{"http:\/\/dx.doi.org\/10.14288\/1.0080145":{"http:\/\/vivoweb.org\/ontology\/core#departmentOrSchool":[{"value":"Science, Faculty of","type":"literal","lang":"en"},{"value":"Mathematics, Department of","type":"literal","lang":"en"}],"http:\/\/www.europeana.eu\/schemas\/edm\/dataProvider":[{"value":"DSpace","type":"literal","lang":"en"}],"https:\/\/open.library.ubc.ca\/terms#degreeCampus":[{"value":"UBCV","type":"literal","lang":"en"}],"http:\/\/purl.org\/dc\/terms\/creator":[{"value":"Parslow, John Stanley","type":"literal","lang":"en"}],"http:\/\/purl.org\/dc\/terms\/issued":[{"value":"2010-03-29T19:36:32Z","type":"literal","lang":"en"},{"value":"1981","type":"literal","lang":"en"}],"http:\/\/vivoweb.org\/ontology\/core#relatedDegree":[{"value":"Doctor of Philosophy - PhD","type":"literal","lang":"en"}],"https:\/\/open.library.ubc.ca\/terms#degreeGrantor":[{"value":"University of British Columbia","type":"literal","lang":"en"}],"http:\/\/purl.org\/dc\/terms\/description":[{"value":"The anomalous phytoplankton seasonal cycle in the Subarctic Pacific has been attributed to grazing control. In simple classical models of the phytoplankton-zooplankton interaction, grazing thresholds are found to be necessary to obtain this type of control. Weathership observations at O.S.P. are analysed to provide a basis for a more realistic model. Phytoplankton are present at O.S.P. in almost uniformly low concentrations (about 0.4 mg Chla.m\u207b\u00b3), have low photosynthetic efficiency (<0.5 mg C.mg Chia\u207b\u00b9.ly\u207b\u00b9), adapt to seasonal changes in solar radiation and show most surface inhibition in the spring. A numerical production model based on these results and driven by physical time series from the weatherships yields low annual production levels compared with previous estimates. Predicted production levels are sensitive to the choice of respiration rate, and introduction of a rapid light response or 'Marra' effect results in a doubling of net production. Predicted year to year variation is low and might be higher if variation in Secchi depth could be accounted for. In a phytoplankton-zooplankton (biomass) model based on the production model, grazing thresholds and over-wintering strategies are both necessary for grazing control. Systems identification techniques are adapted to estimate population parameters for cohorts of the dominant grazers. Cohort structure is introduced into the phytoplankton-zooplankton model using these estimates. As a result, attention is shifted from the spring to late summer and fall where sensitivity and stability problems are associated with the over-wintering departure of the dominant grazers. An approximate mathematical analysis of Steele's (1974) nutrient-phytoplankton-zooplankton model allows the explanation and elaboration of previous authors' numerical results. Stable cyclic solutions are shown to exist under nutrient limitation for constant mortality rates in the absence of grazing thresholds. Attention is focused on the transient (spring bloom) approach to the nutrient-limited cycle and a broader (physiological and behavioural) framework for zooplankton response to declining food concentrations is proposed. Systems identification techniques are also used to estimate zooplankton feeding and growth parameters from CEPEX copepod time series. The estimates are compared with literature values and the statistical and deterministic limitations of the time series discussed with a mind to future experiments. A nutrient-phytoplankton-zooplankton model, based on the parameter estimates, provides a consistent explanation of the observed phytoplankton persistence at low densities as a stable nutrient-limited equilibrium. A mathematical solution in terms of Bessel functions is found for phytoplankton populations undergoing diffusion and sinking in the case of an exponential growth profile. Non-dimensionalization allows a relatively complete discussion of the effects of varying physical and biological parameters on profiles and population growth rates. Subsurface maxima for constant diffusivity and sinking rate, previously reported for an idealised step-function growth profile, are also obtained for the exponential growth profile. Solutions to coupled non-linear phytoplankton-nutrient equations corresponding to subsurface maxima of the nutrient-trap type are also obtained using boundary-layer techniques. The dependence of the depth, shape and magnitude of these maxima on parameters is explored. The approximate theory agrees well with previously published results from a complex simulation model.","type":"literal","lang":"en"}],"http:\/\/www.europeana.eu\/schemas\/edm\/aggregatedCHO":[{"value":"https:\/\/circle.library.ubc.ca\/rest\/handle\/2429\/22896?expand=metadata","type":"literal","lang":"en"}],"http:\/\/www.w3.org\/2009\/08\/skos-reference\/skos.html#note":[{"value":"PHYTOPLANKTON-ZOOPLANKTON INTERACTIONS : DATA ANALYSIS AND MODELLING (WITH PARTICULAR REFERENCE TO OCEAN STATION P (5Q\u00b0N,145\u00b0S) AND CONTR3LLED ECOSYSTEM EXPERIMENTS ) . By JOHN STANLEY PARSLOK B.Sc., U n i v e r s i t y o f Q u e e n s l a n d , 1974 A THESIS SUBMITTED IN PARTIAL FULFIL MENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY i n THE FACULTY OF GRADUATE STUDIES (DEPARTMENT OF MATHEMATICS) We a c c e p t t h i s t h e s i s a s c o n f o r m i n g t o t h e r s ^ u i r e d s t a n d a r d THE UNIVERSITY OF BRITISH COLUMBIA J a n u a r y 1981 \u00a9 John S t a n l e y P a r s l o v , 1981 I n p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t o f t h e r e q u i r e m e n t s f o r an a d v a n c e d d e g r e e a t t h e U n i v e r s i t y o f B r i t i s h C o l u m b i a , I a g r e e t h a t t h e L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e a n d s t u d y . I f u r t h e r a g r e e t h a t p e r m i s s i o n f o r e x t e n s i v e c o p y i n g o f t h i s t h e s i s f o r s c h o l a r l y p u r p o s e s may be g r a n t e d by t h e h e a d o f my d e p a r t m e n t o r by h i s o r h e r r e p r e s e n t a t i v e s . I t i s u n d e r s t o o d t h a t c o p y i n g o r p u b l i c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l n o t be a l l o w e d w i t h o u t my w r i t t e n p e r m i s s i o n . D e p a r t m e n t o f W#7H\u00a3 MflncS The U n i v e r s i t y o f B r i t i s h C o l u m b i a 2075 Wesbrook P l a c e V a n c o u v e r , Canada V6T 1W5 D a t e dint, flfrrtl , t. i i ABSTRACT The anomalous p h y t o p l a n k t o n s e a s o n a l c y c l e i n t h e S u b a r c t i c P a c i f i c h as been a t t r i b u t e d t o g r a z i n g c o n t r o l . In s i m p l e c l a s s i c a l m odels 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 i n t e r a c t i o n , g r a z i n g t h r e s h o l d s a r e f o u n d t o be n e c e s s a r y t o o b t a i n t h i s t y p e of c o n t r o l . W e a t h e r s h i p o b s e r v a t i o n s a t O.S.P. a r e a n a l y s e d t o p r o v i d e a b a s i s f o r a more r e a l i s t i c model. P h y t o p l a n k t o n a r e p r e s e n t a t O.S.P. i n a l m o s t u n i f o r m l y low c o n c e n t r a t i o n s ( a b o u t 0.4 mg C h l a . m \" 3 ) , have low p h o t o s y n t h e t i c e f f i c i e n c y (<0.5 mg C.mg C h i a \" 1 . l y \" 1 ) , a d a p t t o s e a s o n a l c h a n g e s i n s o l a r r a d i a t i o n and show most s u r f a c e i n h i b i t i o n i n t h e s p r i n g . A- n u m e r i c a l p r o d u c t i o n model b a s e d on t h e s e r e s u l t s and d r i v e n by p h y s i c a l t i m e s e r i e s from t h e w e a t h e r s h i p s y i e l d s low a n n u a l p r o d u c t i o n l e v e l s compared w i t h p r e v i o u s e s t i m a t e s . P r e d i c t e d p r o d u c t i o n l e v e l s a r e s e n s i t i v e t o t h e c h o i c e o f r e s p i r a t i o n r a t e , and i n t r o d u c t i o n of a r a p i d l i g h t r e s p o n s e or 'Marra' e f f e c t r e s u l t s i n a d o u b l i n g of n e t p r o d u c t i o n . P r e d i c t e d y e a r t o y e a r v a r i a t i o n i s low and might be h i g h e r i f v a r i a t i o n i n S e c c h i d e p t h c o u l d be a c c o u n t e d f o r . In a p h y t o p l a n k t o n - z o o p l a n k t o n ( b i o m a s s ) model b a s e d on t h e p r o d u c t i o n model, g r a z i n g t h r e s h o l d s and o v e r - w i n t e r i n g s t r a t e g i e s a r e b o t h n e c e s s a r y f o r g r a z i n g c o n t r o l . Systems i d e n t i f i c a t i o n t e c h n i q u e s a r e a d a p t e d t o e s t i m a t e p o p u l a t i o n p a r a m e t e r s f o r c o h o r t s o f t h e dom i n a n t g r a z e r s . C o h o r t s t r u c t u r e i s i n t r o d u c e d i n t o 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 model u s i n g t h e s e e s t i m a t e s . As a r e s u l t , a t t e n t i o n i s s h i f t e d from t h e s p r i n g t o l a t e summer and f a l l where s e n s i t i v i t y and s t a b i l i t y p r o b l e m s a r e a s s o c i a t e d w i t h t h e o v e r - w i n t e r i n g d e p a r t u r e o f t h e dominant g r a z e r s . An a p p r o x i m a t e m a t h e m a t i c a l a n a l y s i s of S t e e l e ' s (1974) 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 a l l o w s t h e e x p l a n a t i o n and e l a b o r a t i o n of p r e v i o u s a u t h o r s ' n u m e r i c a l r e s u l t s . S t a b l e c y c l i c s o l u t i o n s a r e shown t o e x i s t under n u t r i e n t l i m i t a t i o n f o r c o n s t a n t m o r t a l i t y r a t e s i n t h e a b s e n c e o f g r a z i n g t h r e s h o l d s . A t t e n t i o n i s f o c u s e d on t h e t r a n s i e n t ( s p r i n g bloom) a p p r o a c h t o t h e n u t r i e n t - l i m i t e d c y c l e and a b r o a d e r ( p h y s i o l o g i c a l and b e h a v i o u r a l ) framework f o r z o o p l a n k t o n r e s p o n s e t o d e c l i n i n g f o o d c o n c e n t r a t i o n s i s p r o p o s e d . Systems i d e n t i f i c a t i o n t e c h n i q u e s a r e a l s o u s e d t o e s t i m a t e z o o p l a n k t o n f e e d i n g and g r o w t h p a r a m e t e r s from CEPEX c o p e p o d t i m e s e r i e s . The e s t i m a t e s a r e compared w i t h l i t e r a t u r e v a l u e s and t h e s t a t i s t i c a l and d e t e r m i n i s t i c l i m i t a t i o n s o f t h e t i m e s e r i e s d i s c u s s e d w i t h a mind t o f u t u r e e x p e r i m e n t s . A 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, b a s e d on t h e p a r a m e t e r e s t i m a t e s , p r o v i d e s a c o n s i s t e n t e x p l a n a t i o n of t h e o b s e r v e d p h y t o p l a n k t o n p e r s i s t e n c e a t low d e n s i t i e s as a s t a b l e n u t r i e n t - l i m i t e d e q u i l i b r i u m . A m a t h e m a t i c a l s o l u t i o n i n t e r m s of B e s s e l f u n c t i o n s i s f o u n d f o r p h y t o p l a n k t o n p o p u l a t i o n s u n d e r g o i n g d i f f u s i o n and s i n k i n g i n t h e c a s e of an e x p o n e n t i a l g r o w t h p r o f i l e . Non-d i m e n s i o n a l i z a t i o n a l l o w s a r e l a t i v e l y c o m p l e t e d i s c u s s i o n o f t h e e f f e c t s o f v a r y i n g p h y s i c a l and b i o l o g i c a l p a r a m e t e r s on p r o f i l e s and p o p u l a t i o n g r o w t h r a t e s . S u b s u r f a c e maxima f o r c o n s t a n t d i f f u s i v i t y and s i n k i n g r a t e , p r e v i o u s l y r e p o r t e d f o r an i d e a l i s e d s t e p - f u n c t i o n g r o w t h p r o f i l e , a r e a l s o o b t a i n e d f o r t h e e x p o n e n t i a l g r o w t h p r o f i l e . S o l u t i o n s t o c o u p l e d n o n - l i n e a r p h y t o p l a n k t o n - n u t r i e n t e q u a t i o n s c o r r e s p o n d i n g t o s u b s u r f a c e maxima o f t h e n u t r i e n t - t r a p t y p e a r e a l s o o b t a i n e d u s i n g b o u n d a r y - l a y e r t e c h n i q u e s . The dependence o f t h e d e p t h , shape and m a g n i t u d e of t h e s e maxima on p a r a m e t e r s i s e x p l o r e d . The a p p r o x i m a t e t h e o r y a g r e e s w e l l w i t h p r e v i o u s l y p u b l i s h e d r e s u l t s f r o m a complex s i m u l a t i o n m o d e l . V CONTENTS ABSTRACT i i LIST OF TABLES i x LIST OF FIGURES x i 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 I n t r o d u c t i o n .\u2022 1 1.2 P h y s i c a l Oceanography of the S u b a r c t i c P a c i f i c 2 1.3 B i o l o g y of the S u b a r c t i c P a c i f i c 8 1.4 Simple P h y t o p l a n k t o n - Z o o p l a n k t o n Models f o r the S u b a r c t i c P a c i f i c 14 1.5 Pr e v i e w of C h a p t e r s 2-4 33 CHAPTER 2. QUALITATIVE ANALYSIS OF A COMPLEX SIMULATION MODEL. 2.1 I n t r o d u c t i o n ...38 2.2 Model and A n a l y s i s 40 2.3 S i m u l a t i o n R e s u l t s 47 2.4 C o n c l u s i o n s 59 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 68 3.2 Data A n a l y s i s 68 3.2.1 D e s c r i p t i o n of the Data Set 68 3.2.2 C h l o r o p h y l l Data 69 3.2.3 1 4C Data. 75 3.2.4 N i t r a t e Data 101 3.2.5 N i t r a t e C o n c e n t r a t i o n and P r o d u c t i o n 104 3.3 A P h y t o p l a n k t o n Growth Model I l l 3.3.1 I n t r o d u c t i o n I l l 3.3.2 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 113 3.3.3 B i o l o g i c a l B a s i s f o r the Model 117 3.3.4 S i m u l a t i o n R e s u l t s 124 3.3.5 P r i m a r y P r o d u c t i o n and N i t r a t e D e p l e t i o n 148 CHAPTER 4. HERBIVOROUS ZOOPLANKTON AT O.S.P.: DATA ANALYSIS AND MODELLING. 4.1 Parameter 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. 