l e y  phenomenon. in  attributed  equations  the e f f e c t s  treatment  of deep  f o r phytoplankton  of v e r t i c a l  chlorophyll  and n u t r i e n t s  variation in  344  diffusion the  and s i n k i n g  rates  phytoplankton-nutrient  hypothesis. into  dependence the  i s intended  of t h e deep  of a t t r i b u t e s  maximum  regard,  interaction  The a n a l y s i s  t h e phenomenon  are considered,  such  rationalistic  numerical  (1977),  aimed  which  general  c h l o r o p h y l l maximum  as the depth  parameters.  at explaining  approach  insights  and t h e  and c o n c e n t r a t i o n  as complementary  modelling  i s on  and the n u t r i e n t - t r a p to give  on p h y s i c a l a n d b i o l o g i c a l  i t c a n be r e g a r d e d  t h e emphasis  In  of  this  to the detailed, taken  a particular  by J a m a r t  et a l  s e t of  observations.  7.2 A P h y t o p l a n k t o n - N u t r i e n t A allow  simple  physical  concentration  semi-infinite diffusion nutrient t,  grow  light  = K.  column  i s considered  The p h y t o p l a n k t o n N(z,t),  to diffusion.  a t a r a t e ju(z,N)  loss  both  with  functions  ^ P  units,  unit  rate  d  (day" )  - w. Aj> +  each  1  due t o b o t h  equation,  given  of n u t r i e n t  to  eddy  P(z,t) and  o f d e p t h °z a n d t i m e sink  on d e p t h They  at rate w (through  are also  respiration  and  (u(z,N)-d).P  of phytoplankton  uptake.  Some  A  K and w constant, i s :  7.1a  and n u t r i e n t c o n c e n t r a t i o n s unit  a constant  also  i s dependent  first  interaction.  concentration  Phytoplankton  which  The a p p r o p r i a t e  phytoplankton  same  be c o n s i d e r e d  on t h e p h y t o p l a n k t o n - n u t r i e n t  concentration  to a  grazing.  £P  K.  structure will  i n t e n s i t y ) and n u t r i e n t c o n c e n t r a t i o n .  subject  a  rate  are subject  and  If  water  Model.  are expressed  production  fraction  i nthe  corresponds  U of the  to  phytoplankton  345  loss  rate  losses,  d,  corresponding  will  be  conservation  d_N  =  K.  flux  K.  -  r e s p i r a t i o n and  recycled  for  as  nutrient  nutrients  part so  of  that  the  grazing  the  becomes:  (u(z,N)-d.U).P  conditions  at  7.1b  the  surface,  0  at  z=0,  are  given  by  the  zero-  more  problematical  condition:  £j>  -  hz  w.P  Boundary and  equation  £fN  Boundary  rapidly  to  =  0  at  discussion  These  obtaining  a  Progress  i s made  based  the  properties  persistent  a  out for  pair  to a  be  little.  of  coupled  equations  the  prospects  are  not  good.  closed  a  of  simplications  series  deep  form  of  maxima  7.1 in  chlorophyll (Venrick  scales  of  months,  much  allows  us  to  for  and  than  steady-state  on  nature.  maxima  e_t a_l  longer  form  are  ,1973), those  of  solutions  non-linear  and  in  phenomena  look  turn  solution  chlorophyll  that  co  differential  mathematical  of  observation  by  =  represent  partial general  z  z = 0.  i s postponed  equations  second-order  =  bz  conditions  their  on  c3N  ,  and the  approximations observed  First,  the  widespread which  and  change  phytoplankton of  for  7.1  on  time  growth,  which  satisfy:  K.P"  -  w.P'  K.N"  -  (p(z,N)-U.d).P  The  +  ( j u ( z , N ) - d ) .P  problem  is  =  0  7.2a  =0  reduced  7.2b to  that  of  solving  coupled  ordinary  346  differential further closed z=  oo  approximation form,  must  written  =  but  be  integrating  N(z)  equations.  The  will  the  equations  be  problem  addressed  twice,  the  -  P(0)  +  1,  the  are  necessary of  to  defining  first.  