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Predator-prey functional responses and predation by staghorn sculpins (Leptocottus armatus) on chum salmon… Mace, Pamela M. 1983

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PREDATOR PREY FUNCTIONAL  RESPONSES  -  AND PREDATION  BY STAGHORN  SCULPINS  (LEPTOCOTTUS ARMATUS) ON CHUM SALMON FRY (ONCORHYNCHUS KETA) by PAMELA MARGARET MACE B.Sc.  Hons., U n i v e r s i t y Of C a n t e r b u r y , C h r i s t c h u r c h , Z e a l a n d , 1975 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in THE FACULTY OF GRADUATE STUDIES (Department of Zoology) We  accept t h i s t h e s i s required  as conforming standard  to the  THE UNIVERSITY OF BRITISH COLUMBIA M a r c h 1983 ( c ) Pamela M a r g a r e t  Mace,  1983  New  DE-6  In p r e s e n t i n g  this thesis  r e q u i r e m e n t s f o r an of  British  it  freely available  for  Library  shall  for reference  and  study.  I  f o r extensive copying of  understood that for  h i s or  be  her  publication  s h a l l not  be  ^CnQ  10  The U n i v e r s i t y o f B r i t i s h 1956 Main Mall V a n c o u v e r , Canada V6T 1Y3 Date  (3/81)  K  (\f*W  Columbia  1^93  of  make  further this  thesis  head o f  this  my  It is thesis  a l l o w e d w i t h o u t my  permission.  Department o f  the  representatives.  copying or  f i n a n c i a l gain  g r a n t e d by  the  University  the  s c h o l a r l y p u r p o s e s may by  the  I agree that  permission  department or  f u l f i l m e n t of  advanced degree a t  Columbia,  agree t h a t  in partial  written  i  DEDICATION  To  my  parents,  Jim  and  e n c o u r a g e m e n t and p a t i e n c e  Margaret throughout  Mace, this  for  their  endeavour.  support,  i i  ABSTRACT Mathematical prey  interactions  representing A  model  of  prey  armatus),  the  their  keta)  may  of  the  and  new  r e l a t i o n s h i p are  i s d e v i s e d and  qualitatively  of  staghorn  to which they  response for  t o chum  the  sculpin  their  potential  for  limiting  During  p e r i o d s of  fry  migration,  different  small of  low  be  the  There  salinity, major was  important on  a  wide  Juveniles  variable  progressively as  they  grew.  active  80  mm  fry  were  (Oncorhynchus  ( i ) a s c e r t a i n i n g the and of  (ii)  with  juvenile  fauna  salmon. in  Rosewall  the Creek  composed p r e d o m i n a n t l y Tolerance  sculpin  was  in  these  residence the  to  size,  migration  s c u l p i n s to e s t u a r i e s . of  assessing  populations  in length.  that  of  on  waters  found  to  areas. fry  Sculpins  concentrating  of  was  preyed benthic  t h e amphipod Eogammarus c o n f e r v i c o l u s . throughout  more r e s t r i c t e d The  those  salmon  sculpin  governing  particularly  were  C.)  evidence  diversity  crustaceans,  B.  than  in attracting  from  estuarine habitats  survival  which decreased  little  lead  utilize  f o r a g i n g behaviour  Island,  juveniles less  predator  (Leptocottus  e s t u a r i e s o f B i g Q u a l i c u m R i v e r , Salmon C r e e k and Vancouver  proposed.  shown t o  sculpins  p u r p o s e s of  shaping  (on  equations  methods.  extent  examined  evaluated,  predator-  t h a t d i s t i n g u i s h e s between  be  habits  predatory  are  factors  that  feeding  and  vulnerability  using previous  The  and  reviewed  selection  prey  conclusions  produced  are  selected aspects  p e r f o r m a n c e and to  m o d e l s d e s c r i b i n g t h e components of  t h e day,  t o p e r i o d s of  smallest that captured  but  low  f e e d i n g became  light  intensity  f r y were 40-45 mm  in  length. When chum f r y were o f f e r e d t o enclosures, type 3  the response of those  intensity  Predation  from  dawn  l e v e l s during  4-5  times  the rates  to other  usual  to  dusk,  by s c u l p i n s  foraging fry  over  evade  must  of  The reduces  of f r yrequires  run the r i s k combined  alternative thought  to  naive  t o salmon  of being  to  of  the  attack  responsible  In B i g Qualicum R i v e r ,  40,500  were  captured  by  to  that the salmonid  the  need  to  The s c h o o l i n g experienced, p r o c e s s and i n  themselves. schooling  response,  f r y , and  a  hunger  an  estimated  sculpins  in  which  profusion levels,  f o r low f r y c o n s u m p t i o n  situations. chum  and  birds).  prey, which d e c r e a s e s average be  up  a b a l a n c e between t h e  to the attack  eaten  from  Capture  between  s c u l p i n s , even when  attention  intake.  theory.  choice  represents  were  abundance  I t i s suggested  (particularly  effects  incentive  a  with'  f r y , which  f r y increased  of s t a r v a t i o n  that  light  divergence  foraging  trials.  risk  own p r e d a t o r s  the  for  markedly as t h e i r  optimal  invertebrates  devote c o n s i d e r a b l e  so d o i n g ,  to  invertebrates  indicating a  of s c u l p i n s given  of minimizing  their  behaviour  increased,  3-5 two h o u r  benthic  requirement  s c u l p i n s was t y p e  an o v e r a l l p r e f e r e n c e  initially  strategy  and  When b e n t h i c  for fry declined  prey  was  and p o s i t i v e l y c o r r e l a t e d  the night.  predictions  three-fold  o f 80-99 mm  field  in length  more p r o f i t a b l e i n t e r m s o f n e t e n e r g y  However, p r e f e r e n c e relative  80 mm  in  on f r y was i n v e r s e l y r e l a t e d  added, s c u l p i n s e x h i b i t e d were  sculpins  l e s s than  2 ( H o l l i n g 1965) whereas t h a t  (sigmoid).  light  starved  were  i n natural  240,000  1979  of  and  and  1980,  iv  respectively.  This  represents  only  0.51%  and  0.06%  of  t o be  less  than  realized.  Estimated in  1979  one-tenth  Predation  considerably  fry populations,  of  the  rates  greater,  was  potential  on  despite  consumption  and  the  corresponding  coho a  and  fry  fry  estuaries  been  population  (42.97%) and  144,000  were size.  (9.09%)  1980.  populations of  suitable  calculated  (0. k i s u t c h )  S y s t e m s where s c u l p i n s c o u l d consume h i g h e r chum  was  of  t h a t c o u l d have  smaller  817,700  percentages  were  intermediate  benthic  Recommendations  identified  to high  salinity  invertebrates  for reducing  as  and  proportions  small, shallow, with  small  sculpin predation  of warm  relatively numbers  of  few fry.  i n such cases  are  proposed. Birds,  particularly  were f o u n d  t o be  juvenile  salmon  sculpins, appearance  they of  hatchery-reared  even  Bonaparte's g u l l s  more  in  Big  avid  (Larus P h i l a d e l p h i a ) ,  predators  Qualicum  chinook  coho were removed by  estuary.  salmon  (O.  b i r d s i n the  sculpins  on  In  contrast  to  River.  e x h i b i t e d pronounced numerical f r y i n the  than  An  responses  estimated  t s h a w y t s c h a ) and years  1979-81.  to  10-25% of 2-4%  of  the the the  V  TABLE OF  CONTENTS  ABSTRACT  i i  L I S T OF TABLES  x i i  L I S T OF FIGURES  xv  ACKNOWLEDGEMENTS  xxii  PREFACE  xx i v  PART I : THEORETICAL CONCERNS  1 Chapter  1:  GENERAL  INTRODUCTION  TO  PREDATOR-PREY  AND  PARASITOID-HOST RESPONSES  2  1.1 I n t r o d u c t i o n  2  1.2 The Components o f P r e d a t i o n  3  1.3 F u n c t i o n a l  Response  t o Prey Density  9  1.4 F u n c t i o n a l  Response  to Predator  Density  13  1.5 S y n t h e s i s  o f R e s p o n s e s t o P r e y and P r e d a t o r  1.6 E x t e n s i o n  t o M u l t i s p e c i e s Models  Chapter  2:  EVALUATION  AND  Density  16 24  DEVELOPMENT  OF  MODELS  INCORPORATING VARIABLE RATES OF E F F E C T I V E SEARCH  28  2.1  28  Introduction  2.2 C o m p a r i s o n o f M o d e l s D e s c r i b i n g 2.3  Simple Predator-Prey  Formulations  for  Type  3 Responses  Non-Random  and P a r a s i t o i d - H o s t  Systems  Search  ... 34 in 49  vi  B a s i c method  of model d e r i v a t i o n  50  P a r a s i t o i d models  54  P r e d a t o r models  60  B e h a v i o u r and e v a l u a t i o n 2.4  Suggestions  3.1  65  f o r More G e n e r a l M o d e l s  2.5 D i s t i n g u i s h i n g Chapter  o f t h e models  79  R e s p o n s e T y p e s i n Two-prey  3: PREFERENCE, SWITCHING AND  Systems  TYPE 3 RESPONSES  Introduction  94  3.2 M u r d o c h ' s Model o f P r e d a t o r New  3.4  Cases I n v o l v i n g  3.6  Switching  94  P r e f e r e n c e and S w i t c h i n g H y p o t h e s e s  3.5 D e s i g n  91 91  Common symbols u s e d  3.3  .. 