150 4.1.2 Review of Parameter E s t i m a t i o n Techniques 151 4.1.3 A p p l i c a t i o n t o O.S.P. Data 156 4.1.4 S t a t i s t i c a l C o n s i d e r a t i o n s 160 4.1.5 R e s u l t s f o r Calanus plumchrus -..161 4.1.6 R e s u l t s f o r Calanus c r i s t a t u s 168 4.1.7 Other S p e c i e s 174 4.1.8 Secondary P r o d u c t i o n E s t i m a t e s 175 4.2 Biomass Model f o r Zoo p l a n k t o n 180 4.2.1 I n t r o d u c t i o n 180 4.2.2 F o r m u l a t i o n of a Biomass G r a z i n g Model 181 4.2.3 C h o i c e of Zoo p l a n k t o n Parameters 183 4.2.4 S i m u l a t i o n R e s u l t s and D i s c u s s i o n 187 4.3 A Cohort Model f o r Zooplankton 198 4.3.1 I n t r o d u c t i o n 198 4.3.2 Model F o r m u l a t i o n 200 4.3.3 Parameters 201 4.3.4 S i m u l a t i o n R e s u l t s and D i s c u s s i o n 202 4.4 C o n c l u s i o n s 224 CHAPTER 5. PARAMETER ESTIMATION AND STABILITY 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 Zoo p l a n k t o n Growth Model.253 5.3 E s t i m a t i o n R e s u l t s 261 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 I n t e r a c t i o n . 2 9 8 5.5 C o n c l u s i o n s ; 303 CHAPTER 6. DIFFUSION, SINKING AND GROWTH OF PHYTOPLANKTON. 6.1 I n t r o d u c t i o n 310 6.2 Review of a Simple Model 312 6.3 A More R e a l i s t i c Model 321 6.4 E f f e c t of a Mixed L a y e r 327 6.5 A G e n e r a l N e c e s s a r y C o n d i t i o n For Growth 336 6.6 D i s c u s s i o n 337 CHAPTER 7. MATHEMATICAL ANALYSIS OF DEEP CHLOROPHYLL MAXIMA. 7.1 I n t r o d u c t i o n 342 7.2 A P h y t o p l a n k t o n - N u t r i e n t Model 344 v i i i 7.3 E f f e c t of a Mixed Layer 354 7.4 E f f e c t of N u t r i e n t Dependent S i n k i n g Rates 356 7.5 D i s c u s s i o n 358 CHAPTER 8'. CONCLUDING REMARKS 373 BIBLIOGRAPHY 377 Table I . Parameters used i n S t e e l e ' s Model (2.1) 42 LIST OF TABLES. Table I I . 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., 1964 t o 1976 128 Table I I I . Monthly means of p r e d i c t e d d a i l y net p r i m a r 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 a d a p t a t i o n time s c a l e s 134 Table IV. Parameter e s t i m a t e s f o r Calanus plumchrus 163 Table V. Parameter e s t i m a t e s f o r Calanus c r i s t a t u s 169 Table V I . Secondary p r o d u c t i o n e s t i m a t e s 179 Table V I I . Average r e l a t i v e s e a s o n a l abundance of m i c r o z o o p l a n k t o n 233 Table V I I I . F i n a l parameter e s t i m a t e s and SSQ e r r o r s f o r Pseudocalanus 269 Table IX. E x p o n e n t i a l growth r a t e s f o r Calanus 275 Tab l e X. F i n a l parameter e s t i m a t e s and SSQ e r r o r s f o r Calanus 278 T a b l e X I . F i n a l p a r a m e t e r e s t i m a t e s and SSQ e r r o r s f o r P a r a c a l a n u s . x i LIST OF FIGURES. F i g u r e 1. Se a s o n a l c y c l e i n v e r t i c a l s t r u c t u r e a t O.S.P. 5 F i g u r e 2. Schematic diagram of s u r f a c e c u r r e n t s and domains i n the S u b a r c t i c P a c i f i c 6 F i g u r e 3. Se a s o n a l c y c l e i n c h l o r o p h y l l a a t O.S.P. and De p a r t u r e Bay, S t r a i t of G e o r g i a . . . .' 16 F i g u r e 4. Se 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 p r o d u c t i 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 17 F i g u r e 5. Phase p l a n e p o r t r a i t s f o r the system 1.2 21 F i g u r e 6. Type I I I f u n c t i o n a l r e sponses 28 F i g u r e 7. Phase p l a n e p o r t r a i t s f o r the system 1.6 29 F i g u r e 8. Phase p l a n e p o r t r a i t s f o r the system 2.3 45 F i g u r e 9. Comparison of g e n e r a t i o n t i m e s from e q u a t i o n 2.4 and those o b t a i n e d i n n u m e r i c a l s o l u t i o n s of 2.2 48 F i g u r e 10. S t a b l e c y c l i c s o l u t i o n s of the system 2.2 f o r GX=0.05 and (a) F=0.4. (b) F=0.2 49 (c) F = 0.6 51 F i g u r e 11. Be h a v i o u r of the system 2.1 f o r GX=0.05 and B=10. w i t h : (a) F=0.4, D=100. ( s t a b l e c y c l e ) ... 52 (b) F=0.2, D=100. ( u n s t a b l e o s c i l l a t i o n s ) . (c) F=0.2, D=175. ( s t a b l e c y c l e ) 53 F i g u r e 12. S i m u l a t i o n of s p r i n g and summer u s i n g the system 2.1 w i t h : (a) B=10., F=0.4, D=100. and GX=0.05 .....58 (b) B=10., F=0.4, D=175. and GX=0.05 60 (c) B=10., F=0.4, D=175. and GX=0.04 61 (d) B=10., F = 0.2, D=250., GX=0.05 and E=0.3 62 F i g u r e 13. S u r f a c e 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 from the w e a t h e r s h i p s a t O.S.P 70 F i g u r e 14. S u r f a c e c h l o r o p h y l l a ( c r u i s e medians, ann u a l smooth, s e a s o n a l f i t p l u s r e s i d u a l s ) f o r : (a) and (b) 1964-68 72 (c) and (d) 1969-76 73 F i g u r e 15. O b s e r v a t i o n s of C h i b\/Chl a and C h i c \/ C h l a from the w e a t h e r s h i p s a t O.S.P 76 F i g u r e 16. S u r f a c e C h i b\/Chl a ( c r u i s e medians, annual smooth, s e a s o n a l f i t p l u s r e s i d u a l s ) 1969-76 77 F i g u r e 17. S u r f a c e C h i c \/ C h l a ( c r u i s e medians, ann u a l smooth, s e a s o n a l f i t p l u s r e s i d u a l s ) 1969-76 78 F i g u r e 18. 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 per u n i t C h i a from the w e a t h e r s h i p s a t O.S.P 80 F i g u r e 19. Frequency h i s t o g r a m s f o r P(0) 81 F i g u r e 20. P(0) ( c r u i s e medians, annua l smooth, s e a s o n a l f i t p l u s r e s i d u a l s ) f o r : (a) and (b) 1964-68 82 (c) and (d) 1969-76 83 F i g u r e 21. S c a t t e r p l o t of P(0) vs I f o r 1964-68 85 F i g u r e 22. S c a t t e r p l o t of A vs I f o r 1964-68 91 F i g u r e 23. Depth p r o f i l e s of P 92 F i g u r e 24. Monthly a v e r a g e s of ex. 94 F i g u r e 25. S c a t t e r p l o t of B vs I. f o r 1964-68 96 F i g u r e 26. S c a t t e r p l o t of P M A X vs I 0 f o r 1964-68 98 F i g u r e 27. Monthly a v e r a g e s of e s t i m a t e s of l i g h t a d a p t a t i o n p a r a m e t e r s : (a) B 99 (b) B s 100 F i g u r e 28. S u r f a c e o b s e r v a t i o n s of n i t r a t e c o n c e n t r a t i o n from the w e a t h e r s h i p s a t O.S.P. 102 x i v F i g u r e 29. S u r f a c e n i t r a t e c o n c e n t r a t i o n s ( c r u i s e medians, annua l smooth, s e a s o n a l f i t p l u s r e s i d u a l s ) 103 F i g u r e 30. N i t r a t e c o n c e n t r a t i o n s from depth p r o f i l e s ( l a y e r a v e r a g e s , annual smooths, s e a s o n a l f i t s p l u s r e s i d u a l s ) 105 (a) 0 - 20m 106 (b) 20 - 40m 107 (c) 40 - 80m 108 (d) 80 - 130m 109 (e) 130 - 200m 110 F i g u r e 31. P(0) vs low s u r f a c e n i t r a t e v a l u e s , May t o O c t o b e r , 1964-68. 112 F i g u r e 32. Time s e r i e s of t o t a l s o l a r r a d i a t i o n , s u r f a c e t e mperature and mixed l a y e r d e pth used t o d r i v e s i m u l a t i o n model 125 F i g u r e 33. P r e d i c t e d d a i l y net p r o d u c t i o n u s i n g s t a n d a r d parameter s e t 127 F i g u r e 34. P r e d i c t e d net p r o d u c t i o n , mixed l a y e r C h i a and mixed l a y e r C:Chl a r a t i o f o r 1976 u s i n g s t a n d a r d parameter s e t and s e a s o n a l l i g h t a d a p t a t i o n 130 F i g u r e 35. As f o r F i g 34, but w i t h ' i n s t a n t a n e o u s ' l i g h t a d a p t a t i o n 131 XV F i g u r e 36. As f o r F i g 34, but w i t h 3-day a d a p t a t i o n 133 F i g u r e 37. P r e d i c t e d C:Chl a r a t i o i n the mixed l a y e r f o r 1976 u s i n g oc = 0.5 and B =2.0 136 F i g u r e 38. P r e d i c t e d d a i l y net p r o d u c t i o n on i n c r e a s i n g y t o 0.1 139 F i g u r e 39. P r e d i c t e d d a i l y net p r o d u c t i o n and mixed l a y e r C:Chl a r a t i o f o r 0c=l.O, B =2.0 w i t h c o n s t r a i n t V^IOO. ..140 F i g u r e 40. P r e d i c t e d d a i l y net p r o d u c t i o n f o r s t a n d a r d parameter s e t w i t h Marra e f f e c t i n t r o d u c e d 142 F i g u r e 41. Observed S e c c h i depths a t O.S.P. vs time of year 143 F i g u r e 42. P r e d i c t e d d a i l y net p r o d u c t i o n u s i n g s t a n d a r d parameter s e t and (a) upper envelope t o S e c c h i d e p t h s . ...144 (b) lower envelope t o S e c c h i depths 145 F i g u r e 43. P r e d i c t e d s e a s o n a l c y c l e i n mixed l a y e r C:Chl a r a t i o u s i n g o p t i m a l i t y c r i t e r i o n 147 F i g u r e 44. P r o j e c t i o n of approximate 95% c o n f i d e n c e r e g i o n s f o r Calanus plumchrus parameter e s t i m a t e s on (a) (9,R T) p l a n e 164 (b) U,6) p l a n e 165 x v i F i g u r e 45. P r o j e c t i o n of a p p r o ximate 95% c o n f i d e n c e r e g i o n s f o r Calanus c r i s t a t u s parameter e s t i m a t e s 171 on (a) (6,R T) p l a n e 172 (b) (y,B) p l a n e 173 F i g u r e 46. R e g r e s s i o n c o e f f i c i e n t s W,- vs c o r r e s p o n d i n g l e n g t h s 1; on l o g - l o g s c a l e 177 F i g u r e 47. P r e d i c t e d mixed l a y e r C h i a and h e r b i v o r e biomass f o r 1976 u s i n g s t a n d a r d parameter s e t 188 F i g u r e 48. E f f e c t of d e c r e a s i n g D on F i g 47 191 F i g u r e 49. E f f e c t of i n t r o d u c i n g o v e r - w i n t e r i n g s t r a t e g y i n F i g 47 194 F i g u r e 50. E f f e c t of r e d u c i n g s p r i n g r e c r u i t m e n t t o 25 mg Cm\" 3 on F i g 49 195 F i g u r e 51. P r e d i c t e d C h i a and g r a z e r s t a n d i n g s t o c k f o r biomass model w i t h Type I f u n c t i o n a l response and s p r i n g r e c r u i t m e n t ; 199 F i g u r e 52. P r e d i c t e d C h i a and z o o p l a n k t o n biomass f o r 1976, u s i n g weight t h r e s h o l d s f o r d e p a r t u r e i n c o h o r t model 203 F i g u r e 53. As f o r F i g 52, but w i t h f i x e d r e s i d e n c e t i m e s x v i i and lower growth and m o r t a l i t y r a t e s f o r C. C r i s t a t u s . ..209 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 F i g 53 211 F i g u r e 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 F i g u r e 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 Calanus i n F i g 53 214 F i g u r e 57. E f f e c t of d e c r e a s i n g g r a z i n g parameters CO and D i n F i g 53. 216 F i g u r e 58. 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 F i g u r e 59. P r e d i c t e d mixed l a y e r C h i a and t o t a l z o o p l a n k t o n carbon f o r 1964-76, u s i n g the parameters and f i x e d r e c r u i t m e n t l e v e l s of F i g 53 219 F i g u r e 60. Observed z o o p l a n k t o n wet w e i g h t s (10 day means) a t O.S.P. 1956-78 221 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 z o o p l a n k t o n 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 223 F i g u r e 62. Average s e a s o n a l c y c l e s i n (a) obse r v e d x v i i i z o o p l a n k t o n wet w e i g h t s (10 day means, 1956-78) 228 (b) i n g e s t i o n v a r i a b l e VI (10 day means, 1969-78) 229 (c) i n g e s t i o n v a r i a b l e V2 (10 day means, 1969-78) 230 F i g u r e 63. Annual v a r i a t i o n (10 day means) i n (a) i n g e s t i o n v a r i a b l e VI 237 (b) i n g e s t i o n v a r i a b l e V2 238 F i g u r e 64. P h y t o p l a n k t o n carbon (0-20m average) i n CEE5 ..257-F i g u r e -65. J2fj(W), ^(W) f o r WJ-=0.4 jug C, Wj^ =2.0 pq C ...262 F i g u r e 66. F u n c t i o n a l response d a t a f o r P s e u s o c a l a n u s ....264 F i g u r e 67. Observed d e n s i t i e s of Pseudocalanus i n CEE5 ...266 F i g u r e 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 f o r Pseudocalanus (a) t r i a l 1 270 (b) t r i a l 6 271 F i g u r e 69. Observed d e n s i t i e s of Calanus i n CEE5 276 F i g u r e 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 f o r C a l a n u s . (a) t r i a l 1 279 (b) t r i a l 4 281 (c) t r i a l 6 283 (d) t r i a l 7 284 x i x F i g u r e 71. Observed d e n s i t i e s of P a r a c a l a n u s i n CEE5. ...288 F i g u r e 72. Observed d e n s i t i e s of t o t a l n a u p l i i i n CEE5 ...289 F i g u r e 73. 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 f o r P a r a c a l a n u s (a) t r i a l 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 F i g u r e 74. P r e d i c t e d 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 i n CEE5 a f t e r day 30 302 F i g u r e 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. ...316 F i g u r e 76. Contour p l o t of the f u n c t i o n JVioo, Sg ) 317 F i g u r e 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 F i g u r e 78. 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 6.9. ...324 F i g u r e 79. Contour p l o t of the f u n c t i o n A.1 (UJ,\/3) .325 F i g u r e 80. C h a r a c t e r i s t i c p h y t o p l a n k t o n p r o f i l e f o r c o = 0.7, 13 = 20 328 F i g u r e 81. Contour p l o t of the f u n c t i o n _ A 3 ( \/ l , i p f o r (a) j3=0 332 XX (b) (3=1 333 (c) \/?=3 334 (d) (3 = 10 335 F i g u r e 82. Contour p l o t of S,\/(3 vs co,(3 f o r S=0.1 359 F i g u r e 83. C h a r a c t e r i s t i c p r o f i l e s P(S) vs S\/\/5 361 F i g u r e 84. P h y t o p l a n k t o n p r o f i l e s P(