By  s t i l l  non-linear  obtain  solutions in  boundary  combining  conditions  7.2a  and  7.2b  at  depth  z  nutrient concentration  and  at  and  can  be  as:  N(0)  P(z)  +  +  As  long  as  U  impossible condition provided  <  for  N(z)  there. U  <  1,  last  to  The  term  satisfy  i s unbounded any  physically  physical basis  there  is a  net  loss  this  from  problem  the  from  diffusive  flux  of  this  must  approach  any  depth  non-zero the  z  Of fashion  is  value  water  as  column  course, in  the  z -*• oo  to  balance  the  simple  loss  from  the  boundary  is  simple:  balanced  by  diffusion  n u t r i e n t upward  total  some loss  through  positive, of  nutrient  in  above.  ocean  this  and  model  there  which  are  a  account  phytoplankton-nutrient  number  of  other  detritus  to  higher  pools  ranging  trophic  eventually  returned  greatly  in  spatial  lost  sedimentation  to  term  nutrient concentrations  from  a  the  be  i t  combined  pool  Since  must  , making  reasonable  phytoplankton-nutrient below.  which  of  z->oo  as  to  the  levels. inorganic  distribution or  from  and  terrestrial  do  not  number for pool  increase  of  i t .  processes In  pool time  of by  this  and  predators  the  above  enters  nutrient  and  net  dissolved  processes  scale,  this  omitted  reality,  discussed  particulate Much  in  is  varying  that  which  is replaced  by  is  347  coastal  runoff  or s o u r c e s  such as n i t r o g e n  p r o c e s s e s making up t h e l a r g e - s c a l e  nutrient  (Parsons and H a r r i s o n , 1 9 8 1 ) m a i n t a i n concentration layers. the  despite  There  short-term  i s no p o i n t  s i m p l e model a s t h e y  cycle  a constant  fluctuations  in introducing  involve  fixation.  a vast  (Menzel,1974;  nutrient  the  c h l o r o p h y l l maximum, a t a h i g h  the  time  persist,  a t some  finite  i n t h e upper  pools,  into  scales, are s t i l l  e t a_l ( 1 9 7 7 ) ,  i s to f i x  depth z , well  below  L  'deep-water' v a l u e ,  N .  although processes other  i n keeping  than  those  On  L  o f a few months o v e r w h i c h d e e p c h l o r o p h y l l  t h i s i s a reasonable condition,  observations,  nutrient  I c h i k a w a and N i s h i z a w a , 1 9 7 5 ) .  the  scale  deep  range of time  A s i m p l e a l t e r n a t i v e , u s e d by J a m a r t concentration  of the oceans  these processes  some o f w h i c h , p a r t i c u l a r l y t h o s e o f d e t r i t a l controversial  A l l these  maxima  with  represented i n  7.2 a r e needed t o e x p l a i n i t . We  now  seek an a p p r o x i m a t e  corresponding introduction vicinity This  and  t o a deep c h l o r o p h y l l maximum. that  t h e s e maxima a r e t y p i c a l l y  o f a n u t r i c l i n e , below a r e g i o n  suggests that  regions:  steady-state  an upper  growth  t h e water column may layer  are p l e n t i f u l  approach  is facilitated  by u s i n g  = m i n ( j j (z) , J J ( N ) ) 1  2  depletion.  i n t o two  concentrations region  a limiting  factor  and n u t r i e n t s  i n the  i n the  a r e low  i n which  is light-limited.  approach t o the i n t e r a c t i o n of l i g h t  JU(Z,N)  found  be d i v i d e d  i n which n u t r i e n t  and g r o w t h  I t was n o t e d  of n u t r i e n t  i s n u t r i e n t - l i m i t e d , and a l o w e r  nutrients  s o l u t i o n t o 7.2  This  or L i e b i g (Droop,1977) :  348  where,  for simplicity,  growth  i s assumed  light  and n u t r i e n t c o n c e n t r a t i o n  p (z)  = ju . e x p ( - k . z ) ,  Then,  corresponding  exist  some  1  such  that,  N  T  < N  depth z  separately  p (N) = JJ .N/H  s  2  S  t o t h e two  t o depend  for z > z , N > N r  .  K  regions  described  and p ( z , N ) = u ( N ) .  