84  104  Exploitation  of Switching  S w i t c h i n g and Type  112  Experiments  113  3 R e s p o n s e s : An Example  117  Switching  118  Effects  127  o f hunger  Type 3 r e s p o n s e s  131  3 .7 D i s c u s s i o n  PART I I : F I E L D AND  137  EXPERIMENTAL RESULTS  141 Chapter  4: GENERAL INTRODUCTION: F I S H PREDATION STUDIES  4.1  A S h o r t Review o f F i s h F e e d i n g  4.2  P r e d a t i o n a s an A g e n t o f M o r t a l i t y  4.3  I n t e r a c t i o n s Between  Cottids  Studies  ...142 142  i n S a l m o n i d s ....149  and J u v e n i l e  S a l m o n i d s 151  vii  4.4 The P r e s e n t  Study  .  155  SECTION A: EXPERIMENTAL RESULTS  1 58 Chapter  5: IDENTIFICATION  BEHAVIOUR 5.1  AND  OF  FACTORS  AFFECTING  FORAGING  PRELIMINARY EXPERIMENTS  159  Introduction  159  5.2 D e s c r i p t i o n o f S y s t e m 5.3 E x p e r i m e n t a l 5.4 B e h a v i o u r 5.5 D i e l  (ii)  Enclosures  Animals  ...162  i n Enclosures  164  of Feeding  169  samples  Laboratory  5.6 R e d u c t i o n  and E x p e r i m e n t a l  o f F r y and S c u l p i n s  Pattern  (i) F i e l d  161  169  experiments  of F a c t o r s  5.7 P r e l i m i n a r y  170  to Consider  Explicitly  178  Experiments  (i) Enclosure  179  characteristics  179  ( i i ) Time o f day  180  (iii)  182  Water d e p t h  (iv) Digestion  rates  (v) I n t e r f e r e n c e  182  and e x p l o i t a t i o n  186  ( v i ) Maximum f r y d e n s i t i e s  189  5.8 Summary Chapter  6:  EXPERIMENTAL  191 ANALYSIS  OF  FACTORS  AFFECTING  PREDATION  193  6.1  193  Introduction  vi i i  6.2  General  Methods  Collection  and s t o r a g e p r o c e d u r e s .  Experimental Analysis (i) (ii) 6.3  6.4  6.5  197 197  procedure  198  of d a t a  201  S i n g l e - p r e y experiments  201  Two-prey e x p e r i m e n t s  The.Baseline Functional  204 Response:  Effects  of  Fry  Density  209  M e t h o d s and r e s u l t s  210  Discussion  218  Effects  of S c u l p i n  Size  .  ....220  l  M e t h o d s and r e s u l t s  220  Discussion  227  Disaggregation  of  Fry  Density  and  Sculpin  Size  Effects  6.6  6.7  6.8  6.9  ..233  M e t h o d s and r e s u l t s  233  Discussion  240  Effects  of A l t e r n a t i v e  Prey  242  Methods and r e s u l t s  242  Discussion  255  Effects  of S c u l p i n  Size  on S e l e c t i v i t y  261  Methods and r e s u l t s  261  Discussion  271  Effects  of Predator  Experience  on P r e y  Selectivity  .275  M e t h o d s and r e s u l t s  275  Discussion  288  Effects  of L i g h t  Intensity  293  ix  Methods and  results  294  Discussion 6.10  Synthesis  297 and  General Discussion  The  predator  The  fry schooling  The  foraging  SECTION B:  satiation  hypothesis  301  hypothesis  strategy  of  300  307  sculpins  314  F I E L D SAMPLING RESULTS  323 Chapter  7:  DESCRIPTION OF  7.1  Introduction  7.2  Big Qualicum River  SAMPLING AREAS '  324 324 327  ("i ) G e n e r a l d e s c r i p t i o n  327  (ii)  328  Hatchery data  (iii)  Description  ( i v ) Sampling (v) S a l i n i t y  of  estuary  336  stations  339  profiles  339  ( v i ) F r y movements t h r o u g h e s t u a r y  342  (vii)  345  Invertebrate  (viii)  fauna  I n c i d e n t a l catches  in traps  349  7.3  Salmon C r e e k  7.4  R o s e w a l l Creek  356  7.5  Comparison  358  Chapter 8.1  8:  ...352  of A r e a s  UTILIZATION OF  Introduction  ESTUARIES BY ,  STAGHORN SCULPINS  ..359 359  X  8.2  Methods (i)  360  Sampling  (ii)  methods and  areas  sampled  D e t e r m i n a t i o n of s e a s o n a l o c c u r r e n c e and  distribution (iii)  Marking  360 spatial 363  procedures  365  ( i v ) D e t e r m i n a t i o n o f e x t e n t o f movement  366  (v) D e n s i t y c a l c u l a t i o n s  368  8.3  Results (i)  369  Seasonal  (ii)  Spatial  (iii)  369  distributions  377  Movement w i t h i n  (iv) Sculpin 8.4  occurrence  the e s t u a r y  388  abundances  391  Discussion  Chapter  9:  405  FEEDING HABITS OF  STAGHORN SCULPINS  412  9.1  Introduction  412  9.2  Methods  415  (i)  Sampling  (ii) (iii) 9.3  methods and  P r o c e s s i n g o f gut Data  areas  sampled  contents  analysis  422 425  ( i ) Feeding h a b i t s i n Big Qualicum  (iii)  Consumption Consumption  (iv) Feeding 9.4  416  Results  (ii)  Discussion  415  River  of b e n t h i c i n v e r t e b r a t e s of  salmon  fry  habits in other areas  425 433 436 449 456  xi  SECTION C: SYNTHESIS  466 Chapter  10: IMPACT OF STAGHORN SCULPINS ON SALMON FRY  467  10.1  Introduction  467  10.2  Rate of G a s t r i c E v a c u a t i o n  468  10.3 A v e r a g e D a i l y C o n s u m p t i o n  472  10.4  Estimated  Impact  475  10.5  P o t e n t i a l Impact  480  10.6 D i s c u s s i o n Chapter  481  11: SYNTHESIS AND CONCLUDING  Scope o f t h e s c u l p i n - s a l m o n Systems  487  fry interaction  488  susceptible to sculpin predation  Methods o f r e d u c i n g Current  REMARKS  and  492  sculpin predation  future  interactions  research  on  495 predator-prey  :  497  LITERATURE CITED  APPENDIX:  BIRD  504  PREDATION  ON JUVENILE SALMONIDS  IN THE  BIG QUALICUM ESTUARY, VANCOUVER ISLAND  527  xi i  L I S T OF TABLES  Table  1.1 C o m p a r i s o n  disc Table  2.1  Relative  t o type  3.1  Table  Table  several  equations  3 curves  36  3 curves  from  of d i f f e r e n t  example  c a n be i n f e r r e d  several  equations  shapes  showing  42  how  predator  erroneously  103  f o r data  from  (1979) f o r s a t i a t e d  3.4  Hypothetical  motivated  shift  f o r data  from  3.5  123  showing  mode  can  how  lead  a to  predator Table and Table by Table  5.1  shift  hungerapparent .....129  Hypothetical  motivated  Akre and  predators  example  i n search  122  counter-switching Table  Akre and  (1979) f o r s t a r v e d p r e d a t o r s  3.3 C a l c u l a t i o n s o f p r e f e r e n c e  Johnson  from 21  of  3.2 C a l c u l a t i o n s o f p r e f e r e n c e  Johnson  estimated  equations  flexibility  Hypothetical  switching Table  type  values  2.2 P a r a m e t e r v a l u e s e s t i m a t e d  fitted Table  parameter  a n d random p r e d a t o r  describing Table  of  example  i n search  showing  mode  can  how  lead  a to  hungerapparent  switching Effect  130  of time  o f day on c o n s u m p t i o n  o f chum f r y  amphipods by s c u l p i n s 5.2 E f f e c t  of water depth  181 on c o n s u m p t i o n  o f chum f r y  sculpins 5.3  183  Effect  consumption  with  of  sculpin  d e n s i t y on a v e r a g e  10 f r y a v a i l a b l e  chum f r y 187  xi i i  Table  5.4 E f f e c t  consumption Table  5.5  fry Table  6.1  average  chum  fry  20 f r y a v a i l a b l e  188  190  Parameter v a l u e s e s t i m a t e d  the  baseline  functional  f e e d i n g on chum  subsets  of the b a s e l i n e f u n c t i o n a l  response  and 80-99 mm  related  of  from  several  equation 217 equations  s c u l p i n s f e e d i n g on chum  ANOVA  t o the behaviour  61-75 mm 213  the Real  Summary  equations  of  from  47-60 mm 6.4  several  fry  6.3 P a r a m e t e r v a l u e s e s t i m a t e d  for  from  response  6.2 P a r a m e t e r v a l u e s e s t i m a t e d  for  Table  on  Numbers o f chum f r y consumed by s c u l p i n s a t h i g h  sculpins  Table  with  density  densities  for  Table  of s c u l p i n  results  for  six  f r y .226  variables  of s c u l p i n s i n the presence  of  fry  238  Table  6.5 E f f e c t  Table  6.6 P a r a m e t e r v a l u e s e s t i m a t e d  the  response  fry Table  of s c u l p i n  of naive  size  on h a n d l i n g t i m e  61-75 mm  from  p e r f r y .239  two e q u a t i o n s f o r  s c u l p i n s f e e d i n g on chum  and amphipods 6.7 T e s t  search  249  of the hypothesis  f o r amphipods was  that  independent  rate  of  effective  o f amphipod d e n s i t y 250  Table  6.8  from Table  alternative  6.9 L i k e l i h o o d  observed prey Table  ANOVA c o m p a r i n g prey  t h e f i t o f two e q u a t i o n s  experiments  ratio  proportions  t o data  tests  .....251  comparing  predicted  of f r y i n the d i e t  and  in alternative  experiments  6.10 P a r a m e t e r v a l u e s  258 estimated  from  the m u l t i s p e c i e s  xiv  disc on Table  equation  c o h o f r y and 6.11  disc  mm  and  85-99 mm  equation  8.1  sculpins feeding  amphipods  268  Parameter v a l u e s e s t i m a t e d  experience Table  f o r 61-75  for  sculpins  from  of  differing  f e e d i n g on chum f r y and  Effects  of b a i t  on  the m u l t i s p e c i e s levels  of  amphipods  capture  r a t e s of  287 s c u l p i n s by  minnow t r a p s Table  8.2  362  Sampling  recapture  periods  experiments  and  sample  sizes  i n Salmon C r e e k and  for Big  River Table  Qualicum . 