to 33.8%o, and a lower 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 1000m. The t o p of the h a l o c l i n e c o r r e s p o n d s t o the maximum depth of the s u r f a c e mixed l a y e r , a t t a i n e d i n March a t the. end of the p e r i o d of net heat l o s s t h r o u g h the s u r f a c e . A s e a s o n a l t h e r m o c l i n e i s e s t a b l i s h e d over the p e r i o d of net heat g a i n , from A p r i l t o September, w i t h the mixed l a y e r b e i n g t y p i c a l l y about 30m deep a t the end of t h i s p e r i o d . S u r f a c e t e m p e r a t u r e s i n c r e a s e from about 5\u00b0C t o 13\u00b0C over t h i s p e r i o d . The s e a s o n a l t h e r m o c l i n e and 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 eroded over the c o o l i n g p e r i o d , from October t o March, by c o n v e c t i v e o v e r t u r n 4 and s t o r m a c t i v i t y . T h i s s e a s o n a l c y c l e ( F i g 1) i s q u a l i t a t i v e l y c h a r a c t e r i s t i c o f t h e o c e a n i c S u b a r c t i c P a c i f i c a l t h o u g h t h e r e i s some g e o g r a p h i c a l v a r i a t i o n i n t h e m a g n i t u d e and t i m i n g o f t h e c y c l e . Two r e v i e w s of t h e p h y s i c a l o c e a n o g r a p h y of t h e S u b a r c t i c P a c i f i c have been p u b l i s h e d as p a r t o f i n v e s t i g a t i o n s i n t o t h e o c ean e n v i r o n m e n t of salmon by t h e I n t e r n a t i o n a l N o r t h P a c i f i c F i s h e r i e s Commission (Dodimead e t a l ,1963; F a v o r i t e e t a l ,1976). In t h e e a r l i e r r e v i e w , t h e p r i n c i p a l c u r r e n t s y s t e m s and domains were summarized as i n F i g 2. The s o u t h e r n l i m i t o f t h e S u b a r c t i c r e g i o n was d e f i n e d by an a l m o s t v e r t i c a l i s o h a l i n e s u r f a c e of 3 4 % 0 a t a b o u t 42\u00b0N l a t i t u d e . I m m e d i a t e l y t o t h e n o r t h a body o f warmer water formed by m i x i n g of t h e K u r o s h i o and O y a s h i o c u r r e n t s and moving, .eastward a t 2 t o 4 n a u t i c a l m i l e s \/ d a y was i d e n t i f i e d a s t h e t r a n s i t i o n d omain. To t h e n o r t h a g a i n , t h e S u b a r c t i c c u r r e n t was d e s c r i b e d as moving e a s t a t a b o u t 2 n a u t i c a l m i l e s \/ d a y and c o n s i s t i n g o f t h a t p a r t of t h e O y a s h i o w h i c h does n o t mix w i t h t h e K u r o s h i o . T h e s e e a s t w a r d f l o w i n g c u r r e n t s d i v i d e o f f t h e c o a s t o f N o r t h A m e r i c a , p a r t t r a v e l l i n g s o u t h t o form t h e C a l i f o r n i a c u r r e n t and p a r t f l o w i n g n o r t h w a r d a r o u n d t h e G u l f o f A l a s k a , f o r m i n g a r e l a t i v e l y i n t e n s e b o u n d a r y c u r r e n t ( t h e A l a s k a n S t e a m ) . P a r t of t h e A l a s k a n S t r e a m t u r n s s o u t h t o f o r m t h e A l a s k a n G y r e , w h i l e p a r t moves n o r t h i n t o t h e B e r i n g Sea t o e v e n t u a l l y j o i n t h e E a s t Kamchatka c u r r e n t o r t h e O y a s h i o . The c i r c u l a t i o n t i m e f o r t h e e n t i r e s y s t e m i s g i v e n a s a b o u t 4-6 y e a r s . O.S.P. i s l o c a t e d n o r t h o f t h e t r a n s i t i o n z o n e , on t h e s o u t h - e a s t e r l y edge of t h e A l a s k a n G y r e . The f o l l o w i n g q u o t e i s 0 50 CL CD a 100 net h e a t i n g p\\ wind m i x i n g c o n v e c t i v e mix ing seasona l t h e r m o c l i n e \u2014 o p i m a r c n t a ' o c l i n e J F M A M J J A S O N D Month F i g u r e 1. S e a s o n a l c y c l e i n v e r t i c a l s t r u c t u r e a t O.S.P. (Ad a p t e d f r o m T a b a t a , 1 9 6 5 ) . 7 p a r t i c u l a r l y r e l e v a n t f o r m o d e l l i n g the p l a n k t o n i c ecosystem a t O.S.P. a l t h o u g h i t was prompted by c o n s i d e r a t i o n s of s e a s o n a l s a l i n i t y and temperature c y c l e s t h e r e : '. . . the g e o s t r o p h i c f l o w i n the v i c i n i t y of Ocean S t a t i o n \"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 , slow (2 m i l e s \/ d a y ) . Hence, w i t h i n the time p e r i o d c o n s i d e r e d h e r e , ' ( s e a s o n a l ) ' the waters i n the ar e a a r e s u b j e c t e d t o c l i m a t i c c o n d i t i o n s s i m i l a r t o those a t Ocean S t a t i o n \"P\"; t h u s , they can be r e g a r d e d as h a v i n g r e s i d e d t h e r e . . . ' (Dodimead e t a l ,1963) . The second review ( F a v o r i t e e_t a l ,1976) p r e s e n t s a s i m i l a r p a t t e r n f o r the g e n e r a l c i r c u l a t i o n , w i t h f u r t h e r e l a b o r a t i o n of the c u r r e n t and domain s t r u c t u r e s . There i s , however, a new f e a t u r e p r e s e n t e d which may a f f e c t b i o l o g i c a l m o d e l l i n g and d a t a -i n t e r p r e t a t i o n f o r O.S.P. The dynamic topography f o r J u l y shows a s m a l l c l o c k w i s e gyre i n the s u r f a c e c i r c u l a t i o n o f f the Queen C h a r l o t t e I s l a n d s , e a s t of the n o r t h - s o u t h b r a n c h i n g of the S u b a r c t i c C u r r e n t . A s s o c i a t e d w i t h t h i s gyre i s a ' D i l u t e Domain' of water showing the e f f e c t s of c o a s t a l r u n o f f . The rev i e w i s ambiguous as t o the westward e x t e n t of t h i s domain . C i r c u l a t i o n p a t t e r n s suggested by dynamic topography, i n t e g r a t e d t o t a l w i n d - s t r e s s t r a n s p o r t and n u m e r i c a l model r e s u l t s a l l i n d i c a t e t h a t the c o a s t a l gyre l i e s w e l l t o the e a s t of O.S.P. and t h a t the f l o w a t O.S.P. i s e s s e n t i a l l y as d e s c r i b e d above from the e a r l i e r r e v i e w . However, the D i l u t e Domain d e f i n e d on the b a s i s of the 33%\u201e i s o h a l i n e c o n t o u r a t 100m extends westward t o a l m o s t 160\u00b0W and i n c l u d e s O.S.P. In F i g 41 of F a v o r i t e et. a l (1976), the S u b a r c t i c C u r r e n t i s p o r t r a y e d as d i v i d i n g t o the 8 west of the D i l u t e Domain and O.S.P. The D i l u t e Domain i s d i l u t e by comparison w i t h the h i g h e r s a l i n i t y w a t e r s of the Ridge Domain t o the northwe s t (due t o u p w e l l i n g i n the A l a s k a n G y r e ) , the c o a s t a l u p w e l l i n g domains t o the s o u t h e a s t and the t r a n s i t i o n domain t o the s o u t h . I t i s not c l e a r t o what e x t e n t the lower s a l i n i t y a t O.S.P. may be due t o the a n n u a l e x c e s s of p r e c i p i t a t i o n over e v a p o r a t i o n which o c c u r s t h r o u g h o u t the e a s t e r n S u b a r c t i c (Dodimead e t a l , 1 9 6 3 ) , superimposed on a z o n a l f l o w . I f the f l o w t h r o u g h O.S.P. i s p r e d o m i n a n t l y z o n a l and p a r t of the e a s t w a r d - f l o w i n g S u b a r c t i c c u r r e n t , then as i n d i c a t e d by the quote above, a model of s e a s o n a l 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 column or i n the p l a n k t o n i c community may r e a s o n a b l y be c o n s t r u c t e d -without t a k i n g l a r g e - s c a l e h o r i z o n t a l a d v e c t i o n i n t o a c c o u n t . The models p r e s e n t e d i n t h i s t h e s i s have been c r e a t e d on t h i s b a s i s . I f O.S.P. r e a l l y l i e s w i t h i n a s m a l l - s c a l e c o a s t a l c i r c u l a t i o n , which seems u n l i k e l y f o r reasons c i t e d above and o t h e r s ( f o r example, the n i t r a t e s e a s o n a l c y c l e a t O.S.P. i s c h a r a c t e r i s t i c of S u b a r c t i c o c e a n i c r a t h e r than c o a s t a l w a t e r s ) , a more e x p l i c i t t r e a t m e n t of a d v e c t i v e e f f e c t s may be r e q u i r e d . 1.3 B i o l o g y of the S u b a r c t i c P a c i f i c . A v e r y c o n s i d e r a b l e l i t e r a t u r e e x i s t s on the b i o l o g y of the S u b a r c t i c P a c i f i c and t h i s b r i e f s u r v ey makes no p r e t e n c e a t an e x h a u s t i v e r e v i e w . The f o l l o w i n g i n f o r m a t i o n i s p r e s e n t e d p a r t l y as an i n t r o d u c t i o n t o the q u e s t i o n s a d d r e s s e d here u s i n g a m o d e l l i n g approach-. I t i s a l s o i n t e n d e d p a r t l y as an ov e r v i e w of the c u r r e n t s t a t e of b i o l o g i c a l knowledge, so t h a t a reader 9 u n f a m i l i a r w i t h t h i s l i t e r a t u r e can p l a c e the models c o n s i d e r e d h e r e , w i t h t h e i r n e c e s s a r y l i m i t a t i o n s and r a t h e r narrow f o c u s , i n b roader p e r s p e c t i v e . Much of the review w i l l be concerned w i t h s t u d i e s a t O.S.P. which i s by f a r the most i n t e n s i v e l y sampled ocean s t a t i o n i n the r e g i o n . R e l e v a n t i n f o r m a t i o n from s u r r o u n d i n g a r e a s w i l l be mentioned where a p p r o p r i a t e . P r o b a b l y the best-known f e a t u r e of the p l a n k t o n i c ecosystem i n the s u b - A r c t i c P a c i f i c i s the absence of a s p r i n g i n c r e a s e i n p h y t o p l a n k t o n abundance. T h i s was r e p o r t e d f o r the B e r i n g Sea over 20 y e a r s ago (Semina,1958), and c o n f i r m e d by the w e a t h e r s h i p o b s e r v a t i o n s a t O.S.P. ( P a r s o n s , 1 9 6 5 ) . D u r i n g a c r u i s e i n J u l y and August of 1959, m a c r o n u t r i e n t s ( n i t r a t e , s i l i c a t e , p h o s p h a t e ) , p h y t o p l a n k t o n ( c h l o r o p h y l l a) and p a r t i c u l a t e o r g a n i c carbon .were measured ( M c A l l i s t e r et a_l ,1960). N u t r i e n t l e v e l s were c o n s i s t e n t l y h i g h (>6 u g a t . l \" 1 NO\", >16 u g a t . l \" 1 S i O P , >1.2 u g a t . l \" 1 PO^), a t a time of year when n u t r i e n t d e p l e t i o n might n o r m a l l y be e x p e c t e d . T h i s o b s e r v a t i o n , t o g e t h e r w i t h the obs e r v e d e x p o n e n t i a l i n c r e a s e of p h y t o p l a n k t o n i n b a t c h c u l t u r e t o some 40 tim e s the i n i t i a l c o n c e n t r a t i o n once g r a z e r s were removed, s u p p o r t e d the argument, advanced by Semina (1960), t h a t z o o p l a n k t o n g r a z i n g i s r e s p o n s i b l e f o r the c o n s t a n c y of 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 n the S u b a r c t i c . The s e a s o n a l c y c l e i n the S u b a r c t i c P a c i f i c was c o n t r a s t e d w i t h the ' s p r i n g bloom' c y c l e s of c o a s t a l a r e a s and the N o r t h A t l a n t i c by H e i n r i c h (1962). He a t t r i b u t e d the d i f f e r e n c e t o the l i f e h i s t o r y s t r a t e g i e s of the dominant h e r b i v o r o u s copepods, Calanus plumchrus and Calanus c r i s t a t u s , i n the S u b a r c t i c P a c i f i c . A p r e l i m i n a r y p i c t u r e of z o o p l a n k t o n abundance, c o m p o s i t i o n 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 ( 1961), based on s u r f a c e t r a w l s and v e r t i c a l h a u l s made from the w e a t h e r s h i p s from 1956 t o 1958. He d e s c r i b e d a w i n t e r minimum i n z o o p l a n k t o n biomass from December t o March and a summer maximum from A p r i l t o J u l y . S u r f a c e t r a w l s were dominated by copepods i n A p r i l and May and by amphipods i n June and J u l y and i n November and December. V e r t i c a l h a u l s (150m t o s u r f a c e ) were p r e d o m i n a n t l y copepods (ca 75%) and ch a e t o g n a t h s (15%) a t a l l t i m e s . Two l a y e r s of maximum abundance of z o o p l a n k t o n were found, above and below the permanent h a l o c l i n e . Zooplankton wet weight i n the t o p 150m ranged from ca 10 mg.nr 3 i n w i n t e r t o 80 mg.m-3 i n May, 1957. Perhaps the s i n g l e most comprehensive study of the b i o l o g y o f . t h e S u b a r c t i c P a c i f i c t o date i s t h a t of L e B r a s s e u r (1969). T h i s s t u d y of p r e d a t o r - p r e y r e l a t i o n s h i p s i n the G u l f of A l a s k a i n c l u d e d a d e t a i l e d a n a l y s i s of n i n e y e a r s (1956-64) of z o o p l a n k t o n d a t a from O.S.P. The l i f e h i s t o r i e s of the dominant z o o p l a n k t o n s p e c i e s were d i s c u s s e d , based on average s e a s o n a l p a t t e r n s of abundance by stage as r e f l e c t e d i n v e r t i c a l h a u l s and s u r f a c e tows. Next t o the two s p e c i e s mentioned above, a t h i r d l a r g e h e r b i v o r o u s copepod, Eu c a l a n u s b u n g i , makes a s i g n i f i c a n t c o n t r i b u t i o n t o biomass, as noted f o r o t h e r l o c a t i o n s i n the S u b a r c t i c P a c i f i c ( H e i n r i c h , 1 9 6 2 ; S e k i g u c h i , 1 9 7 5 ) . S m a l l e r copepods ( Pse u d o c a l a n u s , Calanus 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 ) a r e i m p o r t a n t i n f a l l and w i n t e r . A mixed c o l l e c t i o n of p l a n k t o n i c p r i m a r y c a r n i v o r e s , i n c l u d i n g c h a e t o g n a t h s ( S a g i t t a e l e g a n s , E u k r o h n i a hamata) and the trachymedusa A g l a n t h a d i g i t a l e were a l s o d i s c u s s e d . An attempt was made t o e s t i m a t e the 11 s t a n d i n g s t o c k s of f o r a g e organisms (myctophid and s q u i d ) and the annual carbon f l u x t h r ough a s i m p l i f i e d food web l e a d i n g up t o the t e r t i a r y consumers (salmon, b a l e e n whales and p o m f r e t ) . Due t o the n a t u r e of the sa m p l i n g program a t O.S.P. , much more i s known of the h e r b i v o r o u s and c a r n i v o r o u s z o o p l a n k t o n than of the t r o p h i c l e v e l s above or below. For example, the p o p u l a t i o n dynamics of s q u i d , an im p o r t a n t p r i m a r y \/ s e c o n d a r y c a r n i v o r e , a r e v i r t u a l l y unknown. 