d i m e n s i o n a l i z e by s c a l i n g  T  there  =N exp(-k.z K  1  I t i s convenient  2  N  above,  a n d ja(z,N)=p ( z ) , w h i l e  T  on  :  and n u t r i e n t c o n c e n t r a t i o n  T  linearly  at this  time and depth as i n the  T  )  for z < z , T  point  t o non-  previous  sect ion:  S=  z.(p /K)°-  T=  ju .t  ,  s  -  2.6U..P'  N"  -  (N/N  P" - 2 .  N"  -  with  P ' ( 0 )  +  i n 7.2, we  (N/N  - U.S  S >S  T  ).P  =  5  ,  obtain  ).P  S=  ,  z .(ju /K)°-  L  L  (3 = 2. (p /K)  °- /k 5  s  for  S < S  T  ,  5  s  . .  :  0  =  7.3a  0  :  . P ' + ( e x p ( - 2 . 5 /f3 ) - c?) . P = 0  ( e x p ( - 2 . 5 /(3  boundary  =  S  -  R  °-  s  5  P"  for  T  s  substituting  and  z . (p /K)  T  uj = w / ( 2 . (ju .K) °- )  On  R  S=  ,  5  s  2.cu  ) U . S ).N -  =  0  7.3b  conditions:  . P ( 0 )  , N'  (  0  )  =  0  , P  0  as  ,  N(S ) = N L  L  ,  349  and  at S =  P, N, P' a n d N' c o n t i n u o u s The  boundary  inconsistent governing  c o n d i t i o n s o n N a n d P may  since N  i s only  P i s independent  for  P c a n be d e f i n e d on  as  5 ^ & $  P'(  applied.  phytoplankton  across  )  L  (0,oo)  S = S  = N  which  L  for S  that,  condition  (0,5 > S  L  e t a l used of zero  , so that  T  diffusive with  on t h e p h y t o p l a n k t o n  will  their  produce  solution P -»  0  condition  flux of their  flux  of the p r e v i o u s large,  a  condition  the boundary  implies a diffusive  The r e s u l t s  appear  ), but t h e equation  and the boundary  sufficiently  L  now  i s inconsistent  L  t h e same d e p t h .  suggest  for 5  assumption  across  N( S  condition  d e f i n e d on  of N  (Jamart  ) = 0 but t h i s  L  ST*  boundary  of  nutrient  chapter  do  boundary  similar  results to  mine) . The  system  obtained  7.3b i s l i n e a r  i n the previous  P = P .exp(cu.(5 - S  f o r P was  chapter:  ) ) . J ( / 0 . e x p ( - S / / ? ))/J (/3 v  r  T  and the s o l u t i o n  v  . e x p ( - S / (3 )) T  7.4 where  j3.( co + S ) °- .  v=  2  Integrating the equation  5  forN  twice  gives:  N = N  The N  T  T  f  + N'(S ).(5-S )+/' T  boundary  condition  ,N' ( Sj ) , N For  S  approximate suggested  T  N( 5  L  ( e x p ( - 2 . s / / 0 )-U.S ) . P ( s ) . d s . d u .  )=N  L  represents a condition  linking  and P .  L  T  < S  r  , the problem  solution  by r e c e n t  must  i s still  be f o u n d .  evidence  that  n o n - l i n e a r and an  The a p p r o a c h  phytoplankton  taken  can take  here i s up  350  nutrients  and grow  (eg Goldman If  r a p i d l y at very  and M c C a r t h y , 1 9 7 8 ) ;  in particular  n = N/N  R  , n  T  << P ,  T  i s , that  ( = exp(-k.z )  R  N  i s very  R  ) , p = P/P  T  Substitution  i n the equations  p"  - 2. u> .p'  + ( n - S ) .p = 0  n"  - (n-U.S ) . p / S An e x a c t  that  concentrations small  define  T  = N /N  low n u t r i e n t  7.3a  T  ,  E*= N / P . R  gives:  =0.  2  outer  T  7.5  solution  o f 7.5, s a t i s f y i n g t h e  surface  boundary c o n d i t i o n , i s  p°  = C .exp(cw.5 ).(cosh(a.S ) +  n°  = U.S  7.6  where a = ( co + (1-U).S" )  0 - 5  satisfy an  inner  o r boundary  ;  where  However,  ^  = (S - S  1  " - n' . ( p ° ( 5 + T  layer  n( $  T  )/£ .  r  solution  ) , n = n° ( S ) +  p ' " - 2. £ . co.p*' ' + ;  .  t h e boundary c o n d i t i o n  p = p°(S ) + p ( J  n  c u . s i n h ( a . S )/a)  1  ; n  this solution ) = n  T  cannot  , so we must  look f o r  by w r i t i n g  (§ )  Substituting  f o r p,n i n 7.5 g i v e s  £\ ( n -(1-U) . S ) . (p° ( S 1  )+p'' ) = 0  We now e x p a n d p' and n' i n powers o f £ :  T  :  + € . J )+p' ) = 0 7.7  351  P  i =  To  Pi ( f )  order  pi"  = o  Using n'  0  ,  the  1.1  n;*• -  becomes  =C .exp(^  ).  2  S  T  ) - n; .p° ' ( S  C  3  R  complete  5 < S  dP/dS  ) .C  ±  so  of order  E  1  p^  i n 7.7  =0,  gives  :  must  P( S  T  P .dp°/dS T  to order  , n' — »  1  ) . ( §  be s a t i s f i e d  -) = P  +  0 as  ^ -*• -oo  )/4.  