367  8.3  Seber-Jolly  survival estuary Table  mark-  and  estimates  recruitment  of  of  population  s c u l p i n s i n the B i g  s i z e , °Qualicum  i n 1980  9.1  Food  397  organisms  found  in  stomachs of  staghorn  sculpins Table  9.2  benthic Table by Table  10.1  417  Estimates  of p r e f e r e n c e by  sculpins  various  invertebrates Estimated  and  437 potential  consumption  s c u l p i n s i n Big Qualicum R i v e r , 10.2  for  Estimated  consumption  Big Qualicum R i v e r ,  1979  and  1979  and  o f coho f r y by 1980  of chum f r y 1980  478  sculpins in 479  XV  L I S T OF  Figure  1.1  The  four  t y p e s of  FIGURES  functional  response  to  prey  density Figure  5  1.2  Functional  e q u a t i o n and Figure  2.1  Eight  effective Figure  2.2  random p r e d a t o r pathways  search Type  2.3  of Figure and  2.4  2.6  and  the  i n Eq. Figure  2.7  and  the  Eq.  2.23  the Figure  2.8  by  to  22 variable  rates  of 31  are  from s e v e r a l  equations  fixed  38 from  the  several  several  asymptote  equations  equations  is fixed  to type  3  ....  44 between r a t e  i n Eq.  Relationship parameter  40  curves  shapes  Relationship  k for  of  effective  search  2.25a between  62 number of  hosts  non-discriminating  attacked  parasitoids  2.22  66  Relationship parameter  k for  between  number of  discriminating  hosts  attacked  parasitoids  in 68  Relationship  parameter 2.9  disc  predators  approach to  fits  prey density  Figure  Figure  Best  different 2.5  leading  points  r a t e of  from t h e  equation  Type 3 c u r v e s g e n e r a t e d  when t h e Figure  by  generated  3 curves generated  when i n f l e c t i o n Figure  responses  k for predators  Functional  parasitoids  between  number of i n Eq.  response curves  prey eaten  and  2.28 from  70 Eq.  2.22  for 73  xvi  Figure  2.10  Functional  response  curves  f r o m E q . 2.28 f o r  predators Figure  75  2.11 F u n c t i o n a l  response curves  generated  from  Eq.  2.35 Figure the Figure the  82 2.12  Type  3 responses t h a t can look  p r e s e n c e o f an a l t e r n a t i v e  Figure  3.1 D e m o n s t r a t i o n  Figure  3.2 L i n e s  Figure  3.3  Johnson Figure  87  3.5  predator  of  like  type  4  in 89  from E q . 3.1  preference switching  96  f r o m E q . 3.2  i n data  98  f r o m A k r e and  (1979) f o r s t a r v e d p r e d a t o r s  3.4 E v a l u a t i o n  Johnson Figure  Evaluation  2 in  prey  of s w i t c h i n g  of constant  type  prey  2.13 Type 3 r e s p o n s e s t h a t c a n l o o k p r e s e n c e o f an a l t e r n a t i v e  like  ( 1979)  of switching  for satiated  Functional exhibits  i n data  from  Akre  and  predators  response  constant  119  curves  preference  124 generated  when a  f o r one p r e y  over  another  134  Figure  5.1 Numbers o f chum f r y c a p t u r e d  Figure  5.2  of Figure  5.3 D i u r n a l v a r i a t i o n  5.4  6.1  sculpins Figure  i n volume o f s t o m a c h  6.2  .167  contents .171  i n number o f m y s i d s  eaten  by  i n l a b o r a t o r y experiments Diurnal  most v o r a c i o u s Figure  sculpins  s c u l p i n s from B i g Q u a l i c u m R i v e r  sculpins Figure  Diurnal variation  by s m a l l  variation  174  i n number o f m y s i d s e a t e n  by  sculpin  Baseline  functional  176 response of  naive  61-75 mm  t o chum f r y Comparison  of d i s t r i b u t i o n  211 o f chum  f r y captures  xvi i  in Figure  baseline 6.3  response with Poisson  model  E f f e c t s o f s c u l p i n s i z e on  214  baseline  functional  response Figure  222  6.4  Percent  different Figure by Figure by Figure of Figure  6.5  6.6  fry Figure  sculpins with Poisson  sculpins with Poisson  of  captures  model  228  6.8  Effects  6.9  model  230  p u r s u i t s a n d s t r i k e s by  s c u l p i n s at three  three  sizes  d e n s i t i e s o f chum f r y  of  alternative  prey  235  (amphipods)  on  f u n c t i o n a l response Relationship  244  between r e l a t i v e  numbers  and amphipods a v a i l a b l e and numbers  of  captured  chum by 61-  sculpins  6.10  246  Preference  by n a i v e  61-75 mm  sculpins  for  chum  o v e r amphipods 6.11  Test  253  for switching  by n a i v e  61-75 mm  sculpins  on chum f r y and amphipods  6.12  sizes, Figure  sculpins  224  6.7 Movements,  feeding Figure  by  C o m p a r i s o n of d i s t r i b u t i o n of chum f r y c a p t u r e s  80-99 mm  75 mm Figure  predation  sizes  47-60 mm  fry  fry  C o m p a r i s o n o f d i s t r i b u t i o n o f chum f r y  baseline Figure  chum  256  Number o f coho f r y c a p t u r e d  by s c u l p i n s  w i t h and w i t h o u t a l t e r n a t i v e p r e y  6.13  Relationship  f r y and amphipods  between  available  relative  and  of  (amphipods) numbers  numbers  two ...263  of c o h o  captured  by  s c u l p i n s o f two s i z e s Figure  6.14  fry  over  Preference amphipods  266 by  two s i z e s o f s c u l p i n s  f o r coho 269  xvi i i  Figure  6.15 T e s t  feeding Figure  6.17  Effects  6.19  with Figure  of  of  two  sizes  in enclosures  experience  61....278  on chum f r y  experience  280  on  chum  fry  i n t h e p r e s e n c e o f many amphipods o f amphipods  levels  Tests levels  eaten  282  by 61-75 mm  sculpins  of experience by 61-75 mm  of experience  various  by i n i t i a l l y - n a i v e  o f few amphipods  predator  6.20 P r e f e r e n c e  6.21  272  trials  predator  i n the presence  Number  various  levels Figure  successive  6.18 E f f e c t s o f  consumption Figure  sculpins  o f chum f r y e a t e n  s c u l p i n s over  consumption Figure  by  on coho f r y and amphipods  6.16 Number  75 mm Figure  f o r switching  285 sculpins  with  various  f o r chum f r y o v e r amphipods  for  switching  of experience  by 61-75 mm  feeding  on  289  s c u l p i n s of  chum  f r y and  amphipods Figure  6.22  291 Effects  of  light  coho f r y by 60-75 mm Figure  6.23 E f f e c t s o f l i g h t  Figure  6.25  being Figure fry Figure  295  intensity  effects  consumption  of 298  o f hunger on p r e f e r e n c e  by  f o r f r y and amphipods Effects  of  within various  and s u c c e s s  density  distances  6.26 H y p o t h e s i z e d  304 on t h e p r o b a b i l i t y  of  of f r y  of each other  r e l a t i o n s h i p between c l o s e n e s s  310 of  r a t e s of s c u l p i n s  6.27 T o t a l e n e r g y c o n s u m p t i o n  levels  on  sculpins  6.24 H y p o t h e s i z e d  sculpins  on c o n s u m p t i o n o f  sculpins  chum f r y by 60-75 mm Figure  intensity  experience  feeding  312 by s c u l p i n s o f v a r i o u s on  chum  f r y , w i t h and  xix  without  amphipods p r e s e n t  Figure  7.1 L o c a t i o n  Figure  7.2 D i s c h a r g e  and Figure  7.3  rates  study  areas  325  i n B i g Qualicum  River  in  1979 329  Numbers  of  chum  fry  descending  B i g Qualicum  i n 1979 and 1980  7.4 Numbers o f  River Figure  showing  1980  River Figure  map  316  332  coho  fry  descending  Big  Qualicum  i n 1979 and 1980  7.5  Location  334  of  sampling  stations  in  the Big  Qualicum estuary Figure  7.6  Tidal  stations Figure  7.7  Figure at Figure  the  in  of  Incidental  Eogammarus  catches  confervicolus  in  minnow  traps  350 set at  s t a t i o n s i n the B i g Qualicum estuary  B i g Qualicum estuary less  8.1b S c u l p i n s g r e a t e r Length-frequency  Salmon C r e e k  of  sculpins  353 in  1979, 1980, 1981  Figure  than  120 mm  than  sampling area,  8.2a - S c u l p i n s  less  than  Figure  8.2b S c u l p i n s  greater  370  in length  90 mm  distributions  Figure  Figure  347  s t a t i o n s i n the B i g Qualicum estuary  8.1a S c u l p i n s  the  340  8.1 L e n g t h - f r e q u e n c y d i s t r i b u t i o n s  8.2  sampling  i n v e r t e b r a t e s a t sampling  of  Figure  Figure  at  i n the B i g Qualicum estuary  sampling 7.9  salinities  benthic  7.8 S i z e d i s t r i b u t i o n s  sampling Figure  variation  i n the B i g Qualicum estuary Numbers  stations  337  371  in length  372  of s c u l p i n s i n  1978-79 120 mm  than  in length  90 mm  8.3 L e n g t h - f r e q u e n c y d i s t r i b u t i o n s  374 375  i n length of  sculpins  376 at  XX  Big Figure Big Figure Big Figure Big Figure Big Figure  Qualicum 8.4  sampling  Length-frequency  Qualicum 8.5  sampling  Qualicum 8.6  sampling  8.7  sampling  Qualicum 8.8  8.9  sampling  Proportion  in  the  8.10  Figure  in Figure  types Figure  Abundance  in length Percent  found  9.2  of  sculpins  i n the B i g Qualicum distributions reaches  estimates  of  sculpins  that  estuary  of  the  moved ..389  sculpins  Big  Qualicum 392  for  estimates  sculpins  sculpins  and  for  frequency found  and  for  ....399  sculpins  1980  401  sculpins  exceeding  e s t u a r y i n 1978  volume  ....395  exceeding  e s t u a r y i n 