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 of p h y t o p l a n k t o n a t O.S.P. i s a l s o c o m p a r a t i v e l y p o o r l y known. W e a t h e r s h i p p h y t o p l a n k t o n samples have not 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 of net p h y t o p l a n k t o n i s a v a i l a b l e from the s h i p s of o p p o r t u n i t y program ( V e n r i c k , 1 9 7 1 ) and from w e a t h e r s h i p m i c r o -z o o p l a n k t o n d a t a ( d i s c u s s e d l a t e r ) . O b s e r v a t i o n s a t O.S.P. i n J u l y , August, 1959 ( M c A l l i s t e r e t a l , 1 9 6 0 ) , d u r i n g the Transpac c r u i s e of 1969 (Parsons,1972) 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 s t a t i o n s (R. Waters, p e r s . comm. ) suggest t h a t p h y t o p l a n k t o n a re dominated i n biomass by s m a l l f l a g e l l a t e s , l e s s than 10 pm i n d i a m e t e r . There have been a number of t h e o r e t i c a l a n a l y s e s of 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 of c h l o r o p h y l l a and p r i m a r y p r o d u c t i o n a t O.S.P. (based on 1 4 C uptake measurements) was d e s c r i b e d by Parsons (1965). In t h a t paper, S v e r d r u p ' s c r i t i c a l depth model (Sverdrup,1953) was used t o e x p l o r e the i n t e r a c t i o n of s e c c h i d e p t h , 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 . A l l macro-n u t r i e n t s were d e s c r i b e d as n o n - l i m i t i n g t hroughout the G u l f of A l a s k a and an annual p r i m a r y p r o d u c t i o n of ca 60 g Cm\" 2 was 12 e s t i m a t e d , based on 1 4 C measurements and the 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. The c r i t i c a l depth approach was l a t e r expanded t o cov e r the f u l l G u l f of A l a s k a by Parsons and L e B r a s s e u r (1968). The l a r g e - s c a l e p a t t e r n i n the t i m i n g of the s p r i n g i n c r e a s e i n z o o p l a n k t o n s t a n d i n g s t o c k was p r e d i c t e d i n t h i s s t u d y . R e s u l t s of p r i m a r y p r o d u c t i o n s t u d i e s on the Transpac c r u i s e and 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 between S e a t t l e and Yokohama were r e p o r t e d by Parsons and Anderson (1970). A d e p t h - i n t e g r a t e d form of S t e e l e and Menzel's (1962) e q u a t i o n was found t o o v e r e s t i m a t e p r o d u c t i o n on the Transpac c r u i s e . Reducing the p h o t o s y n t h e t i c e f f i c i e n c y parameter from 0.24 t o 0.17 (ug C.ug C h i a ^ . l y \" 1 ) i n t h i s e q u a t i o n p r o v i d e d a b e t t e r f i t f o r the Transpac d a t a , but v a l u e s r a n g i n g from 0.07 t o 3.1 (jug C.ug C h i a ^ . l y \" 1 ) were n e c e s s a r y t o o b t a i n agreement w i t h d a t a from the AML c r u i s e s . The c o n v e r s i o n of p r i m a r y p r o d u c t i o n t o secondary p r o d u c t i o n a t O.S.P. has been s t u d i e d by M c A l l i s t e r (1969,1972). D a i l y p h y t o p l a n k t o n p r o d u c t i o n ( e s t i m a t e d from 1 4 C measurements), minus e s t i m a t e d dark r e s p i r a t i o n , was assumed t o be i n g e s t e d by z o o p l a n k t o n . Zooplankton r e s p i r a t i o n was c a l c u l a t e d as a f r a c t i o n of z o o p l a n k t o n s t a n d i n g s t o c k . A s s i m i l a t e d r a t i o n minus r e s p i r a t i o n was taken as secondary p r o d u c t i o n . Because of u n c e r t a i n t y i n z o o p l a n k t o n r e s p i r a t i o n r a t e s (eg S t e e l e , 1 9 7 4 ) , e s t i m a t e s of secondary p r o d u c t i o n r e p r e s e n t e d the d i f f e r e n c e of two 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 v a l u e s t o a maximum of 23g C.m~ 2.yr _ 1. An e s t i m a t e of 13 g C . i r r 2 . y r ~ l was chosen as most l i k e l y . 13 For the purposes of t h i s t h e s i s , an i n t e r e s t i n g summary of the c u r r e n t s t a t e 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 be o b t a i n e d by a s s e s s i n g i t s s t r e n g t h s and weaknesses as a b a s i s f o r a d e t a i l e d , r a t i o n a l m e c h a n i s t i c model ( P i a t t e_t a l ,1975) of the ecosystem t h e r e . Whether a model of t h i s k i n d i s a d e s i r a b l e g o a l , p a r t i c u l a r l y i f we wi s h t o make s u c c e s s f u l q u a n t i t a t i v e p r e d i c t i o n s , i s d e b a t a b l e ( P i a t t e t a_l ,1975), but i t does p r o v i d e a u s e f u l way t o s t r u c t u r e an approach t o e x i s t i n g d a t a . There i s a l o n g time s e r i e s 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, 1 4 C p r o d u c t i v i t y and 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 of p h y t o p l a n k t o n dynamics, a knowledge of 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 r a t i o s , i n the p h o t o s y n t h e s i s vs l i g h t r e l a t i o n s h i p (on s h o r t and l o n g time s c a l e s ) , i n p h y t o p l a n k t o n r e s p i r a t i o n r a t e s and i n l i m i t i n g e f f e c t s of m i c r o n u t r i e n t s , i f any, would a l l be d e s i r a b l e . An attempt w i l l be made t o i n f e r some of the s e i n d i r e c t l y from the a v a i l a b l e o b s e r v a t i o n s i n Chapter 3. I n v e s t i g a t i o n of s i z e and\/or s p e c i e s dependent e f f e c t s i n b i o l o g i c a l i n t e r a c t i o n s would r e q u i r e 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 . The l o n g time s e r i e s of 150m v e r t i c a l h a u l s i s the p r i n c i p a l z o o p l a n k t o n d a t a s o u r c e , p r o v i d i n g i n f o r m a t i o n on t o t a l wet we i g h t , s p e c i e s c o m p o s i t i o n and some st a g e and\/or s i z e s t r u c t u r e . These d a t a a r e s u p p o r t e d by s t u d i e s on s e a s o n a l and d i u r n a l 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 (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 ) . Some measurements have been made of g r a z i n g r a t e s , c h e m i c a l c o m p o s i t i o n and r e s p i r a t i o n of the dominant h e r b i v o r e s p r i m a r i l y i n c o a s t a l l o c a t i o n s f a r from O.S.P. (Parsons 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). However, i n f o r m a t i o n on the f u n c t i o n a l and n u m e r i c a l responses of thes e dominant copepods i s poor compared w i t h b e t t e r s t u d i e d c o a s t a l 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) or Pseudocalanus ( P a f f e n h o f f e r and H a r r i s , 1 9 7 6 ) . The average s e a s o n a l p a t t e r n of 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. has been r e p o r t e d ( L e B r a s s e u r and Kennedy,1972), but the a u t e c o l o g y of most of these organisms i s v e r y p o o r l y known. A s i m i l a r s t a t e of i g n o r a n c e e x i s t s f o r most of the p r i m a r y c a r n i v o r e s mentioned above, w h i l e even the abundance of f o r a g e organisms such as s q u i d i s p o o r l y known. The study of f o r a g e organisms i s c o m p l i c a t e d by t h e i r v e r y l a r g e 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 ( m u l t i - y e a r ) g e n e r a t i o n t i m e s . For such l o n g - l i v e d o r g a n i s m s , 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 whales, salmon and po m f r e t , a model of c o n d i t i o n s at O.S.P. becomes m e a n i n g l e s s and the l a r g e - s c a l e v a r i a b i l i t y and c i r c u l a t i o n of the S u b a r c t i c P a c i f i c would have t o be m o d e l l e d . 1.4 Simple P h y t o p l a n k t o n - z o o p l a n k t o n Models f o r the S u b a r c t i c Pac i f i c . As the summary of S e c t i o n 1.3 demonstates, an attempt t o c o n s t r u c t a d e t a i l e d , complete ecosystem model f o r O.S.P. , or r a t h e r f o r the G u l f of A l a s k a or the whole S u b a r c t i c P a c i f i c , would be premature, t o say the l e a s t . For m u l t i p l e t r o p h i c l e v e l s and l a r g e h o r i z o n t a l s c a l e s , the s i m p l e r tropho-dynamic arguments of L e B r a s s e u r (1969) and Sanger (1972) a r e more a p p r o p r i a t e a t p r e s e n t . A much narrower range of q u e s t i o n s i s 15 a d d r e s s e d h e r e , c e n t e r i n g on the observed c o n s t a n c y of 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. These q u e s t i o n s can be ad d r e s s e d u s i n g s i m p l i f i e d s i m u l a t i o n models which a re more commensurate w i t h the p r e s e n t s t a t e of b i o l o g i c a l knowledge. The s e a s o n a l c y c l e of C h i a a t O.S.P. i s p r e s e n t e d i n F i g 3. The c y c l e i n the S t r a i t of G e o r g i a i s a l s o p r e s e n t e d f o r c o n t r a s t . W h i l e the de c r e a s e i n mixed l a y e r depth and i n c r e a s e i n s o l a r r a d i a t i o n a t O.S.P. i n the s p r i n g does r e s u l t i n an i n c r e a s e i n p r i m a r y p r o d u c t i v i t y (Parsons,1965) t h i s i s not r e f l e c t e d 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 s t a n d i n g s t o c k (as measured by C h i a ) . I n s t e a d , the biomass (wet weight) of z o o p l a n k t o n above 150m v a r i e s s e a s o n a l l y i n a s i m i l a r manner t o p r i m a r y p r o d u c t i v i t y ( 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 the h y p o t h e s i s t h a t z o o p l a n k t o n g r a z i n g r e s u l t s i n a p h y t o p l a n k t o n m o r t a l i t y r a t e which b a l a n c e s p h y t o p l a n k t o n growth throughout the y e a r . 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 i n t u r n r a i s e s o t h e r q u e s t i o n s . Why s h o u l d i t o c c u r i n the o c e a n i c S u b a r c t i c P a c i f i c and not i n c o a s t a l a r e a s , nor i n the o c e a n i c N o r t h A t l a n t i c ? Perhaps even more t r o u b l e s o m e , i n view of the r a p i d growth r a t e 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 i n t h i s growth r a t e , and the observed v a r i a b i l i t y i n z o o p l a n k t o n s t a n d i n g s t o c k , i s the t i g h t b a l a n c e 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 growth by t h i s h y p o t h e s i s . The f i r s t q u e s t i o n was answered by H e i n r i c h (1962) as f o l l o w s . The dominant g r a z e r s i n the N o r t h A t l a n t i c and 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 Calanus p a c i f i c u s , o v e r - w i n t e r as l a t e c o p e p o d i t e s t a g e s or a d u l t s and 3 0 CO JL 2 0 D Q_ 10 O _o U 0 i \" i s 11 \/1 M M v O N D M o n t h Figure 3. Seasonal cycle in chlorophyll a at O.S.P. (solid line) and i n Departure Bay, Strait of Georgia (dashed l i n e ) . (Adapted from Parsons, 1965). J F M A M J J A S O N D MONTH Figure 4. Seasonal cycle at O.S.P. i n (a) primary production and (b) zooplankton standing stock, (from McAllister, 1969) . 18 cannot reproduce i n the s p r i n g u n t i l they have accumulated egg t i s s u e by f e e d i n g on adequate 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 . A f u r t h e r p e r i o d i n which n a u p l i i do not f e e d f o l l o w s r e p r o d u c t i o n , so t h a t a s i z e a b l e l a g o c c u r s i n the n u m e r i c a l response of z o o p l a n k t o n t o the s p r i n g i n c r e a s e i n p h y t o p l a n k t o n growth. The dominant copepods i n the S u b a r c t i c , C. plumchrus and C. c r i s t a t u s , reproduce a t depth i n the s p r i n g , u s i n g f a t s t o r e s l a i d down the p r e v i o u s summer. 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 h e r e b y r e c r u i t e d t o the s u r f a c e i n the s p r i n g w i t h o u t any l a g i n response 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 r a t e s . There a r e some c o a s t a l a r e a s such as the S t r a i t of G e o r g i a (Parsons,1965) and the Sea of Japan ( H e i n r i c h , 1 9 6 2 ) where C. plumchrus and\/or C. c r i s t a t u s dominate but a p h y t o p l a n k t o n s p r i n g bloom does o c c u r . H e i n r i c h a t t r i b u t e d t h i s t o an e a r l i e r i n c r e a s e i n p h y t o p l a n k t o n growth r a t e i n s t r a t i f i e d c o a s t a l w a t e r s and a consequent f a i l u r e 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 the s e copepods. W h i l e t h i s argument seems t o o f f e r a s o l u t i o n t o the f i r s t q u e s t i o n (a s u f f i c i e n t l a g i n z o o p l a n k t o n response i n the s p r i n g w i l l c l e a r l y ensure a s p r i n g p h y t o p l a n k t o n bloom), i t does not add r e s s the second q u e s t i o n . In the remainder of t h i s s e c t i o n , t h i s q u e s t i o n of b a l a n c e i s a d d r e s s e d by c o n s i d e r i n g the s t a b i l i t y p r o p e r t i e s of some s i m p l e biomass models of p h y t o p l a n k t o n - z o o p l a n k t o n i n t e r a c t i o n s . These s h o u l d not be regarde d as r e a l i s t i c or p r e d i c t i v e models but r a t h e r as sta t e m e n t s of paradigm, i n the sense of Kuhn(1970). T h e i r c o n s i d e r a t i o n w i l l a l l o w us t o r e l a t e the q u e s t i o n of b a l a n c i n g g r a z i n g 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 i s s u e s i n m a r i n e e c o l o g y . A c l a s s i c a l s t a r t i n g p o i n t f o r s i m p l e m o d e l s of p r e d a t o r -p r e y i n t e r a c t i o n s (as t h e z o o p l a n k t o n - p h y t o p l a n k t o n i n t e r a c t i o n w i l l be r e g a r d e d f o r t h e r e s t o f t h i s s e c t i o n ) i s t h e d i f f e r e n t i a l e q