Thus,  (S )  T  >  at  a matter N  <  5  T  ~)  .(§*  T  the c o n t i n u i t y  i s simply T  =0  ) . ( 1 + S.p°'  2  = P .(dp°/dS  that,  yields  imply  that  to order  ,  . ( c o s h ( a . S ) + co. s i n h ( a . 5 ) / a )  + C .exp(£  f o r P, N  so that  =  §  is :  T  the s o l u t i o n ,  , dN/dS  2  p  T  for  ) = N .U.S  , C  ~°°  i  o.  0 as  )  T  conditions  2  condition  =  terms  ni = C .p°'(5 ).exp(J  solution  dP/dS  S.n' (§ ) +...  . pi ' = 0  T  To  ) + P ; )  T  ) •+  :  p j , n^ ->  P( S ) = P . e x p ( a j . S  N(S  1  Collecting  the matching =0,  s  ;.(p ( o  n <  the conditions  n; " - n i .p°(  p;  , n ' = n'Jtj  +  £°,  p | " - 2.  and  £.p|( J ) ...  +  -f  conditions  S =5  T  .  The  of d e f i n i n g = N  )/4)  T  .  f o r P,  continuity the constants  S < S  For  ( l / c ).(dp:/dj + f.dpi/df  T  ,  +0(£: )) 2  + 0(€ )  7.8  z  £°, t h e c o n t i n u i t y  N,  condition  f o r dP/dS  at  S =  352  Sj  becomes  (a. s i n h ( a . S  :  T  + exp(-S //3  S < 5T  dN/dS  R  N'(5 )  T  . ( ( n - U . S )/£ T  T  L  f  i n equation  .exp(-2.  T  v?  for  f  2  T  2  5  large,  N  =  On  collecting  these  in 0 < S  P  P  . e x p ( c o AS  / P  (cosh(a.S  = P .exp(o;.(S T  T  7.4  //? ) +  ( e x p ( - 2 . s / / c ? )-U.S  £°  T  )/J  v  (/? .ekp(-5 / c3 r  T  + dn^/dj)  )-U.S  results < S  /  £.P  - S  7  +  - Sj  T  )) = 0  7.9  ),  5 >5  condition  T  T  , gives  :  )-U.ff ).(S  T  S = S  at  + 0{6  ).P(s).ds.du  co.sinh(a.S  )  2  )  .(exp(-2..? //ff  together  +0(£ )  , so  2  then,  we  - S  becomes  L  )  have  >  T  :  7.10  a  solution  to  :  ) . (cosh(a. 5 )  +  c o . s i n h ( a . S )/a)  for S < S  )/a)  ) ) . J ( ft . e x p ( - S /ft v  £\dn;/dj  0(€  +  of t h e form  L  )  )  for N(5  the boundary  order  =  )  r  + (l/£ ) . ( d n i / d j +  £ . P . ( e x p ( - 2 . S //3  £ .P  so,  r  (dnVdS  Substituting  N=  ^ .exp(-o° //J  P . £  =  T  ) + cu. s i n h ( a . S ) / a )  T  ,  = N . =  )/(cosh(a.S  T  )  T  For  ) + ct».cosh(a. S )  T  ) ) / J ( /3 . e x p ( - S /^S  ))  r  v  for  5  > S  T  353  N  = N .(U.S  + exp(-(2.5  R  >  ) - U . S ).exp((S - S  //8  T  )/£ )  T  S  for N  = N .exp(-2. S R  /0  r  +/* /  £*.P . ( e x p ( - 2 . 5  ) +  ) - U. S ) . P ( s ) . d s . d u  (exp(-2.s//?  r  ) - U . S ). (5  //3  T  T  < S - 5  )  T  for S > S , T  7.11 with  S  determined  r  by e q u a t i o n  7.9 a n d P  determined  r  by  equation  7.10. This  approximate  complicated, have  been  maximum the  assumed  rates  layer  nutrient  within uptake  this  of  phytoplankton  enough,  outer  limited  compared  layer  which,  solution  (N = N  below.  R  the  light-limited  kinetics  a l l net growth  grazing  losses.  does  smaller  not a f f e c t  )  level  the depth to this  by  with  rate  i s then  The n u t r i e n t  layer  t o deepen needed  matching the  light-  and shape o f by  losses,  as  region.  the phytoplankton to balance  c o n c e n t r a t i o n i n deep  o r shape  this  approximation  i n the l i g h t - l i m i t e d  growth  at  on t h e d i s t r i b u t i o n  and g r a z i n g and r e s p i r a t i o n  tend  the  N^.U.6" When  the layer  will  with  phytoplankton  i s obtained  rate  the depth  by  influence  above  near  result,  varies  scales,  recycling.  order,  As a  at  equations are  reduced  i s determined  occurs  the grazing  as a  .U.S  a n d grow  actually  length  In p a r t i c u l a r ,  maximum  maximum  rapidly  to first  phytoplankton  Decreasing  mixing  i t has a n e g l i g i b l e  solution  rate  by n u t r i e n t  the  almost  growth  with  Phytoplankton  concentrations.  to the equilibrium  i s balanced  thin  up n u t r i e n t  (and the governing  concentration being  is  the  low n u t r i e n t  algebraica