1979  size-class  estimates  exceeding  e s t u a r y i n 1980  for  e s t u a r y i n 1979  remains  at  heights  f r e q u e n c y and  fish  sculpins  386  of  i n t h e stomachs o f B i g Q u a l i c u m  of  at  TP  i n t h e Salmon C r e e k  Percent  categories Qualicum  tidal  of s c u l p i n s  384  recaptured  upper  at  LR  Abundance e s t i m a t e s by  8.13  9.1  station  sculpins  382  i n the B i g Qualicum  the B i g Qualicum  of  SP  i n the B i g Qualicum  Abundance  in length  8.12  60 mm Figure  in length  8.11  45 mm Figure  Abundance  380  distributions  stations  estuary at d i f f e r e n t  45 mm  station  of  of s c u l p i n s at  distributions  Length-frequency  captured  Figure  station  Length-frequency  378  FP  distributions  Length-frequency  Qualicum  and MR  distributions  station  Length-frequency  between s a m p l i n g Figure  s t a t i o n s UR  volume  various sculpins of  ....403 prey ...426  various  i n t h e stomachs o f  Big 431  xxi  Figure in  9.3  Numbers  the  s t o m a c h s of  estimated Figure  of  9.4  Big  descending  the  9.5  9.6  compared  to 434  compared  i n the to  s t o m a c h s of  estimated  Big  numbers  river  438 i n the  s t o m a c h s of  Big  sculpins  442  Comparison  sculpins  found  abundances  Numbers of c o h o f r y f o u n d  Qualicum  invertebrates  sculpins,  Numbers o f chum f r y f o u n d sculpins  Figure  benthic  Qualicum  invertebrate  Qualicum  Figure  various  f r o m two  of  numbers o f chum  different  habitats  fry i n the  consumed Big  by  Qualicum  estuary Figure  9.7  444 Comparison  sculpins  f r o m two  of  numbers of  different  c o h o f r y consumed  habitats  i n the  Big  by  Qualicum  estuary Figure  9.8  types Figure  Figure  Percent  found  9.9  types  447  starved  i n the  Percent  found  10.1  f r e q u e n c y and  R a t e s of after  s t o m a c h s of frequency  i n the  an  volume  gastric initial  Salmon C r e e k  and  stomachs of  of  volume o f  various sculpins various  Rosewall Creeek  evacuation meal o r  by  prey  prey  sculpins  sculpins  e l s e given  ...450  either  continuous  access to a d d i t i o n a l food Figure  10.2  feeding  Average i n cages  daily  454  470 fry  consumption  by  sculpins 473  xxi i  ACKNOWLEDGEMENTS Many study.  people  I thank my  greatly  exposure  Dr.  C.  J.  sections  to  Walters  when  Dr.  benefitted  A. I  am  taught  and  me  of  the  as  throughout.  He  Holling  me  Dr.  Center  who  R.  Y e s a k i who t o Ted  Grant  s u p p o r t i v e of t h i s  guidance  G.  of  designed and on  Bill  built  giving  my  Northcote  and  text. Grant  hatchery and  other  Humphries Fergus  who  O'Hara  programming;  figures. the  who  s e v e r a l p i e c e s of  of the o r i g i n a l  t o the B i g Qualicum and  also  Webb o f t h e B i o l o g y  computer  in  of  supplied fry  and  I  Harvey,  Bob  and  project  the t h e s i s  Dick  J o h n s o n and  for overseeing  project  theoretical  members  reviewed  I  pursuits.  absence.  T.  that  interpretation  Creek study a r e a ;  advice  Perry  who  of the  other  Parsons,  drew a number o f t h e  e s t u a r y , and  land.  of  the  expertise  Dave Z i t t e n  Holling,  academic  general  leave of  typed a l a r g e p o r t i o n  a l s o due  their  on  the Rosewall  this  began s u p e r v i s i n g t h i s  salmon b i o l o g y and  gave h e l p f u l  Sanson who  Qualicum  the  workshop who  equipment;  is  for  as  of  ensured  s u g g e s t i o n s on  critically  experiments;  about  Zoology  i d e a s and  other  well  advice  Parsons  S.  o t h e r p e r s o n n e l a t the B i g Qualicum  field  Itsuo  went  the  much a b o u t  materials ° for informed  results,  grateful  Ladouceur  helpful  c o m m i t t e e , D r s . T.  Larkin.  of  C.  provided valuable c r i t i s m  from  supervisory  initial  range  i n t h e t h e s i s and  encouragement  P.  broad  t o the development  s u p e r v i s o r , Dr.  many of my  a  experimental  1981  contributed  original  influenced  had  of  have  manuscript;  Data Bing and  My a p p r e c i a t i o n  work  in  the  Big  I n d i a n Band f o r b e i n g  permission  to  work  on  XXI 1 1  I  thank  assisting stomach the  Jeneen  with  Weekes  f i e l d w o r k i n 1979  laboratory, identifying Jung  Griffiths  for  also  k e y e d out  some of  the  otolith  burn  fish  the a q u a t i c  technique.  Morrison  constructively  indebted  students need.  They  Crawford,  include  Roberta  i s Dr. left  people,  P. for  A.  numerous  Rachelle  seemed  an  summer o f  1980.  for  continuous  programs f o r m i n g have been  work,  those  the c e n t r a l  performed.  the  and  Peter  thesis.  fellow  graduate  help  i n times  Eric  when  thesis  thoroughly,  editorial  field  second during  The Dr.  time  f o c u s of  and  to He  this is  the  spring  thesis  and  his make also  project  person  experiments this  first  from  d e d i c a t i o n , good humour and field  Joan  Walters  comments. of  Terry  Woodsworth.  supervisor  somehow f o u n d  of  Reader,  mention.  He  to  the  special  The  i n the  Without her  practical  when c o m p l e t i o n  goal.  me  Richards  P e a c o c k , Dave  and  encouragement a t times  Laura  C h r i s Wood and  1982.  suggestions  assisted  demonstrated  f r i e n d s and  deserve  offered  who  identified  B o u f f a r d , Bruce Crawford,  the e n t i r e  unattainable  Carveth  some s e c t i o n s o f  and  became my  in  Bob  Campana  distances.  above a l l ,  to read  Hamilton,  Ron  s a m p l e s and  Olenick, Adrienne  sabbatical  helful  coding data.  coding  Webb s u p p l i e d a computer p r o g r a m  support  L a r k i n who  busy s c h e d u l e  and  insects.  S h a r p , M i k e S t a l e y , Mary T a i t t , Two  fish  data;  criticised  moral  analyse  r e m a i n s and  t o a number of o t h e r  for their  to  algae,  samples i n  Steve  Tim  nearest-neighbour  helping  marine  for processing f i e l d  counting  analyse  am  fish  samples.  identifying  and  c o n t e n t s ; Debbie M i l l e r  Granger  I  for  Maria and  ability sampling  would  never  xxiv  PREFACE . This aspects  thesis  of p r e d a t o r - p r e y  concerns,  predator  i n f o r m a t i o n on  relationships,  experimental  interaction,  prey  presents  analysis  determination  stomach  of  contents,  could  be  between  s e c t i o n s , so t h a t t h e decide  Part  I,  and  Here,  reader  used  models of  to  Concerns  describe  the p r e d a t o r  restricted  to  the  specific  situations  p a r a s i t o i d s and  predators  and  (parasitism) are describing another The  density, rate,  and  In c h a p t e r  identified  in  terms covers the  and  of  and  their  the  the e f f e c t s  as  topics  of  linkage  in  specific  investigates  develops  as  new  It covers  rather  well  presented  response  of prey  mathematical  and  examined  the e x i s t i n g  utility,  generalizes results  where  than  a  being  i n subsequent  those  between between  1, t h e components of p r e d a t i o n  functional  functional  the  consider relationships  hosts  e a c h component a r e  review  density,  prey.  chapters  their  1-3),  response.  interactions  insect  only  interactions,  possible  two  systems  degree  (chapters  the  first  of  interested  of  The  of  t h e p r o p o r t i o n of  summarize  wide r a n g e  sections.  the  analysis  systems, e v a l u a t e s  the  some components o f  of  tackle.  t h e p r o p e r t i e s of p r e d a t o r - p r e y models  nature  of  the  different  theoretical  numbers,  I  describe  which p a r t s to  Theoretical  the  identification  intense.  i n each chapter  can  of  computations  covered  areas  including  predator  removed by p r e d a t o r s , and  predation  a number of  mathematical  and  compared  simplicity response to  to  predator  d e p l e t i o n on  t o systems  and  to  one  generality. prey  (host)  (parasitoid)  the o v e r a l l  involving  models  attack  more t h a n  one  XXV  prey  type.  The  1.4)  i s not  discussed  Chapter  functional  of  the  scope  chapter  of  in  to  3, m o d e l s of  these  experiments  are  commonly-used model also  evaluation involved in  Part concerned system:  of  and  most  it is  indicative  three  later  are  (sigmoid)  omitted  of  of  components of  2.2,  used 3  types  from  models compared  responses.  consideration  and  selection  prey  used  to  preference,  that  distinguish  vulnerability  are devised.  for predator  switching in  to test An  predator  example  by  to  the  literature  switching using a previous, the  demonstrate and  from  new  how  examine  between t y p e s o f  method.  