u a t i o n model : x = r . x - a . x . y 1.1a y = e.a.x.y - m.y 1.1b ( L o t k a , 1 9 2 5 ) . In t h i s model, t h e p a r a m e t e r r r e p r e s e n t s an i n t r i n s i c r a t e of g r o w t h of p r e y , x; t h e p a r a m e t e r a i s t h e s u c c e s s f u l e n c o u n t e r r a t e by a s i n g l e p r e d a t o r p e r u n i t p r e y d e n s i t y so t h a t a.x.y i s t h e t o t a l r a t e o f c o n s u m p t i o n o f p r e y by p r e d a t o r s ; e i s t h e e f f i c i e n c y o f c o n v e r s i o n of .consumed p r e y t o p r e d a t o r so t h a t e.a.x.y r e p r e s e n t s t h e c o n s e q u e n t r a t e of i n c r e a s e i n p r e d a t o r , y, w h i l e m i s a c o n s t a n t p e r c a p i t a l o s s r a t e f o r p r e d a t o r s i n t h e a b s e n c e o f f o o d . The o s c i l l a t o r y s o l u t i o n s of s y s t e m 1.1 o r i g i n a l l y a t t r a c t e d some i n t e r e s t , but t h e i r n e u t r a l s t a b i l i t y p r o p e r t i e s and c o n s e q u e n t l a c k o f r o b u s t n e s s under s m a l l s t r u c t u r a l c h a n g e s i n t h e model l e d t o i t s r e p l a c e m e n t . The model c a n be i m p r o v e d by i n t r o d u c i n g f u r t h e r non-l i n e a r i t i e s t o a c c o u n t f o r r e s o u r c e - l i m i t a t i o n o f p r e y and s a t u r a t i o n of p r e d a t o r s ( H o l l i n g , 1 9 5 9 ) . One p o s s i b l e f o r m f o r s u c h a model i s x = r . x . ( l - x \/ K ) - i M . x . y \/ ( D + x ) 1.2a y = e . i M . x . y \/ ( D + x ) - m.y 1.2b 20 H e r e , K i s a c a r r y i n g c a p a c i t y f o r p r e y i n t h e a b s e n c e of p r e d a t o r s , i M i s t h e maximum r a t e o f c o n s u m p t i o n o f p r e y p e r p r e d a t o r and D i s t h e p r e y d e n s i t y a t w h i c h p r e y c o n s u m p t i o n r e a c h e s h a l f i t s maximum v a l u e . The r a t i o i M \/ D i s c o m p a r a b l e t o t h e p a r a m e t e r a i n e q u a t i o n 1.1. The q u a l i t a t i v e b e h a v i o u r o f t h i s model has seen much d i s c u s s i o n ( f o r a r e v i e w , s e e May,1974). Two t y p e s o f phase p l a n e p o r t r a i t s a r e shown i n F i g 5. The p r e y i s o c l i n e a l w a y s forms a q u a d r a t i c 'hump' and t h e p r e d a t o r i s o c l i n e a v e r t i c a l l i n e . T h e i r i n t e r s e c t i o n ( x , y ) i s a c r i t i c a l p o i n t of t h e s y s t e m and a s i m p l e g e o m e t r i c r u l e d e t e r m i n e s i t s s t a b i l i t y p r o p e r t i e s . When t h e p r e d a t o r i s o c l i n e (x = x) l i e s t o t h e r i g h t of t h e 'hump', ( F i g 5 a ) , (x,y) i s a s y m p t o t i c a l l y s t a b l e and t r a j e c t o r i e s s p i r a l i n t o i t . When t h e p r e d a t o r i s o c l i n e l i e s t o t h e l e f t of t h e hump, t r a j e c t o r i e s s p i r a l o u t w a r d t o a s t a b l e l i m i t c y c l e s o l u t i o n . To r e l a t e t h e s e r e s u l t s t o t h e O.S.P. s e a s o n a l c y c l e , a way must be f o u n d t o i n c o r p o r a t e s e a s o n a l e f f e c t s i n t o t h i s p r e s e n t l y homogeneous mo d e l . The o b v i o u s a p p r o a c h i s t o a l l o w t h e p h y t o p l a n k t o n g r o w th p a r a m e t e r s r and K t o be f u n c t i o n s o f t i m e , t , r e f l e c t i n g t h e s e a s o n a l c y c l e i n p r i m a r y p r o d u c t i v i t y ( F i g 4 ) . The r e s u l t i n g s y s t e m i s no l o n g e r homogeneous and c o n s e q u e n t l y d i f f i c u l t t o t r e a t a n a l y t i c a l l y . Suppose f o r t h e moment t h a t t h e p a r a m e t e r s i n 1.2 a t any p a r t i c u l a r t i m e t a r e s u c h t h a t a s t a b l e n o n - t r i v i a l e q u i l i b r i u m ( x ( t ) , y ( t ) ) e x i s t s f o r t h e c o r r e s p o n d i n g homogeneous s y s t e m . I f t h e t i m e s c a l e f o r a p p r o a c h t o t h i s e q u i l i b r i u m i s s u f f i c i e n t l y f a s t compared w i t h t h e ( s e a s o n a l ) t i m e s c a l e s o v e r w h i c h r and K change, i t seems r e a s o n a b l e t h a t t h e s o l u t i o n t o t h e non-homogeneous s y s t e m , i f i t s t a r t s n e a r 21 C Q) D \"O ^ ^ isocl ine predator isocl ine a \\ o prey dens i t y x K p r e y dens i ty x Figure 5. Phase plane portraits, for system 1.2. (a) equilibrium stable. (b) equilibrium unstable. 22 ( x ( t ) , y ( t ) ) , w i l l remain c l o s e t o t h i s q u a s i - e q u i l i b r i u m s o l u t i o n . By t h i s s e p a r a t i o n of time s c a l e s (Ludwig e t a l ,1978), a s e a s o n a l c y c l e can be e n v i s a g e d which i s c l o s e l y a p p r o x i m a t e d by 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 . Now x ( t ) , y ( t ) are d e t e r m i n e d by e. i M . x \/ ( D + x ) = m 1.3a y ( t ) = r ( t ) . ( l - x \/ K ( t ) ) . ( D + x ) \/ i M 1.3b There a r e two im m e d i a t e l y e n c o u r a g i n g a s p e c t s of these e x p r e s s i o n s . 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 t i m e , depending o n l y on z o o p l a n k t o n p a r a m e t e r s . 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 c o n c e n t r a t i o n i s p r o p o r t i o n a l t o the p h y t o p l a n k t o n growth r a t e . .Phytoplankton 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 s which f o l l o w e d ( x ( t ) , y ( t ) ) c l o s e l y would behave as observed 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 p r i m a r y p r o d u c t i v i t y . There i s however a s e r i o u s problem w i t h t h i s ' e x p l a n a t i o n ' . The s t a b i l i t y c r i t e r i o n g i v e n above can be w r i t t e n as 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 s t o c k of p h y t o p l a n k t o n a t which r e s o u r c e l i m i t a t i o n causes the growth r a t e t o drop t o z e r o . There a r e two ob v i o u s p o t e n t i a l l y l i m i t i n g r e s o u r c e s f o r p h y t o p l a n k t o n , namely n u t r i e n t s u p p l y and a v a i l a b l e l i g h t . In s e c t i o n 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. were d e s c r i b e d as 23 n o n - l i m i t i n g . The v a l u e of K reached i n the c u l t u r e experiment of M c A l l i s t e r e t a l (1960) was about 40 times x. In f a c t , the u s u a l n o n - l i n e a r M i c h a e l i s - M e n t e n r e l a t i o n s h i p between growth and n u t r i e n t c o n c e n t r a t i o n means t h a t any n e g a t i v e feedback on growth r a t e of s m a l l i n c r e a s e s i n x above x w i l l be much s m a l l e r than a v a l u e of x\/K of 1\/40 i n the l o g i s t i c model would su g g e s t . I n c r e a s e s i n p h y t o p l a n k t o n d e n s i t y can d e c r e a s e p h y t o p l a n k t o n growth r a t e s t h r o u g h s e l f - s h a d i n g . T h i s e f f e c t 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 p o p u l a t i o n by employing 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, and c h l o r o p h y l l , x, k = k\u201e + k t .x (adapted from Parsons et a l , 1977), combined w i t h a . p h o t o s y n t h e s i s (P) vs l i g h t ( I ) r e l a t i o n s h i p (eg P= #.I.exp(-I \/ I MAX)' S t e e l e , 1 9 6 2 ) . I n t e g r a t i n g over depth g i v e s an e x p r e s s i o n f o r growth i n the mixed l a y e r as a f u n c t i o n of mixed l a y e r d e p t h , z M , s u r f a c e l i g h t i n t e n s i t y I 0 , P vs I parameters a and I M A X , the parameters k 0 and k 4 and the C h i a c o n c e n t r a t i o n , x. D i f f e r e n t i a t i n g t h i s e x p r e s s i o n w i t h r e s p e c t t o x and s u b s t i t u t i n g f o r x = x y i e l d s a v a l u e s u i t a b l e f o r i n s e r t i o n as 1\/K i n 1.2. For z M =' 30m ( l a t e summer), k 4 = .02 m2.mg C h i a \" 1 , (Lorenzen,1980;Megard et a l ,1980), k 0 = .1 n r 1 , and I 0 \/ I M A x = 2. ( S t e e l e , 1962), t h i s y i e l d s a v a l u e of K, j g h t = 8 mg C h i a . n r 3 or about 20 t i m e s x. T h i s v a l u e a g r e e s w i t h the r e s u l t s of T a kahashi and P a r s o n s ( 1 9 7 2 ) . Both 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 s e l f - s h a d i n g 24 suggest t h a t ,at O.S.P. ', the p h y t o p l a n k t o n p o p u l a t i o n i s b e i n g m a i n t a i n e d 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 c a p a c i t y or l e s s . A c c o r d i n g t o the 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 about 20 t i m e s x which would imply t h a t z o o p l a n k t o n a t O.S.P. a r e growing and r e p r o d u c i n g a t 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 s u f f i c i e n t t o s u p p l y 1\/40 of t h e i r maximum r a t i o n . W h i l e l a r g e v a l u e s of the 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 of t h i s o r d e r have been r e p o r t e d , such a low r e l a t i v e i n g e s t i o n r a t e seems u n l i k e l y t o cov e r even the b a s a l m e t a b o l i c r e q u i r e m e n t s of the z o o p l a n k t o n . Of c o u r s e , the p a r t i c u l a r v a l u e 1\/40 can be q u e s t i o n e d as i t depends on the s p e c i f i c form of the g r a z e r s ' f u n c t i o n a l r e s p o n s e , as w e l l as the assumptions u n d e r l y i n g the e s t i m a t e of K. However, the problem i s more fundamental than the s t a b i l i t y c r i t e r i o n a l o n e would s u g g e s t . For example, a p i e c e w i s e l i n e a r , or Type I ( H o l l i n g , 1 9 6 5 ) f u n c t i o n a l response on the p a r t of z o o p l a n k t o n g r a z e r s has r e c e i v e d support ( M u l l i n and Brooks,1975). I f a f u n c t i o n a l response of t h i s type i s s u b s t i t u t e d i n t o 1.2, the e q u i l i b r i u m i s always a s y m p t o t i c a l l y s t a b l e , no matter how s m a l l x\/K may be. The l o c a l r a t e of approach t o e q u i l i b r i u m , on the o t h e r hand, i s then g i v e n by r.x\/K, and the 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 depends not o n l y on (x,y) b e i n g s t a b l e , but on the f u r t h e r c o n d i t i o n t h a t approach 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 time s c a l e compared w i t h changes i n ( x ( t ) , y ( t ) ) . V a l u e s of x\/K of o r d e r l \/ 2 0 t h and a v a l u e of r of 0.2 d a y : 1 , e s t i m a t e d from M c A l l i s t e r et a l (1960), g i v e a c h a r a c t e r i s t i c time of approach t o e q u i l i b r i u m of o r d e r 50 days. T h i s i s not s h o r t compared w i t h the s e a s o n a l time 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 ) . L a r g e o s c i l l a t i o n s 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 abundance would o c c u r i f s u c h a w e a k l y - s t a b l e s y s t e m was s u b j e c t e d t o s e a s o n a l c y c l e s i n p r o d u c t i v i t y . A l t h o u g h t h e homogeneous s y s t e m 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 i s s t a b l e , when x\/K i s v e r y s m a l l , i t i s l i t t l e b e t t e r t h a n n e u t r a l l y s t a b l e i n t e r m s of i t s a b i l i t y t o t r a c k a s e a s o n a l l y v a r y i n g e q u i l i b r i u m . The s y s t e m 1.2 c a n n o t p r o v i d e a c o n s i s t e n t e x p l a n a t i o n of o b s e r v a t i o n s a t O.S.P. However, t h e q u a s i - e q u i l i b r i u m a p p r o a c h seems p r o m i s i n g and one m i g h t t r y t o p r o c e e d by m o d i f y i n g t h e s y s t e m so as t o overcome t h e s t a b i l i t y t i m e s c a l e p r o b l e m , w h i l e r e t a i n i n g t h e q u a l i t a t i v e b e h a v i o u r of 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 ( x , y ( t ) ) . One a p p r o a c h - w h i c h d o e s n ' t work i s m e n t i o n e d h e r e f o r i n t e r e s t s a k e . I n t r o d u c i n g a q u a d r a t i c l o s s t e r m f o r p r e d a t o r s i s known t o b r i n g a b o u t a s y m p t o t i c s t a b i l i t y i n s i m p l e p r e d a t o r -p r e y m o d els ( B a z y k i n , 1 9 7 4 ) and h a s been s u g g e s t e d i n o t h e r p h y t o p l a n k t o n - z o o p l a n k t o n models ( L a n d r y , 1 9 7 6 ) . A model of t h i s k i n d , w i t h t y p e I f u n c t i o n a l r e s p o n s e , can be w r i t t e n as x = r . x - a . x . y 1.5a y = e.a.x.y - u . y 2 1.5b f o r x a t sub-maximal r a t i o n l e v e l s . (An i n c r e a s e i n p e r c a p i t a l o s s r a t e (u.y) w i t h p r e d a t o r d e n s i t y c o u l d r e s u l t from i n t r a s p e c i f i c c o m p e t i t i o n f o r r e s o u r c e s o t h e r t h a n f o o d , o r from s w i t c h i n g o r a g g r e g a t i v e r e s p o n s e s i n h i g h e r c a r n i v o r e s . ) T h i s model c a n be r e j e c t e d , w i t h o u t c o n s i d e r i n g i t s s t a b i l i t y 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 v a r y i n g r ( t ) i s y ( t ) = r ( t ) \/ a x ( t ) = u . y ( t ) \/ ( e . a ) 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 x ( t ) i s not c o n s t a n t but v a r i e s w i t h y ( t ) i n t h i s model. T h i s i s c e r t a i n l y not c o n s i s t e n t w i t h o b s e r v a t i o n s 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 esponses have a l r e a d y been c o n s i d e r e d i n 1.2 : a h y p e r b o l i c (Type I I ) and p i e c e w i s e l i n e a r (Type I ) response ( H o l l i n g , 1 9 6 5 ) . W i t h i n the c o n t e x t of s i m p l e biomass models such as 1.2, the s e can be c h a r a c t e r i z e d as d e s t a b i l i z i n g 