This  h u n g e r can  affect  the  difficulties  functional  responses  circumstances. II, Field with  that  of  on chum salmon to  are  i s re-analyzed  in distinguishing  certain  1 and type  reported in Part I I .  is  new  although  section  prey  used  which c l a i m s to demonstrate  example  thesis,  In  describe  interactions  models  for  given to p a r t i c u l a r  chapter  between p r e d a t o r p e r f o r m a n c e of  (section  2.  In c h a p t e r  Several  suggestions  the  sections.  ability  Parasitoid-host  rest  importance  presented  their  after  the  i n other  originally  with  parasitoid-host  of  relative  predation  for  and  to predator d e n s i t y  further.  2 i s concerned  predator-prey beyond the  response  examine  and  the  Experimental  investigation  staghorn  sculpins  R e s u l t s ( c h a p t e r s 4-11), i s of a s p e c i f i c  ( L e p t o c o t t u s armatus)  f r y (Oncorhynchus k e t a ) . the  feeding  L. a r m a t u s t o d e t e r m i n e  habits  t h e most  predator-prey  and  The  major o b j e c t i v e  foraging  important  feeding  behaviour  components  was of  affecting  xxvi  their fry  interaction populations  locations,  and  lost the  could potentially Chapter  with  to  the size  4 provides  that  form  outlines  the areas  a  a  covered  Part  performed  t o a s c e r t a i n which  chapter basis  contain  of  previous in field  laboratory patterns.  enclosures,  Preliminary  Chapter components  synthesized  individually into  an  in sculpins.  density,  sculpin  prey,  are  the results sampling  also  results  important  to In the  observations  of  sampling  programs diel  and  feeding  t o the design  of the  6 are also detailed. E f f e c t s of  the  i n chapter  and  results  5  are  the c a u s a l i t y of foraging  explicitly  investigated are f r y  inhibition  predator  and f a c i l i t a t i o n ,  experience  included.  a r e brought  6  on  Some e x p e r i m e n t s on s c u l p i n p r e d a t i o n  (0. k i s u t c h )  interpreting  4),  enclosures,  f r ysocial  and  of experiments  s e l e c t e d f o r study  of  5  are identified  of the t h e s i s .  in field  Factors  size,  r e s e a r c h , and  determine  i n chapter  evaluation  presence of a l t e r n a t i v e intensity.  to  feeding  mode o f o p e r a t i o n .  experiments relevant  6 forms the n u c l e u s  behaviour  Field  and f i e l d  of sculpin predation  examined  fish  Chapters  results  (chapter  conducted  keystone experiments r e p o r t e d  fry  the  and t h e i r  literature  experiments  the  5-11.  5, components o f t h e i n t e r a c t i o n  behaviour  are  of  f a c t o r s a r e t h e most  interaction,  particular  populations.  f o r the present  i n chapters  II)  in  o f chum  o f s y s t e m s where s c u l p i n s  review  background  A  sculpin-fry  predation  o f chum  brief  (Section  the  of  sculpin  characteristics  limit  studies  j u v e n i l e chum, t h e p r o p o r t i o n s  Models  foward  (Section  B)  and  on c o h o that  from c h a p t e r s are  light salmon  aid in 1-3.  presented  in  xxvi i  chapters  7-9.  important  characteristics  on V a n c o u v e r and  Chapter  I s l a n d , B.  Rosewall  faunal  7 i s mostly  C.  Creek).  composition  of t h e  is  II.  8 i s concerned  Chapter  abundance  distributions Creek  data  from  and  of  sculpin  It  chapter  of  7 only.  to  generated  from  chapters  5  facilitate  interpretation  forms  10 and  the  do  not  10). from  realize Major  and  limit  sizes,  from  and  of t h e  chapters  the  their  t h a t from potential of t h e  s e c t i o n s and biological  within Part and  spatial  and  Salmon  experiments and  and  utilizes  to a n a l y s i s  three  study  areas.  general c h a r a c t e r i s t i c s  6.  experimental Data  from  of  of  results  chapters  5-8  r e p r e s e n t a s y n t h e s i s of  the  results.  5-9.  of B i g Qualicum and  This  their  9 i s devoted  test  ( S e c t i o n C)  elements  previous  physical could  coho),  11  to  salinities,  occurrence  Information  from  chapters  and  potential  b a s i s f o r e s t i m a t i o n of t h e a c t u a l  (maximal) impact chum and  chapters  distributions,  determine  Creek  migrations.  mark-recapture  Chapter  and  7-9  different  the  sampled  Salmon  water  seasonal  of s c u l p i n s from  were  results  fry  the  involves  habits  important  on  a l l remaining  with  feeding  Chapters  data  of  length-frequency  objectives  River,  movements w i t h i n t h e B i g Q u a l i c u m  stomach c o n t e n t s  The  by  sculpins  estuaries.  determination  the  accessed  of  includes  sizes  information  detailing  t h r e e e s t u a r i n e systems  (Big Qualicum  It  and  descriptive,  s c u l p i n s on  chapters  5-6  salmon  suggests  fry why  (both  sculpins  under n a t u r a l c o n d i t i o n s ( c h a p t e r sculpin-fry  summarized  attributes  salmon p r o d u c t i o n a r e  of  in  interaction chapter  systems  identified.  11,  are and  drawn the  i n which s c u l p i n s P o s s i b l e remedies  xxvi i i  for  s i t u a t i o n s where  sculpin  predation  is  intense  are  also  proposed. The  Appendix,  Bird  Big Qualicum Estuary, sculpin similar  study  table  Vancouver  and c a n be r e a d  questions  estimating  Predation  and u s e s  impacts  of contents,  on J u v e n i l e S a l m o n i d s  Island,  is  an  adjunct  separately, although  some  of  the  on salmon p o p u l a t i o n s . summary and r e f e r e n c e s .  same  i n the to  the  i t addresses  techniques  It contains  for  i t s own  1  PART I : THEORETICAL CONCERNS  2  CHAPTER  1: GENERAL  INTRODUCTION TO PREDATOR-PREY  AND PARASITOID-  HOST RESPONSES  J_.J_ I n t r o d u c t i o n A  comprehensive  predators  and  evaluation  their  prey  how  consumption  by i n d i v i d u a l  the  density  the  of  the developmental into  predator  first  of  effects In  density  reproduction.  the  the f u n c t i o n a l  situations  response  of  are  interactions  affects  translated with the  has  three  to prey d e n s i t y , the 1961)  and  the  rate.  responses  been  used  to  1.3-1.5). O n l y  parameters  presented.  Methods  includes  systems  ( s e c t i o n 1.2), describe  each  models t h a t a r e  having of  in  a  biological  generalizing  one s p e c i e s o f p r e y a r e  1 . 6 ) . The r e v i e w in addition  o f p r e y , and  which  attack  possible  (sections with  how  t h e components o f t h e f u n c t i o n a l  have  i n v o l v i n g more t h a n  (section  prey  is  (Holling  on t h e o v e r a l l  types  tractable,  response  i s concerned  response,  to predator density  of t h e p r o c e s s and  and  o n l y one s p e c i e s o f p r e y and p r e d a t o r  interpretation  host  This thesis  functional  as w e l l as t h e models t h a t  briefly  how p r e y  t h i s c h a p t e r , I review and  predators  consumption  response  containing  simple  of  functional  response  the  between  p r e d a t o r s , the numerical response  of prey d e p l e t i o n  response  element  r e s p o n s e s . The  to the density  elements:  functional  interaction  p r e d a t o r s responds  these,  distinct  the  i n v o l v e s d e t e r m i n a t i o n of f u n c t i o n a l ,  n u m e r i c a l and d e v e l o p m e n t a l describes  of  (insect)  outlined  parasitoid-  to predator-prey interactions,  many o f t h e u n d e r l y i n g mechanisms a r e common t o b o t h  to  cases.  as  3  j_.2 The Much  of  the  m o d e l s of t h e carried  out  1968).  Components of  initial  functional  by  Holling  Holling  Predation  development  response  of p r e d a t o r s  (1959a, b,  emphasized  of b i o l o g i c a l l y  1961,  the  to t h e i r  1963,  need  for  an  understanding  ecological  of  first  of by  into  each the  each  component  e x p e r i m e n t a l l y and of t h e i r  qualitatively began  considering  his  the  representation  analysis  of  most  predation:  t h e d e n s i t y of  predators  (Holling  associated  with  1961).  the e f f e c t s  of prey  (ii)  r a t e of e f f e c t i v e  (iii)  predator  and  the  and  prey  h a n d l i n g time each  of  effects  components identifying  the  process.  process  variables the  a  that  density  defined eight  by  affect of  the  components  density: exposed  (product  likelihood  of area  that a prey  will  covered be  per  captured  (time  taken  to pursue,  subdue,  eat  and  prey)  (iv)  level  (v)  facilitation  of p r e d a t o r  hunger  by  prey  experience  leading to  inhibition  by p r e y  in  He  are  search  the  then  predation  and  complex  encountered)  digest  (vi)  prey  1959a,  time  once  the  fundamental the  (i)  time  and  distinct  two  of  o r g a n i z i n g the  occurrence;  and  disaggregating  i t s c o n s t i t u e n t components, d e t e r m i n i n g  universality  Holling  1966  was  experimental  and  process  prey  1965,  components a p p r o a c h t o t h e a n a l y s i s processes. This consists  realistic  contacts  improved c a p t u r e  contacts  recognizing distasteful  (e.g.  success)  (e.g. predator prey)  predator  experience  4  (vii)  (viii)  prey  social  easier  t o c a p t u r e as p r e y  prey  social  difficult and  five  facilitation  inhibition  (individual  become  density increases)  (individual  t o capture as prey  components  prey  associated  prey  density  become  more  increases),  with the e f f e c t s  of p r e d a t o r  density: (ix)  exploitation  (depletion  (x)  interference  between  (xi)  social  (xii)  d i s t u r b a n c e of prey  (xiii)  avoidance  Components basic  facilitation  (i)—(iii)  effects,  as  exposed  to i t s prey  1959b),  and e v e r y  in  s o d o i n g , must  eight a 2  to  between  predators  by  prey.  (ix)-(x)  were  e v e r y p r e d a t o r must and spend t i m e p r e d a t o r must  considered  be  search f o r i t s prey,  handling  exploit  i t s prey  (deplete) i t s prey  o r may  be  (Holling  w i t h o t h e r p r e d a t o r s . The and may  to  and,  remaining  n o t be p r e s e n t i n  p r e d a t o r - p r e y system. T h e r e f o r e , t h e r e a r e p o t e n t i a l l y  different  f o r m s o f p r e d a t i o n ; however, t h e s e c a n  four q u a l i t a t i v e l y  1,2,  predation)  by p r e d a t o r s  and  interfere  through  predators  components a r e s u b s i d i a r y  given 8  learning  of prey  3  and  responses  4,  distinct  Fig.  1.1)  responses and  to predator density ( F u j i i  When  the  basic  two  t o prey  be  reduced  density  qualitatively  (types  distinct  e_t a l _ . 1978).  components o f t h e r e s p o n s e  t o prey d e n s i t y  a r e t h e o n l y components o p e r a t i n g , a t y p e  2  1959b)  components m o d i f y t h e  is  generated.  baseline  response  average  hunger  The f i v e  in different level  subsidiary  ways. As p r e y  of the p r e d a t o r s w i l l  response  (Holling  density  increases,  decrease  and  result  cr o I-  Type  < Q LU  cr cr LU Q_  Type 2  CO  <  rh-  < LL, O  cr  LU  Type 3  m  CO  o LU <  Type 4  f* CO  PREY  DENSITY  7  in  a corresponding decrease  1966).  If  predators  negligible  amount o f t i m e  accurately, (Fig.  for  Richman  some  which  response  type  origin  Parsons  activity more  threshold  efficient  increases.  crossing  at  and  This  or i f  happen t h a t  their  density  produce  a  In  increases with density (Murdoch a n d O a t e n  shown  i f reward maintain  prey  t o the  of  type 2  baseline density, a right  of  d e s c r i b e d by  as  r a t e s a t low constant  become the  prey  searching  progressively of encounter  a t low p r e y d e n s i t i e s  few p r e y a r e  as  and c a u s e s  1975).  to  rate  to  realize  that  the predators learn (Murdoch  by some k i n d s  situations,  such  the  first  of  prey v e l o c i t y  i n c r e a s e s (Haynes a n d S i s o j e v i c  stimulus,  1 response  s p e c i e s of z o o p l a n k t o n .  predators  be p r o d u c e d  some  a  ( o r , more  effect  low p r e y  was  attack at a correspondingly higher rate  facilitation.  the  to  a t some  variant  to  capturing  can a l s o  If  the a b s c i s s a  p r e y , whereas a t h i g h d e n s i t i e s  3 responses  been  spend  ( s e e , f o r example,  similar  c o n t a c t e d and p r e d a t o r s take a l o n g time are  a type  has  (Holling  and  a hunger  1967).  are generated  1978),  I t may  effect  Rigler  insufficient  (Hassell  until  i s reached,  I f grazing ceases  obtained.  are  rate  crustaceans  e_t a_l. (1967) f o r s e v e r a l  densities  attack  constant  qualitatively  Type 3 r e s p o n s e s  of  p r o g r e s s i v e , a somewhat d e p r e s s e d  2 response is  a  rate  h a n d l i n g prey  and  is is  results.  modified  at  filter-feeding  hunger  response,  the  This threshold  1958 and B u r n s  decreasing  the  "satiation")  1.1) r e s u l t s .  exist  feed  in  they  quickly  1973). prey  Type social  i n c r e a s e s as  1966).  Prey  may  a c h e m i c a l , whose c o n c e n t r a t i o n the predator  to  hunt  faster  8  If  pronounced  r e s p o n s e s can from  a  p r e y . As  kinds  result.  Holling  simulation  model  the  contacts  between t h e became  inhibitory  r a t e of stimulus  established  and  developed  Alternatively, bumping  into  The density prey  as  basic  ( ( i x ) and  attacked be  (xiii),  to  produce  declines  r e g u l a r l y with  Most  rises  modified  and  determining  describe for  describing  below. D e t a i l e d are and  provided Hassell  the  density  (Fujii  may  1972). by  number This  types:  of  basic (xi)-  one  that  one  that  predator-prey  and  density  has  and  1978).  concentrated the  response  have been u s e d e x t e n s i v e l y  the  other  are  of  on to to  potentially useful  components a r e  behaviour  Royama  predator  components,  components o f  that  to  increases.  e t §_1.  the  i n r e v i e w s by  prey  attack  the  distinct  models t h a t  (1978).  make  subsidiary  basic  the  an  response  to  r e s p o n s e s , and  a c c o u n t s of  Some  progressively  during  the  responses  e f f e c t s of  The  (Tostowaryk  a n a l y t i c a l work on  Models  declined.  becomes  increasing predator  functional  only.  these  the  then d e c l i n e s  e f f e c t s of  density  of  qualitatively  e x p e r i m e n t a l and  parasitoid-host  prey  two  distastefulness  1966).  predator by  responses  association  increases  (x)) would both t e n d  d e c l i n e as  4  learned  predation.  which  components  r e s p o n s e can  initially  Chant  4  type  distasteful  of a t t a c k s  inhibit  density  type  its  d i s t u r b a predator  i t ( M o r i -and  operating,  a mildly  the  and  number  reaction  t h e y may  two  prey  also  their  on  increased,  the  possess a group defense better  predation  the  of p r e y b e h a v i o u r may  are  (1965) p r o d u c e d  of  from  effects  some o f  (1971), H a s s e l l  presented  the  models  e t a l . (1976)  9  I'l The  Funct i o n a l  Response  t o Prey  t h r e e e q u a t i o n s most commonly  predator-prey  Density  used  to describe  type  2  responses a r e :  A(N ) = W l " e- )  (1.1)  aN0  0  AnaxNo  A(N ) =  C1.2)  0  Km  N  +  aTN  0  0  A(N ) =  and  (1.3)  0  1 + ahN  where  N  = initial  0  = total  a  = rate  of e f f e c t i v e  h  = time  spent  =  0  The  enzyme  effect  p r e d a t o r and prey  a r e exposed  search  h a n d l i n g each  o f . prey  prey  density  on  of a t t a c k s p e r p r e d a t o r d u r i n g time  the instantaneous  T  Amax  = maximum number o f a t t a c k s d u r i n g t i m e  Km  = the prey  first  Ivlev  time  density  T  A(N ) number  prey  0  the second  kinetics,  equivalent situations  a t w h i c h A ( N ) = 0.5Amax. 0  e q u a t i o n was d e r i v e d i n d e p e n d e n t l y by Gause  (1961),  Holling  density  T  the  and t h e t h i r d  (1959b). and  is  Equations  differ  t o which  they  only  Michaelis-Menten  i s the disc 1.2 in  have been  and their  (1934) and  equation  of  e q u a t i o n d e v e l o p e d by 1.3  are  structurally  derivations  applied.  and  the  10  A of  number  t h e Gause  Fujii  and d i s c  21 c a s e s  were  and  obtained  f o r the d i s c  re-analysis  equation  seem  was  Although responses,  similar that  for f i t t i n g  E q s . 1.1-1.3 c a n n o t  they are a c l e a r  (Lotka  1925, V o l t e r r a  which  assume  a  1928)  linear  rate  o f a t t a c k and p r e y  the  rate  of  equation  the prey  0  t o 21  (1969) d a t a i n most  results. either type  of the  (1970)  the d i s c  of the cases  studied.  However,  it  does  responses. types  over  3  or  Oaten  1975)  response. prey  In t h i s  item  encounters, equivalent  has  (1935)  the  type  models  instantaneous 3  responses,  increasing  function  function AmaxN  0  B + N  0  n  (1.4)  is  a  n  constants  frequently  with  been u s e d  e x p r e s s i o n , the time decreasing  r a t h e r than expression  (Murdoch  to d e s c r i b e the  of  function  proposed  n > 1  necessary  function  a linear  i s that  4  the L o t k a - V o l t e r r a  between  be an  1,  0  B and n a r e p o s i t i v e  not  is consistently  A(N ) =  where  in  that  and N i c h o l s o n - B a i l e y  relationship  squared  in nine  found  describe  improvement  The  s e t s of d a t a  sums o f  equation 2  abilities responses.  