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 . A t h i r d form of f u n c t i o n a l response d i s c u s s e d by H o l l i n g (1965) i s the s i g m o i d or Type I I I f u n c t i o n a l r e sponse, i n c o r p o r a t i n g a reduced c l e a r a n c e r a t e by copepods a t low food d e n s i t i e s . T h i s type of f u n c t i o n a l response has a s t a b i l i s i n g e f f e c t i n s i m p l e biomass models. A number of s t u d i e s of p e l a g i c copepods have suggested g r a z i n g r esponses of t h i s t y p e , 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 f e e d i n g below some t h r e s h o l d food d e n s i t y ( P a rsons et_ a_l ,1969; F r o s t , 1 9 7 2 ) , or a r e d u c t i o n i n f i l t e r i n g r a t e a t low food d e n s i t i e s ( F r o s t , 1 9 7 5 ) . T h e o r e t i c a l c o n s i d e r a t i o n s suggest t h a t a copepod t r y i n g t o o p t i m i z e energy i n t a k e s h o u l d reduce i t s f i l t e r i n g a c t i v i t y a t low food d e n s i t i e s (Lam and F r o s t , 1 9 7 6 ; S t e e l e and F r o s t , 1 9 7 7 ) . The 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 response can be seen i n a phase p l a n e a n a l y s i s of the system 27 k = r.x - f ( x ) . y 1.6a y = e . f ( x ) . y - m.y 1.6b where the g r a z i n g f u n c t i o n f ( x ) has the t h r e s h o l d form ( F i g 6a) or s i g m o i d form ( F i g 6 b ) . (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 term (-x\/K) has been dropped as i t i s n e g l i g i b l e under the c o n d i t i o n s p r e v a i l i n g a t O.S.P.). Phase p l a n e p o r t r a i t s f o r the system 1.6 a r e g i v e n i n F i g 7. For the case of a g r a z i n g t h r e s h o l d a t x=x\u201e , the prey i s o c l i n e asymptotes t o the v e r t i c a l l i n e x=x c a t the l e f t , has a minimum a t some p o i n t x*, and a s y m p t o t e s ' t o the l i n e y = r . x \/ i M as x approaches oo. (The parameter i M r e p r e s e n t s the maximum z o o p l a n k t o n r a t i o n . ) The z o o p l a n k t o n i s o c l i n e i s a v e r t i c a l l i n e as. f o r system 1.2. These i n t e r s e c t a t the 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 l i n e a r i z i n g i n the neighbourhood of ( x , y ) , i t can be shown t h a t t h i s e q u i l i b r i u m i s a s y m p t o t i c a l l y s t a b l e i f and o n l y i f x < x*; t h a t i s , the z o o p l a n k t o n i s o c l i n e must l i e t o the l e f t of the minimum i n the p h y t o p l a n k t o n i s o c l i n e . G iven t h a t t h i s c o n d i t i o n i s s a t i s f i e d , t h e r e a r e s t i l l two q u a l i t a t i v e l y d i s t i n c t phase p o r t r a i t s . When x i s v e r y 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 M . y 1.8a y = ( e . i M - m ) . y 1.8b 28 a x o p rey dens i t y x p rey dens i t y x Figure 6. Type III functional responses: (a) threshold. (b) sigmoid. 29 0 prey density x F i g u r e 7. Phase p l a n e p o r t r a i t s f o r s y s t e m 1.6. (a) e . l M ~ m > r . (b) e . i ^ - n K r . 30 I f e.i^-m > r , o r , e q u i v a l e n t l y , i f the maximum growth r a t e of z o o p l a n k t o n exceeds t h a t of p h y t o p l a n k t o n , t r a j e c t o r i e s s t a r t i n g from l a r g e x and s m a l l y w i l l always c y c l e back t o low 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 i g 7 a ) . In t h i s sense, i t i s i m p o s s i b l e f o r p h y t o p l a n k t o n t o escape z o o p l a n k t o n c o n t r o l p e r m anently. I f e . i M - m < r , (as seems more l i k e l y ) , the be h a v i o u r of t r a j e c t o r i e s f o r l a r g e x depends on i n i t i a l c o n d i t i o n s , (x, ,y,- ). I f yi i s l a r g e enough, the t r a j e c t o r y w i l l c y c l e , but o t h e r w i s e , x w i l l i n c r e a s e i n d e f i n i t e l y and the t r a j e c t o r y w i l l approach (+00,+ 00). i n the c o r r e s p o n d i n g phase p l a n e p o r t r a i t ( F i g 7b)- t h e r e i s a s e p a r a t r i x which d i v i d e s t r a j e c t o r i e s which c y c l e from those which don't. The phase p l a n e p o r t r a i t s f o r s i g m o i d f u n c t i o n a l responses a r e q u a l i t a t i v e l y s i m i l a r , the p r i n c i p a l d i f f e r e n c e b e i n g t h a t the p h y t o p l a n k t o n i s o c l i n e asymptotes t o x=0. The l o c a l 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 , can be found by l i n e a r i z i n g about ( x , y ) . In f a c t , i f h(x) i s the c l e a r a n c e r a t e of z o o p l a n k t o n ( g i v e n by f ( x ) \/ x ) , the l o c a l r a t e of approach t o e q u i l i b r i u m i s g i v e n by r . x . h ' ( x ) \/ h ( x ) The s t a b i l i t y c r i t e r i o n x < x* i s s i m p l y e q u i v a l e n t t o a p o s i t i v e r a t e of approach, x* b e i n g g i v e n by h'(x*)=0. That i s , the e q u i l i b r i u m (x,y) i s l o c a l l y s t a b l e p r o v i d e d the c l e a r a n c e r a t e i n c r e a s e s w i t h x a t x=x. In the case of a t h r e s h o l d , h y p e r b o l i c f u n c t i o n a l response as used by S t e e l e (1974), 31 f ( x ) = i M . ( x - x 0 ) 4 \/ ( D + ( x - x e ) + ) , and e x p l i c i t f o r m u l a e 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 a p p r o a c h : x * - x 0 = \/D7X0 r a t e o f a p p r o a c h = r . ( ( x * - x 0 ) 2 - ( x - x \u201e ) 2 ) \/ ( ( x - x e ) . D + ( x - x \u201e ) 2 ) T h a t i s , t h e c l e a r a n c e r a t e i s a maximum a t a d e n s i t y x* w h i c h i s g r e a t e r t h a n t h e t h r e s h o l d by t h e g e o m e t r i c mean of t h e h a l f -s a t u r a t i o n c o n s t a n t , D, and t h e t h r e s h o l d x e . P r o v i d e d t h e e q u i l i b r i u m x does n o t l i e t o o c l o s e t o x*, t h e r a t e of a p p r o a c h t o e q u i l i b r i u m i s of t h e same o r d e r as t h e p h y t o p l a n k t o n g r o w th r a t e , r . I t i s p o s s i b l e t h e n f o r t h e a p p r o a c h t o e q u i l i b r i u m i n 1.6 t o o c c u r on r e l a t i v e l y f a s t t i m e s c a l e s , of t h e same o r d e r as p h y t o p l a n k t o n g r o w t h , and c o n s e q u e n t l y f o r t h e q u a s i - e q u i l i b r i u m a s s u m p t i o n t o be v a l i d f o r s e a s o n a l v a r i a t i o n i n p h y t o p l a n k t o n g r o w t h r a t e , r . Note t h a t t h e q u a s i - e q u i l i b r i u m c y c l e g i v e n by 1.7 p r e d i c t s , as b e f o r e , a c o n s t a n t p h y t o p l a n k t o n s t a n d i n g s t o c k and a z o o p l a n k t o n s t a n d i n g s t o c k w h i c h v a r i e s w i t h p r i m a r y p r o d u c t i o n . The s i m p l e b i o m a s s model 1.6, a c c o r d i n g t o t h i s a p p r o x i m a t e argument b a s e d on s e p a r a t i o n o f t i m e s c a l e s , s h o u l d be c a p a b l e of q u a l i t a t i v e l y r e p r o d u c i n g t h e o b s e r v e d s e a s o n a l c y c l e a t O.S.P. R e f l e c t i o n on t i m e s c a l e s a l l o w s an i n t e r e s t i n g , i n t u i t i v e p e r s p e c t i v e on t h e d e v e l o p m e n t so f a r . The t i m e s c a l e s one t r a d i t i o n a l l y a s s o c i a t e s w i t h p h y t o p l a n k t o n g r o w t h a r e s h o r t and 32 one would expect an agent which r e g u l a t e s p h y t o p l a n k t o n d e n s i t y t i g h t l y t o o p e r a t e on a s i m i l a r time s c a l e . The time s c a l e s one t r a d i t i o n a l l y a s s o c i a t e s w i t h copepod dynamics a r e much l o n g e r and t h i s appears t o be a problem w i t h a g r a z i n g c o n t r o l h y p o t h e s i s a t O.S.P. Where n u t r i e n t s a r e l i m i t i n g , feedback e f f e c t s on p h y t o p l a n k t o n growth can t i g h t l y r e g u l a t e p h y t o p l a n k t o n d e n s i t y , as d i s c u s s e d i n Chapter 2, but t h i s i s a p p a r e n t l y not the case a t O.S.P. The s i g m o i d or t h r e s h o l d f u n c t i o n a l response 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 time s c a l e , a z o o p l a n k t o n b e h a v i o u r a l time s c a l e , 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 of the 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 i n the S u b a r c t i c and e l s e w h e r e , based on the l i f e - h i s t o r y s t r a t e g i e s of the dominant g r a z e r s . He a t t r i b u t e d the s p r i n g bloom i n the N o r t h A t l a n t i c t o a d e l a y i n the n u m e r i c a l response of the dominant g r a z e r t o the s p r i n g i n c r e a s e i n p h y t o p l a n k t o n growth and abundance. I f the maximum growth r a t e of p h y t o p l a n k t o n exceeds t h a t of z o o p l a n k t o n biomass, a phase p l a n e p o r t r a i t as i n F i g 7b i s a p p r o p r i a t e . The s p r i n g i n c r e a s e i n p h y t o p l a n k t o n growth r a t e i s e q u i v a l e n t t o a v e r t i c a l s h i f t i n the p h y t o p l a n k t o n i s o c l i n e . A d e l a y i n z o o p l a n k t o n response can then e a s i l y r e s u l t i n the system b e i n g o v e r t a k e n by the s e p a r a t r i x , l e a v i n g the l o c a l s t a b i l i t y domain about (x,y) and e n t e r i n g a r e g i o n i n which p h y t o p l a n k t o n have escaped z o o p l a n k t o n c o n t r o l . A c c o r d i n g t o 1.6, the t r a j e c t o r y w i l l a pproach ( + oo, + oo). In 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 the s p r i n g bloom t e r m i n a t e s . H e i n r i c h ( 1 9 6 2 ) a t t r i b u t e d the o c c u r r e n c e of s p r i n g blooms i n 33 c o a s t a l r e g i o n s of the S u b a r c t i c where C. plumchrus i s p r e s e n t t o a f a i l u r e i n t i m i n g of r e c r u i t m e n t . A l t h o u g h the s p r i n g bloom does o c c u r e a r l i e r i n the S t r a i t of G e o r g i a than i n o c e a n i c l o c a t i o n s ( P a r s o n s , 1 9 6 5 ) , the n a u p l i i of C. plumchrus r e a c h the s u r f a c e w a t e r s of the S t r a i t of G e o r g i a i n F e b r u a r y and March ( F u l t o n , 1 9 7 3 ) and i t i s not c l e a r t h a t the t i m i n g i s i n a p p r o p r i a t e . The phase p l a n e p o r t r a i t i n F i g 7b sug g e s t s another e x p l a n a t i o n . In c o a s t a l w a t e r s s u b j e c t t o r u n - o f f , s t r a t i f i c a t i o n and a consequent i n c r e a s e i n p h y t o p l a n k t o n growth r a t e can occur very r a p i d l y . T h i s may r e s u l t i n the system s t a t e ( x ( t ) , y ( t ) ) f a i l i n g t o t r a c k the r a p i d l y s h i f t i n g q u a s i -e q u i l i b r i u m c y c l e ( x , y ( t ) ) and a g a i n b e i n g o v e r t a k e n by the s e p a r a t r i x , r e s u l t i n g i n a s p r i n g bloom. The 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 perhaps more c o n v i n c i n g e x p l a n a t i o n : namely, t h a t r e c r u i t m e n t of C. plumchrus i n the s t r a i t f a i l s t o c o i n c i d e i n space, r a t h e r than t i m e , w i t h the s p r i n g bloom. S u c c e s s f u l o v e r - w i n t e r i n g of C. plumchrus o c c u r s o n l y i n water deeper than 300m, which o c c u p i e s o n l y o n e - f o u r t h of the a r e a of the s t r a i t . A r r i v a l of C. plumchrus i n the r e m a i n i n g a r e a s i s presumably d e l a y e d , s u b j e c t t o h o r i z o n t a l a d v e c t i o n . 1.5 Pr e v i e w of C h a p t e r s 2-4. Each of the s i m p l e models c o n s i d e r e d above can be r e g a r d e d as a composite h y p o t h e s i s c o n c e r n i n g t r o p h i c i n t e r a c t i o n s a t O.S.P. There i s c l e a r l y a need f o r independent e x p e r i m e n t a l t e s t s of a s p e c t s of the s e h y p o t h e s e s . For example, the f u n c t i o n a l r esponses of the dominant copepods a t O.S.P. a r e not well-known and the 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 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 i s an e s s e n t i a l p a r t of the 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 have h a r d l y been exhausted by the above a n a l y s i s . G i v e n the c o m p a r a t i v e w e a l t h of i n f o r m a t i o n i n the l o n g time s e r i e s of o b s e r v a t i o n s a t O.S.P. , more than rough q u a l i t a t i v e agreement of a model w i t h average s e a s o n a l c y c l e s can be demanded. 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 and 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. For example, the s t a t e v a r i a b l e s x and y above have been used r a t h e r l o o s e l y 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 z o o p l a n k t o n s t a n d i n g s t o c k , a l t h o u g h they have been i m p l i c i t l y i d e n t i f i e d w i t h C h i a i n mg.irr 3 and z o o p l a n k t o n wet weight i n mg.nr 2 r e s p e c t i v e l y . As the 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. v a r i e s s e a s o n a l l y i n v e r t i c a l d i s t r i b u t i o n and ( h y p o t h e t i c a l l y ) i n C:Chl a r a t i o ( M c A l l i s t e r , 1 9 6 9 ) , the use of C h i a i n mg.m\"3 as a s t a t e v a r i a b l e c l e a r l y needs c a r e f u l e x a m i n a t i o n . Zooplankton wet weight i s a rough summary v a r i a b l e , r e p r e s e n t i n g c o n t r i b u t i o n s 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 dominated by h e r b i v o r o u s copepods a t most t i m e s ( L e B r a s s e u r , 1 9 6 9 ) . More d e t a i l e d 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 a v a i l a b l e f o r O.S.P. and t h i s c e r t a i n l y d e s e r v e s a t t e n t i o n i n view of r e c e n t t h e o r e t i c a l r e s u l t s ( S t e e l e , 1 9 7 4 ; S t e e l e and F r o s t , 1 9 7 7 ) . These problems a r e ad d r e s s e d i n Chapter 3 and Chapter 4. In Chapter 3, an attempt i s made a t a more r e a l i s t i c q u a n t i t a t i v e s i m u l a t i o n model of p h y t o p l a n k t o n growth a t O.S.P. The model i s p a r t l y based on an o r i g i n a l a n a l y s i s of the p h y t o p l a n k t o n d a t a , p a r t i c u l a r l y 1 4 C uptake r a t e s , o b t a i n e d from the w e a t h e r s h i p s . 