i n 12 c a s e s . G l a s s  s e a r c h , a, must  density, N .  2  equation  d e n s i t y . To g e n e r a t e  of e f f e c t i v e  the  type  residual  f o r t h e Gause  to conclude  the other  Smallest  descriptor  (1959b) o b t a i n e d  than  to describe  Griffith's  the b e t t e r  possible  better  of  have compared  t h e two e q u a t i o n s  from the l i t e r a t u r e .  deviations  Holling  authors  equations  et_ a _ l . (1978) f i t t e d  obtained  his  of d i f f e r e n t  of  by R e a l  sigmoid  to l o c a t e  the prey  number  and  of  each prey  density.  (1977) t h r o u g h  An an  11  analogy  between  feeding  and t h e k i n e t i c s  of a l l o s t e r i c  enzyme  reactions:  bTN n 1 + bhN  A(N ) = 0  where  b i s a c o n s t a n t . In t h i s  search  i s expressed  (1.5) n 0  equation, the rate  of  effective  by  a(N ) = bNo""  1  Q  Real  called  interpreted with  this  He  learning  Hassell  often  that  et  it  of  potential  i s maximally type  3  detection  a predator effective  response  must  result  (1977) a n a l y z e d  sigmoid  density  functional search  several  responses  could  by a r e l a t i o n s h i p  bN  sets  be  and found  that  expressed  of the  +  N  form  0  this  expression for a  (1.3) y i e l d s  bTN  2 0  A(N ) =  (1.8)  Q  g + N  Fujii  a  0  b and g a r e c o n s t a n t s . S u b s t i t u t i n g equation  as  (.1.7) g  the disc  from  from  Q  into  prey  of data  a(N ) =  where  have  on t h a t  to  and  by t h e p r e d a t o r .  of e f f e c t i v e  of p r e y  rate  of encounters  the  a_l.  exhibit  the r a t e  function  before  considered  associative  species  the  n a s t h e number  i t s prey  item.  term  ( 1 > 6 )  e t a _ l . (1978) u s e d  a similar  0  + bhN 2 Q  method  to  develop  another  12  model d e s c r i b i n g  type  3 responses.  They assumed  that  a(N ) = be^O  (1.9)  0  where  b  and  d a r e c o n s t a n t s , and s u b s t i t u t e d  E q . 1.9 i n t o E q .  1.3 t o g i v e be °TNo dN  A(N )  =  0  Here,  the parameter  constant, density  with  The  to  relative  be  thought  indicating  abilities  of each  i n shape h a s  an  detailed  in section  well  evaluation  as  type  of  effect  1.5,  responses  of prey effect  comparative  The d i s c  1.8  that  investigated.  2.2). A l l are capable 3 responses.  facilitation  o f E q s . 1.4,  been  their  a  effect.  functional  not  as  d > 0 a facilitative  and d = 0 no  to describe predator-prey  be s i g m o i d  of  an i n h i b i t o r y  ( v i ) or ( v i i i ) ) ,  (v) or ( v i i ) )  reason,  as  can  d < 0  (components  (components  1.10  d  (1.10)  1 + be^OhNg  appear  (For  of d e s c r i b i n g equation  type  describe  cannot  i s regained i f  densities.  4  1.10 i s t h e o n l y e x p r e s s i o n s p e c i f i c a l l y a l l four types  generate  plateau,  are  2  and  respectively. Equation  to  this  flexibilities is  n = 1 i n E q s . 1.4 and 1.5 o r i f g = 0 and d = 0 i n E q s . 1.8 1.10,  and  i t  a true type  can  produce  This condition  generated responses  of responses. 1 response  a response occurs  i f 0 < d < bh, t y p e i f  d < 0.  Although  the equation  with a l i n e a r that  i s linear  when d = bh. Type 3 responses  In t h e l a t t e r  case,  developed  rise  to  a  a t low p r e y 2  responses  i f d > bh and t y p e t h e maximum  attack  1 3  rate  occurs  cases,  when N  and  estimates  for  0  = - 1/d and A = T / ( h - d e / b ) .  biologically  i n E q s . 1.5 and  approached  as N  produce  "type  allowed  to  4"  tends  0  1.8,  Equation  responses  (biologically  N  0  i f the  Equations  for  describing  considerable  unrealistic)  The  to  prey  work  and May  negative  response  the  predators. component  in  density, this  to predator and s o c i a l of  their  attack  although  area 1969,  density  in  values. generated  responses  there  recent  Hassell  years  1971,  or  by  i n simple  in  empirically  ways:  (1959) and H a s s e l l and  al.  1977).  of  avoid  to  predators  o f t e n be more i m p o r t a n t ,  well-quantified  et  1975,  direct  p r e d a t o r s , as  learning  in detail.  1971,  Beddington  prey  between  been  Royama  by  interference  those  (see, f o r  i n t e r f e r e n c e caused  t h a t h a s been c o n s i d e r e d may  between  as  to  has  i n v o l v e s the e f f e c t s  facilitation  indirect  prey  Direct  components  two  are  Density  1973, R o g e r s and H a s s e l l 1974,  effects  disturbing  other  1) i s  e t a l . 1975, H a s s e l l e t a l . 1976 and F r e e  interference as  to Predator  instantaneous  example, H a s s e l l and V a r l e y  Lawton  however,  parameters  d e n s i t y have n o t been n e a r l y so w e l l d e v e l o p e d  responses  Hassell  will,  is  = - 2g.  j_.4 F u n c t i o n a l R e s p o n s e  predator  other  by A = T/h  1.8  The maximum number o f a t t a c k s , A = 4bgT/(4bgh when  a l l  ranges of the parameter  an a s y m p t o t e g i v e n  to i n f i n i t y .  (dome-shaped)  assume  feasible  In  Even they  well  predators their  i s the only though  the  have n o t been  m o d e l s . I n t e r f e r e n c e h a s been  modelled  a s e x e m p l i f i e d by t h e models o f Watt  Varley  (1969),  and  behaviourally,  as  1 4  exemplified  in  t h e works  Beddington  (1975).  of  Rogers  O b s e r v a t i o n s on p a r a s i t o i d - h o s t rate  per  parasitoid (1969)  parasitoid density  to  declined  l e d b o t h Watt  propose  an  and  Hassell  systems  where  regularly  the  with  (1959) a n d H a s s e l l  equation  (1974) a n d  attack  increasing and  i n which t h e a t t a c k  Varley  r a t e was  e x p r e s s e d by A(P)  where P  = parasitoid  = qP-  (1.11)  m  (predator)  density  A(P) = e f f e c t of p a r a s i t o i d d e n s i t y number o f a t t a c k s  In a  = constant  m  = mutual i n t e r f e r e n c e  attack  searching  equation  available Hassell there  are  linear  constant  gives  ^ 0.  a linear relationship  i n many  several  theoretical  (Royama  1971, H a s s e l l and  much o f t h e i r d a t a  (1.12)  description cases  o f much  i s not s u r p r i s i n g associated  a n d May 1973, Cheke (1969)  of the  ( e . g . H a s s e l l 1971,  problems  Varley  between  density:  = log(q) - mlog(P)  1972) i t does n o t . T h i s  a l . 1977). H a s s e l l  that  i n the absence o f i n t e r f e r e n c e  a reasonable  although  and Rogers  formulation et  provides  data  instantaneous  e f f i c i e n c y and i t s p o p u l a t i o n  log(A(P))  This  rate  f o r m t h e model  parasitoid's  the  per parasitoid  q  logarithm  on  themselves  would have been b e t t e r  fitted  with i t s 1974, F r e e mentioned by a non-  m o d e l . On t h e o r e t i c a l g r o u n d s t h e r e l a t i o n s h i p s h o u l d  curvilinear  with  an  increasingly  negative  slope  as  be  as p a r a s i t o i d  15  density  rises.  These problems, a l o n g  nature  of  Hassell-Varley  the  equation,  t o p r o p o s e more complex b e h a v i o u r a l R o g e r s and for  interference,  adult  amount of  and time  unprofitable egg).  given  fixed  period  depends  on  efficiency tend  to  for hosts  searching between  the  time. U n l i k e  the  being able  this  eventually of  parasitoids  M o d e l A,  the  logarithm linear  of  only  the  number of  at  very  were of  levels  and  for  Model By  a B  varying able  to  searching  parasitoids  high  an  host  search  density.  a  in  a  in  logarithm  an  then  ceasing  the  Hassell  an  time. For  encountering  host  b o t h m o d e l s , R o g e r s and  makes  results  interference  and  certain  other  of  parasitoids  r e l a t i o n s h i p between t h e  become  period  a  to deposit  each  fixed  number  a  parasitoid  for a  both p a r a s i t o i d density  and  interference  of  models  between  been p a r a s i t i z e d t h e n abandon  p a r a m e t e r s of that  for  terms  In M o d e l B,  already of  (in  authors  behavioural  interference  (B)  parasitoids  search.  has  two  i s wasted each time a  between  that  empirical  models.  p a r a s i t o i d s which encounter  of  equilibrium  show  (tw)  search  number  resuming  another  encounter  the  for  purely  a p a r a s i t i z e d h o s t . B o t h assumed t h a t  In M o d e l A,  abandon  (A)  the  have l e d s e v e r a l  (1974) d e v e l o p e d  one  p a r a s i t o i d s and  parasitoid  the  Hassell  with  of  would mutual  interference. The  assumptions  interference:  of  Beddington's  (1975)  model  of  16  aT A(P) =  (1.13) 1 + r t ^ C P " 1)  where  r  is  similar  to  relationship linear  only  smaller Eq.  the  rate  those  in  values.  