35 In Chapter 4, a parameter e s t i m a t i o n t e c h n i q u e d e v e l o p e d f o r copepod time s e r i e s i s a p p l i e d t o e s t i m a t e p o p u l a t i o n parameters f o r the dominant h e r b i v o r e s a t O.S.P. Two more e l a b o r a t e models 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 i n t e r a c t i o n a r e c o n s t r u c t e d , u s i n g t h e s e r e s u l t s and those of Chapter 3. Some e f f e c t of 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 i s c o n s i d e r e d i n the second model. The models 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 and 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 c o n c e r n i n g 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 i n t e r a c t i o n a t O.S.P. have 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 e x p l a i n the s e a s o n a l c y c l e s 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 the system's a b i l i t y t o c l o s e l y t r a c k a q u a s i - e q u i l i b r i u m s e a s o n a l c y c l e . The hypotheses d i f f e r i n the mechanisms r e s p o n s i b l e f o r the s h o r t term s t a b i l i s i n g feedback n e c e s s a r y f o r t h i s 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 e q u i l i b r i u m . The model 1.2, i n v o l v i n g r e s o u r c e - l i m i t a t i o n of p h y t o p l a n k t o n growth, i s c e r t a i n l y c a p a b l e 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 , but has been r e j e c t e d on the b a s i s of independent e x p e r i m e n t a l e v i d e n c e . The model 1.5, i n v o l v i n g a q u a d r a t i c l o s s r a t e f o r g r a z e r s , cannot reproduce the o b s e r v e d s e a s o n a l c y c l e . The model 1.6, which assumes a r e d u c t i o n i n g r a z i n g 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 , i s c a p a b l e of r e p r o d u c i n g the observed s e a s o n a l c y c l e q u a l i t a t i v e l y . As noted e a r l i e r , the a c t u a l f u n c t i o n a l r e s p o n s e s of g r a z e r s a t O.S.P. a r e not known. The s i m u l a t i o n s c o n s i d e r e d i n Chapter 4 w i l l be p r i m a r i l y based upon the system 1.6 and the assumption of g r a z i n g 36 t h r e s h o l d s . However, t h e c o m p o s i t e h y p o t h e s i s i m p l i c i t i n 1.6 w i l l become m o d i f i e d i n t h e p r o c e s s of model e l a b o r a t i o n . In p a r t i c u l a r , an e x p l i c i t t r e a t m e n t of c o p e p o d l i f e - h i s t o r y s t r a t e g i e s w i l l show t h a t t h e , s p r i n g r e c r u i t m e n t o f n a u p l i i o v e r an e x t e n d e d p e r i o d can s t a b i l i s e 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 i n t e r a c t i o n and p r e v e n t a s p r i n g bloom, even i n t h e a b s e n c e o f g r a z i n g t h r e s h o l d s . As a r e s u l t , a t t e n t i o n w i l l f o c u s on t h e .problem of m a i n t a i n i n g g r a z i n g c o n t r o l i n t h e summer and f a l l a g a i n s t t h e d e s t a b i l i s i n g e f f e c t of o v e r - w i n t e r i n g d e p a r t u r e by t h e d o m i n a n t c o p e p o d s . A l a c k of model r o b u s t n e s s d u r i n g t h i s p e r i o d and d i s c r e p a n c i e s i n d e t a i l between model p r e d i c t i o n s and o b s e r v a t i o n s w i l l f o r c e a r e - e v a l u a t i o n of t h e g r a z i n g t h r e s h o l d h y p o t h e s i s and a s e a r c h f o r a l t e r n a t i v e s . The p o s s i b i l i t y o f n u t r i e n t l i m i t a t i o n o f p h y t o p l a n k t o n g r o w th i n l a t e summer and f a l l w i l l be r e c o n s i d e r e d and t h e p o t e n t i a l i m p o r t a n c e of s p a t i a l v a r i a t i o n i n m a i n t a i n i n g g r a z i n g c o n t r o l w i l l be d i s c u s s e d . C h a p t e r 2 i s not d i r e c t l y c o n c e r n e d w i t h e v e n t s a t O.S.P. but i s of more g e n e r a l t h e o r e t i c a l i n t e r e s t . The i d e a t h a t f e e d i n g t h r e s h o l d s may be i m p o r t a n t i n p l a n k t o n e c o s y s t e m s i s n o t a new one. In a n u m e r i c a l model of p h y t o p l a n k t o n - z o o p l a n k t o n i n t e r a c t i o n s i n t h e N o r t h Sea, S t e e l e ( 1 9 7 4 ) f o u n d t h a t t h r e s h o l d s were n e c e s s a r y t o o b t a i n r e a l i s t i c b e h a v i o u r . An a l t e r n a t i v e was p r o p o s e d by L a n d r y ( 1 9 7 6 ) , who f o u n d t h a t i n s e r t i n g a q u a d r a t i c l o s s t e r m f o r h e r b i v o r e s i n t o S t e e l e ' s model e l i m i n a t e d t h e need f o r t h r e s h o l d s . B o t h a u t h o r s b a s e d t h e i r t h e o r e t i c a l c o n c l u s i o n s on s i m u l a t i o n r e s u l t s . A 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 o f t h e s e m o d e l s , b a s e d on s e p a r a t i o n o f t i m e s c a l e s , i s g i v e i n C h a p t e r 2. The a n a l y s i s p r o v i d e s i n s i g h t s i n t o t h e r e s u l t s o f 37 S t e e l e and L a n d r y w h i c h a p p e a r t o have i n t e r e s t i n g i m p l i c a t i o n s f o r m a r i n e e c o s y s t e m models i n g e n e r a l , and t h e m o d e l s c o n s i d e r e d h e r e f o r O.S.P. i n p a r t i c u l a r . 38 CHAPTER 2 QUALITATIVE ANAYLSIS OF A COMPLEX SIMULATION MODEL 2.1 I n t r o d u c t i o n . In a t h e o r e t i c a l t r e a t i s e on the p l a n k t o n i c ecosystem of the N o r t h Sea, S t e e l e (1974) examined the r e l a t i v e importance of i n t e r a c t i o n s between d i f f e r e n t 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 system'. H i s c o n c l u s i o n s were based on a s i m u l a t i o n model of a s i m p l i f i e d n u t r i e n t - p h y t o p l a n k t o n - c o p e p o d food c h a i n which c o n t i n u e s t o be of t h e o r e t i c a l i n t e r e s t , both as a b a s i s f o r more complex r e a l i s t i c models ( S t e e l e and F r o s t , 1 9 7 7 ) and as a f i r s t s t e p i n c o m p l e x i t y and r e a l i s m above the h i g h l y s i m p l i f i e d p r e d a t o r - p r e y models of the 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 based p r i m a r i l y on comparisons of a number of computer s i m u l a t i o n s of the model i n v o l v i n g v a r i o u s a ssumptions c o n c e r n i n g the form and magnitude of t r o p h i c i n t e r a c t i o n s . A key f i n d i n g was t h a t the model c o u l d not p r e d i c t t i m e streams which agreed q u a l i t a t i v e l y w i t h o b s e r v a t i o n s of the N o r t h Sea 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 f u n c t i o n a l response ( H o l l i n g , 1 9 5 9 ) , was i n v o k e d f o r h e r b i v o r o u s copepods. Landry (1976) o b t a i n e d r e a l i s t i c b e h a v i o u r from S t e e l e ' s model w i t h o u t t h r e s h o l d s by i n t r o d u c i n g a per c a p i t a p r e d a t i o n r a t e on h e r b i v o r e s which i n c r e a s e d i n p r o p o r t i o n t o h e r b i v o r e numbers a t low d e n s i t i e s . T h i s was p a r t l y i n t r o d u c e d by Landry as a s i z e - d e p e n d e n t term ( s m a l l e r copepods b e i n g more abundant) but i t s s i g n i f i c a n c e seemed t o l i e i n the r e s u l t i n g q u a d r a t i c l o s s term f o r h e r b i v o r e s ( S t e e l e , 1 9 7 6 ) ; the 39 i n t r o d u c t i o n of such a l o s s term i s well-known t o produce s t a b l e e q u i l i b r i a i n o t h e r w i s e u n s t a b l e s i m p l e L o t k a - V o l t e r r a models ( B a z y k i n , 1 9 7 4 ) . Both S t e e l e and Landry p r e s e n t e d t h e i r r e s u l t s as a t l e a s t s u g g e s t i v e e v i d e n c e f o r the e x i s t e n c e and importance of t h r e s h o l d s and d e n s i t y - d e p e n d e n t p e r - c a p i t a p r e d a t i o n r a t e s r e s p e c t i v e l y i n r e a l ecosystems. T h i s f o c u s i n g of a t t e n t i o n on p a r t i c u l a r b i o l o g i c a l q u e s t i o n s i s r e c o g n i s e d as a v a l u a b l e 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 s t u d i e s i n g e n e r a l , and i t appears t o have been s u c c e s s f u l here i n view of the d i s c u s s i o n a r o u s e d c o n c e r n i n g the e x p e r i m e n t a l o b s e r v a t i o n of f e e d i n g t h r e s h o l d s ( M u l l i n et a l ,1975; F r o s t , 1 9 7 5 ) . However, any argument t h a t some a s p e c t of a model i s n e c e s s a r y i f r e a l i s t i c r e s u l t s a r e t o be o b t a i n e d must, always be viewed w i t h c a u t i o n . One reason f o r t h i s , common t o a l l m o d e l l i n g s t u d i e s , i s t h a t c o n c e p t u a l e l a b o r a t i o n of a model i s always p o s s i b l e (as i n the case of Landry v e r s u s S t e e l e ) , so t h a t the n e c e s s i t y of any p a r t i c u l a r concept can never be e s t a b l i s h e d . A second reason a p p l i e s p a r t i c u l a r l y t o s t u d i e s such as S t e e l e ' s and L andry's which a r e based on computer s i m u l a t i o n . G iven the l a r g e u n c e r t a i n t y 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 not 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 f u n c t i o n a l form s h o u l d be judged t o be i n c a p a b l e of 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 c r i t e r i a on the b a s i s of a s m a l l number of\" n u m e r i c a l s o l u t i o n s u s i n g p a r t i c u l a r parameter v a l u e s . T h i s c r i t i c i s m can be a d d r e s s e d by the t h e o r e t i c i a n , a t l e a s t i n p r i n c i p l e , as 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 the model can p r o v i d e i n f o r m a t i o n about the b e h a v i o u r of s o l u t i o n s over r e g i o n s i n 40 p a r a m e t e r s p a c e , r a t h e r t h a n a t p o i n t s w i t h i n them. A n a l y s i s o f t h i s t y p e c a n a l s o l e a d t o a b e t t e r u n d e r s t a n d i n g o f t h e b e h a v i o u r o f t h e model and i t s dependence on p a r a m e t e r s and t h e r e b y a l l o w u s e f u l b i o l o g i c a l i n s i g h t s . A d d i t i o n a l m o t i v a t i o n f o r a q u a l i t a t i v e a n a l y s i s of S t e e l e ' s model i s p r o v i d e d by t h e s t u d y o f s i m p l e L o t k a - V o l t e r r a models i n C h a p t e r 1 w h i c h s u g g e s t e d t h a t g r a z i n g t h r e s h o l d s a r e i m p o r t a n t i n t h e S u b a r c t i c P a c i f i c e c o s y s t e m . W h i l e t h i s i s s u p e r f i c i a l l y c o n s i s t e n t w i t h S t e e l e ' s c o n c l u s i o n f o r t h e N o r t h Sea, a p u z z l i n g d i s c r e p a n c y e x i s t s . I t was n e c e s s a r y t o i n v o k e t h r e s h o l d s i n s i m p l e m o d els of t h e S u b a r c t i c P a c i f i c o n l y b e c a u s e p h y t o p l a n k t o n t h e r e a r e n o t n u t r i e n t - l i m i t e d . In t h e N o r t h Sea s i m u l a t i o n m o d e l, p h y t o p l a n k t o n a r e n u t r i e n t - l i m i t e d t h r o u g h o u t much of t h e summer, and i t i s not c l e a r why t h r e s h o l d s s h o u l d be needed f o r s t a b i l i t y . An e x p l a n a t i o n o f t h i s d i s c r e p a n c y i s so u g h t h e r e t h r o u g h t h e q u a l i t a t i v e a n a l y s i s o f S t e e l e ' s m o d e l . 2.2 Model and A n a l y s i s The model d e v e l o p e d by S t e e l e (1974) and u s e d , w i t h c e r t a i n a l t e r a t i o n s , by L a n d r y ( 1 9 7 6 ) i s : R = V.(RO-R) + U. ( E . ( P - P l )\/(D+P) + F).Z.W 0- 7 - A.R.P\/(B+R) 2.1a ( r a t e o f change of n u t r i e n t e q u a l s m i x i n g t h r o u g h t h e r m o c l i n e + z o o p l a n k t o n e x c r e t i o n - p h y t o p l a n k t o n u p t a k e ) P = A.R.P\/(B+R) - V.P - C.Z.W 0 7.(P-P1)\/(D+P) 2.1b ( r a t e of change of p h y t o p l a n k t o n = g r o w t h - m i x i n g - g r a z i n g ) 41 W = ( (0.7.C-E) . (P-P1)\/(D+P) - F).W 0- 7 2.1c ( r a t e o f i n d i v i d u a l g r o w t h = n e t a s s i m i l a t i o n - a c t i v e and b a s a l m e t a b o l i s m ) Z = -GW.(W-Wl).(Z-Z1)\/(H+Z.W) - GX.Z 2 . I d ( z o o p l a n k t o n m o r t a l i t y r a t e = n o n l i n e a r , w e i g h t - d e p e n d e n t t e r m + 1 i n e a r t e r m ) . When c o p e p o d s r e a c h a d u l t w e i g h t , W2, growth i s d i v e r t e d t o r e p r o d u c t i v e s t o r e , S, f o r a p e r i o d of J d a y s , a f t e r w h i c h ZO n a u p l i a r e r e l e a s e d , ZO b e i n g g i v e n by ZO = X.S\/WO \u2022 2.1e where X i s e f f i c i e n c y and WO t h e i n i t i a l n a u p l i a r w e i g h t . The n o t a t i o n h e r e ( T a b l e I) f o l l o w s t h a t o f S t e e l e ( 1 9 7 4 ) , e x c e p t t h a t GW i s u s e d i n 2 . I d t o a l l o w G t o be r e s e r v e d f o r z o o p l a n k t o n b i o m a s s . The r e a d e r i s r e f e r r e d t o S t e e l e f o r a d e t a i l e d d e r i v a t i o n of t h e mo d e l . The s y s t e m 2.1 r e p r e s e n t s a s e t o f f o u r s i m u l t a n e o u s , n o n l i n e a r , d i f f e r e n t i a l e q u a t i o n s i n v o l v i n g t h r e s h o l d s and t h e d i s c o n t i n u o u s r e c r u i t m e n t o f n a u p l i i and t h e r e i s l i t t l e p o i n t i n l o o k i n g f o r n o n - t r i v i a l s o l u t i o n s i n c l o s e d f o r m . A s e r i e s o f s i m p l i f i c a t i o n s and a p p r o x i m a t i o n s i s employed h e r e t o o b t a i n some u n d e r s t a n d i n g of t h e b e h a v i o u r of t h e model o v e r c o r r e s p o n d i n g r e g i o n s i n p a r a m e t e r s p a c e . T h e s e a p p r o x i m a t e r e s u l t s a r e t h e n c h e c k e d and e x t e n d e d by computer s i m u l a t i o n . T h e . b a s i c s t e p i n t h i s q u a l i t a t i v e a n a l y s i s i s t h e 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) i n mixed layer. RO...nutrient concentration (carbon equivalent) below mixed layer. V...mixing rate through thermocline. U...fraction of excreted nutrient recycled. P...phytoplankton carbon concentration i n mixed layer. 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. ..fixes maximum zooplankton ingestion rate. Pi..threshold for zooplankton grazing on phytoplankton carbon. D. ..fixes zooplankton grazing rate above PI. E. ..fixes component of metabolic rate proportional to ingestion. F. ..fixes basal metabolic rate. GW. .maximum of weight and density-dependent mortality rate. Wl..weight threshold for mortality. Zl..number threshold for mortality. H...'half-saturation' constant for mortality. GX..constant mortality rate. ZO. . i n i t i a l number of nauplii i n cohort. WO..initial naupliar weight. W2..adult weight. S...reproductive store. J...period over which reproductive store accumulates. 43 two a s p e c t s of the s e a s o n a l b e h a v i o u r s t u d i e d by S t e e l e and Landry; namely, the t r a n s i e n t response t o h i g h i n i t i a l n u t r i e n t c o n c e n t r a t i o n s (the s p r i n g bloom), and the approach t o a s t a b l e c y c l i c 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 f o l l o w s . The l a t t e r i s more l i k e l y t o be amenable t o q u a l i t a t i v e a n a l y s i s and i s t r e a t e d here f i r s t . The a n a l y s i s proceeds t h r o u g h the r e c o g n i t i o n of t h r e e d i s t i n c t time s c a l e s i n the model under n u t r i e n t l i m i t a t i o n . (For an i n s t r u c t i v e example of the use of m u l t i p l e time s c a l e s 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 s c a l e f o r n u t r i e n t t u r n o v e r can be o b t a i n e d by d i v i d i n g the source term, V.RO, by the 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 n u t r i e n t u p t a k e , B. For S t e e l e ' s v a l u e s of V ( 0 . 0 1 , d a y 1 ) and RO (760 pg C ( e q ) . ! - 1 ) , V.RO e q u a l s 7.6. S t e e l e used a r a t h e r h i g h v a l u e of B (96 p g C ( e q ) . l _ 1 a c c o r d i n g t o L a n d r y ) . 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 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 the oceans, suggest t h a t 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 s h o u l d be s m a l l e r than t h i s , of o r d e r 0.1 )jg a t N . l \" 1 or a p p r o x i m a t e l y 10 jug C ( e q ) . ! - 1 (McCarthy and Goldman,1978). T h i s r e s u l t s i n a time s c a l e f o r n u t r i e n t t u r n o v e r of o r d e r 1 day, much s h o r t e r than t h a t of p h y t o p l a n k t o n (maximum growth r a t e 0.2 d a y 1 , y i e l d i n g a time s c a l e of 5 days) or z o o p l a n k t o n . We proceed t h e r e f o r e by t r e a t i n g R as a f a s t v a r i a b l e ; t h a t i s , by assuming t h a t n u t r i e n t c o n c e n t r a t i o n a d j u s t s r a p i d l y t o changes i n o t h e r s t a t e v a r i a b l e s so t h a t R R(P,Z,W), where R makes the r i g h t - h a n d s i d e of e q u a t i o n 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.W0-.7 . ( (C-U.E) .f (P) - U.F) , 2.2a where f ( P ) s t a n d s f o r (P-P1)\/(D+P), and the term V.R has been n e g l e c t e d s i n c e R i s assumed t o be of o r d e r B or a p p r o x i m a t e l y one-hundredth RO. When combined w i t h e q u a t i o n s 2.1 c,d,e, e q u a t i o n 2.2a forms a system ( 2 . 2 ) , from which n u t r i e n t s have been e l i m i n a t e d . The second and t h i r d time s c a l e s can be d i s t i n g u i s h e d p r o v i d e d the 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 compared w i t h p o t e n t i a l growth r a t e s of P and W. Then Z can be t r e a t e d as a slow v a r i a b l e and the b e h a v i o u r of P and W c o n s i d e r e d w i t h Z f i x e d . The system P = V.RO - V.P - Z.W0-7 .( (C-U.E) . f ( P ) - U.F) 2.3a W = ( (0.7.C-E) .f (P) - F).W\u00b0-7 2.3b has the phase p l a n e p o r t r a i t shown i n F i g 8. The n o n - t r i v i a l e q u i l i b r i u m s o l u t i o n (P,W(Z)) of 2.3 i s s t a b l e p r o v i d e d f ' ( P ) i s 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 r e g a r d l e s s of the v a l u e of P I . Then, a c c o r d i n g t o the s l o w - v a r i a b l e a p p r o x i m a t i o n , as Z ( t ) d e c r e a s e s t h r o u g h m o r t a l i t y , P ( t ) and W(t) s h o u l d t r a c k 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 ( P , W ( Z ( t ) ) ) . A c c o r d i n g t o 2.3, P and Z ( t ) . W 0 , 7 are both c o n s t a n t . I t f o l l o w s t h a t a c o h o r t w i l l r e a c h a d u l t w e i g h t , W2, from an i n i t i a l w e i g h t , WO, when the d e n s i t y has dropped by a f a c t o r (WO\/W2) 0 7. For example, i f the per c a p i t a m o r t a l i t y i s c o n s t a n t (GW = 0, GX f 0 ) , Z ( t ) = ZO.exp(-GX.t) and, a l l o w i n g f o r the i n c u b a t i o n p e r i o d J , the g e n e r a t i o n time i s g i v e n by F i g u r e 8. Phase p l a n e 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. 2.4 So f a r , o n l y the growth of a s i n g l e c o h o r t has been d e a l t w i t h but an approximate t r e a t m e n t of r e p r o d u c t i o n i s a l s o p o s s i b l e . D u r i n g the p e r i o d of J days over which r e p r o d u c t i v e s t o r e s a r e acc u m u l a t e d , W i s f i x e d a t a d u l t weight W2, so t h a t e q u a t i o n 2.3b and the e q u i l i b r i u m v a l u e P a r e not r e l e v a n t . I t i s c o n s i s t e n t w i t h the s l o w - v a r i a b l e a p p r o x i m a t i o n t o assume t h a t P i s a p p r o x i m a t e l y e q u a l t o P ( Z ) , where P makes P = 0 i n 2.3a f o r W = W2. S u b s t i t u t i n g P = P(Z) i n the e q u a t i o n f o r S and i n t e g r a t i n g g i v e s , a f t e r J days: S = SO - Z.W\u00b0 7 . S I where Z.W07 i s e v a l u a t e d a t the b e g i n n i n g of the r e p r o d u c t i v e 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) . ( l . - e x p ( - G X . J ) ) \/ ((C-E.U).GX) . But a c c o r d i n g t o the approximate t r e a t m e n t of growth, Z.W0-7 = Z03.W00-7 where ZO 3 i s the i n i t i a l s i z e of the g t h c o h o r t . U s i n g e q u a t i o n 2.1e, i t f o l l o w s t h a t Z 0 9 + 1 = S0.X\/W0 - Z0 9.S1.X\/W0\u00b0- 3 . 2.5 T h i s c o n s t i t u t e s a d i f f e r e n c e e q u a t i o n f o r ZO 3. I f the c o e f f i c i e n t of ZO 9 i n e q u a t i o n 2.5 i s l e s s than 1 i n magnitude, 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 s o l u t i o n ZO* and, a c c o r d i n g t o 47 t h i s a p p r o x i m a t e t h e o r y , t h e r e i s a c o r r e s p o n d i n g s t a b l e c y c l i c s o l u t i o n 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 s y s t e m 2.2. I f t h e c o e f f i c i e n t i s g r e a t e r t h a n 1 i n m a g n i t u d e , ZO i s u n s t a b l e and a s t a b l e c y c l i c s o l u t i o n t o 2.2 h a v i n g c o n s t a n t a m p l i t u d e c a n n o t be e x p e c t e d . T h i s c o e f f i c i e n t i s p r o p o r t i o n a l t o t h e b a s a l m e t a b o l i c r a t e , F. 2.3 S i m u l a t i o n R e s u l t s . The a p p r o x i m a t e t h e o r y p r e d i c t s a s t a b l e c y c l i c s o l u t i o n t o S t e e l e ' s model under c o n d i t i o n s of n u t r i e n t - l i m i t a t i o n w i t h o u t any need f o r t h r e s h o l d s o r q u a d r a t i c p r e d a t i o n t e r m s . T h i s i s i n k e e p i n g w i t h t h e r e s u l t s of C h a p t e r 1 but somewhat s u r p r i s i n g i n view of S t e e l e ' s s i m u l a t i o n r e s u l t s . The p r e d i c t i o n has been t e s t e d by computer s i m u l a t i o n of b o t h t h e s i m p l i f i e d 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 2.2 and t h e f u l l model 2.1 f o r t h e s i m p l e s t c a s e of no t h r e s h o l d s , c o n s t a n t m o r t a l i t y r a t e and f i x e d m e t a b o l i c r a t e (P1=E=GW=0). The f i r s t s e t of s i m u l a t i o n s were o b t a i n e d by f i x i n g t h e m e t a b o l i c r a t e F and v a r y i n g t h e m o r t a l i t y r a t e GX i n t h e s i m p l i f i e d model 2.2. F o r F = 0.4 and GX r a n g i n g from 0.02 t o 0.06 d a y \" 1 , s t a b l e c y c l i c s o l u t i o n s were a p p r o a c h e d w i t h g e n e r a t i o n t i m e d e c r e a s i n g a s GX i n c r e a s e d , i n q u a l i t a t i v e a greement w i t h t h e s l o w - v a r i a b l e t h e o r y . However, q u a n t i t a t i v e a greement between s i m u l a t e d g e n e r a t i o n t i m e s and thos,e p r e d i c t e d a c c o r d i n g t o 2.4 i s n o t p a r t i c u l a r l y good ( F i g 9 ) . P a r t , o f t h e e x p l a n a t i o n f o r t h e p o o r agreement can be seen i n F i g 10a, where a s i m u l a t e d c y c l e i s p o r t r a y e d . The p h y t o p l a n k t o n d e n s i t y i s f a r from b e i n g c o n s t a n t o v e r t h e p e r i o d o f c o p e p o d g r o w t h , due p a r t l y 48 F i g u r e 9. C o m p a r i s o n o f g e n e r a t i o n t i m e s p r e d i c t e d by e q u a t i o n 2.4 ( s o l i d l i n e ) and t h o s e o b t a i n e d i n n u m e r i c a l s o l u t i o n s o f t h e s y s t e m 2.2 ( d o t s ) . 49 300 200 U O) 100 r15 50 100 150 200 TIME (days) 250 300 350 400 400n 3004 200A U CJ) 3. 100H 150 200 250 TIME (days) 300 350 400 F i g u r e 10. S t a b l e c y c l i c s o l u t i o n s o f the system 2.2 f o r : (a) F=0.4, GX=0.05; (b) F=0.2, GX=0.05. 50 t o the l i m i t a t i o n s of the s l o w - v a r i a b l e a p p r o x i m a t i o n and p a r t l y t o the p e r t u r b a t i o n imposed by the r e l e a s e of n a u p l i i a t the end of each r e p r o d u c t i v e p e r i o d . In s p i t e of t h i s , f u r t h e r q u a l i t a t i v e agreement between the a p proximate t h e o r y and s i m u l a t i o n was found. When F i s d e c r e a s e d t o 0.2, the c y c l e p e r i o d i s not a f f e c t e d , b u t , as shown i n F i g 10b, the s i m u l a t e d c y c l e i n v o l v e s lower P and h i g h e r ZO, as p r e d i c t e d by the s l o w - v a r i a b l e t h e o r y . A l s o , i n c r e a s i n g F t o 0.6 i n the s i m u l a t i o n r e s u l t s i n an approach t o a r a t h e r c u r i o u s c y c l e i n v o l v i n g a l t e r n a t e l y h i g h and low n a u p l i a r r e c r u i t m e n t s ( F i g 1 0 c ) . T h i s b e h a v i o u r c o r r e s p o n d s i n the a p p r o ximate t h e o r y t o a s t a b l e s o l u t i o n of p e r i o d 2 i n the d i f f e r e n c e e q u a t i o n 2.5, ZO* h a v i n g been d e s t a b i l i z e d by i n c r e a s i n g F. The phenomenon of p e r i o d i c and a p e r i o d i c s o l u t i o n s t o d i f f e r e n c e e q u a t i o n s i n s i m p l e e c o l o g i c a l models has a r o u s e d c o n s i d e r a b l e i n t e r e s t (eg May,1975). The v a l i d i t y of u s i n g the f a s t - v a r i a b l e a ssumption t o e l i m i n a t e n u t r i e n t s has been t e s t e d by computer s i m u l a t i o n of S t e e l e ' s f u l l model 2.1. For F = 0.4, GX = 0.05 day\" 1 and low B (10 jug C ( e q ) . ! \" 1 ) , n u m e r i c a l s o l u t i o n s of the f u l l model approach a s t a b l e c y c l i c s o l u t i o n ( F i g 1 1 a ) , which i s almost i d e n t i c a l t o t h a t o b t a i n e d from system 2.2 ( F i g 10a). Throughout the c y c l e , R remains low (of o r d e r B) and the 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 s q u i t e a c c u r a t e . However, i t can be v i o l a t e d i n a number of ways. For example, i f F i s reduced t o 0.2, w i t h o t h e r parameters unchanged, l a r g e , d i v e r g i n g o s c i l l a t i o n s i n R, P and Z appear ( F i g l i b ) . The e x p l a n a t i o n f o r t h i s can be found i n the n u t r i e n t e q u a t i o n 2.1a and the f a c t t h a t , f o r F=0.2, the c y c l i c s o l u t i o n .51 TIME (days) Figure 10c. Stable cyclic solution of the system 2.2 for F=0.6, GX=0.05. O 50 100 150 200 250 300 350 400 TIME (days) Figure 11a. Stable cyclic solution of the system 2.1 for F=0.4, GX=0.05 and B=10. 53 400 r20 100 150 200 TIME (days) KI 300 r15 c r 7.15 0 = ( c7*. s i n h ( a * . f )+ co? ^ cosh(a* . (fr )) \/ ( c o s h ( a * . fT )+ ^Ta\/.sinMa*. \u00a3V)\/a* ) + \/3*.Q- \u00b0-5 .exp(- c2JT) . jy ( fl* e - \u00b0-5 .exp(-p r) ) \/Jv(\/3* e -\u00b0- 5.exp(-c^)). 7.16 where a* = ( u;* 2 + (1-U). 8*). 0 5 , V= \/?*. ( co* 2 + \u00a3*) \u00b0-5 are independent of 8 . The cases of i n t e r e s t are those with P T \u00bb P(0) or a*. . fi . (f -