than  be  used  may  direct  or  the  between p a r a s i t o i d s  Of that the prey  by  response should  their t o prey not  in  be  curvilinear realistic  than  i t s application.  It  other  The H a s s e l l - V a r l e y model  (Eq.  other  factors  qualitatively  interference, that  for  to predator density  approximation  some  form  to  (components ( x i ) -  suggests  and  a  value  of m  an e f f e c t  negative  of s o c i a l  of  value  facilitation  i s operating.  of Responses t o Prey listed  and P r e d a t o r  in section  be d e s c r i b e d r e a l i s t i c a l l y  above models  and  in providing a f i r s t  data  t h e 13 components  cannot  rtw  i s more b i o l o g i c a l l y  where  suggests  j_.5 S y n t h e s i s  t o be  of  Model  t o be o p e r a t i n g . H e r e , a p o s i t i v e  indirect  qualitatively  A ( P ) and l o g (P - 1) was shown  interference.  responses  from  The  to d e s c r i b e responses  a r e thought  estimated  Hassell's  restricted  be more u s e f u l  describing (xiii))  more  between p a r a s i t o i d s , a r e A.  T h i s model  t h a t of d i r e c t  1.11)  log  and  large values  1.11, b u t i s a l s o  cannot  encounter  Rogers  between f o r very  of  exploitation,  predators.  Expressions  density  0  used  in  the  or e m p i r i c a l l y  i s t h a t of  (A(N ))  1.2,  Density only  by any o f  i . e . , depletion for  the  and t o p r e d a t o r  t h e forms p r e s e n t e d  one  of  instantaneous  density  to determine  (A(P)) prey  17  consumption very  (host a t t a c k )  small  prey  (less  about  r a t e of prey  predator  to  the  m o d e l s assume their  is  in  zero  the  where  0  instantaneous  responses  A(N ,P)  Poisson is  to host  number o f a t t a c k s . Hence  there  Most  (parasitoids)  the p r o p o r t i o n of h o s t s of  combined  and p a r a s i t o i d  occurrence with of  the  density  on  the  t h e number o f h o s t s  attacked,  = N ( l - -A(N ,P)P/No)  A  0  e  NA  (1.14)  0  i s no i n t e r f e r e n c e between p a r a s i t o i d s  0  i f A(N ) i s given  N  The most by  (Griffiths distribution distribution  widely  A  is  ( E q . 1.3)  (1.1  Q  used  Holling assumed  so t h a t  equation  = N ( 1 - e-aPT/d+ahNo))  parasitoids and  (1.15)  0  by t h e d i s c  0  search  mean  effect  A(N ,P) = A(N ) and  the  exploitation  distribution  the  of  from  N  If  i t s prey.  a s one minus t h e p r o b a b i l i t y  A(N ,P)P/N ,  calculated  p o p u l a t i o n ) or  between p r e d a t o r s  a  is  are eaten.  (hosts). For p a r a s i t o i d s ,  of  total  of  encounters  expressed  0  exploitation  d e p l e t i o n d e p e n d s on t h e r e s p o n s e  category  0  of  10% o f t h e i n i t i a l  distribution  random  prey  killed  is  t h e amount  a r e r e p l e n i s h e d c o n t i n o u s l y as they The  and  than  unless  model  is  the  1969,  for  representing  negative May  binomial  1978).  t o be i d e n t i c a l  t h e number o f h o s t s  The  6 )  non-random distribution  mean  of  the  t o that of the Poisson attacked  is  expressed  18  by  A(N ,P)P 0  N  where This of  = N { 1 - (1 +  A  k i s the d i s p e r s i o n c o e f f i c i e n t equation  h a s o f t e n been  the d i s t r i b u t i o n  May  good  Griffiths attacks  fits  to  Neodiprion  experiments,  exhibited  that  the  grounds  practice  that  obtained.  distribution  Pleolophus  basizonus,  11 c o n f o r m e d  t o E q . 1.17  binomial  with  on  response  k > 0  (k < 0 ) . I n o t h e r  distribution  of  1.17.  and  cases,  h a s been c r i t i c i z e d on  k i s an i n c o n s i s t e n t measure o f a g g r e g a t i o n  1.14,  because,  as they  number o f p r e y equation  1.16  and  1.17  unlike parasitoids,  attack  mean o f t h e P o i s s o n  density,  were  for  in  ( T a y l o r e t a l . 1978, 1979).  Equations predators  the  studies  parasitoid-host functional  regular distributions  of the n e g a t i v e  disc  found  binomial  description Anderson and  parasitoid-host  negative  binomial.  s e r t i f e r was d e s c r i b e d w e l l by E q .  14 o t h e r  of which  use  prey  (1969)  a good  attacks per host.  by t h e i c h n e u m o n i d p a r a s i t o i d ,  a l s o surveyed  three  the  of t h e n e g a t i v e  to provide  15 e m p i r i c a l  and H o l l i n g  sawfly,  They  found  of p a r a s i t o i d  (1978) t a b u l a t e d  which  the  (1.17)  0  them. F o r r a n d o m l y distribution  removed (Eq.  must  from t h e system. 1.3)  and  t h e a p p r o p r i a t e mean i s  there  are  inappropriate  predators searching be  a  remove  for their  p r e d a t o r s , the  function  If A(N ) i s given 0  i s no e f f e c t  of  the  by t h e  of p r e d a t o r  19  u = a(PT - hN )  (1.18)  A  (Rogers  1972). The  (1971)  method o f  as  time  equation,  t , and prey  result  expressing  a differential  at  same  equation  can  the  consumption  obtained  instantaneous  i n Nt,  integrating  be  over  number  t h e number o f time.  w o u l d be  For  by  Royama's of  prey  example,  calculated  by  attacks  remaining  f o r the  disc  solving  the  equation dN  - aPN  t  dt  between yield  the  the  so-called  An  i s analogous  expression  derived.  (The  not  values data or  of on  the the  predators 1.17  of  response  parameters initial  predictions  known p a r a m e t e r  of  - NA.  Both  (Rogers  methods  1972):  (1.20)  h N  and  (a and final  numbers of  that  for  exploitation  type  0  equation"  "random p a r a s i t e  a p p r o p r i a t e model  a l t e r the  N  0  0  Eq.  Incorporating  = N ,  = N ( l - e-a( " A))  t o the  to  Nt  p T  A  for  corresponding  T and  t  "random p r e d a t o r  N  which  1 + ahN  l i m i t s t = 0,  (1.19)  t  equation" search  parasitoids,  (Eq.  non-randomly, has  not  i s derived in section into predator-prey produced, h)  but  i f these  prey  eaten  v a l u e s . When t h e d i s c  models  i n an  does the from  experiment,  are  (Eq.  been  2.3).  estimated  i f these  equation  yet  i t does a f f e c t  are  numbers of p r e y  1.16).  based  1.3)  and  on the  20  random of  predator  data,  larger  the  value  (Table  substantial effective the  increasing  the  t h a t when p r e y  the  disc  reaches  equation  from  equation  is  greater  half-saturation  the  prey  densities.  When  i n t e r f e r e n c e between p r e d a t o r s  these  effects  incorporating i.e. disc the  with  equations disc  response  total 0  equation  can  1.3) be  the  to  response  A(N ,P). (Eq.  but  larger  the  or  The are  i n t o an  substituted  for q  combining  the  disc  than  converge  response to  density  combine  prior  exploitation (Eq. by  i n Eq.  at  model,  1.11)  assuming 1.11.  to  and that  Hence,  0  number of a t t a c k s  equation  equation  (1-21)  1 + ahN  total  from  density  0  the  same s e t of  disc  A(N ,P)P = —  Alternatively,  exp(-aT)),  curves  combined  the  produced  The  Hassell-Varley readily  initial  some o t h e r  prey  them  than  (1 -  prey  two  equation,  The  operating, i t i s necessary  the  the  to c a l c u l a t e  is  1.1).  is  r a t e of  describe  from  (0.5Amax) a t a lower  high  density  there  to  1.2).  random  predator  slightly  disc  response  (Fig.  equation,  is  when  equation  the  to  predator  same s e t  the  always  is calculated  equation,  is  case  (Table  random p r e d a t o r  i n each  to the  f o r example, a h i g h  generated  (aT)  consumption  values  latter  exploitation  equation  s l o p e of  parameter  (through, are  fitted  i s greatest  random p r e d a t o r  disc  are  i n the  This difference  the  with the  initial so  1.1).  of  1.20)  of a e s t i m a t e d  search). If data  decreases of  (Eq.  exploitation  ability  slope  equation  with  can  Beddington's  be  derived equation  by (Eq.  21  T a b l e 1.1. P a r a m e t e r e s t i m a t e s and sums o f s q u a r e d deviations from predicted values (sse) obtained by f i t t i n g t h e random p r e d a t o r e q u a t i o n , R.P.E. ( E q . 1.20) t o t h r e e sets of data g e n e r a t e d from t h e d i s c e q u a t i o n ( E q . 1.3) w i t h T = 1 and N = 5, 10, 15, ... 320.  Parameters used i n disc equation  Parameters estimated f r o m R.P.E.  a h sse  0.1000 0.0500 approx. 0  0.1043 0.0508 0.0024  a h sse  0.5000 0.0500 approx. 0  0.581 5 0.0510 0.4394  a h sse  0.9000 0.0500 approx. 0  1.1504 0.0511 1.6992  22  Figure  1.2.  F u n c t i o n a l responses  equation  (Eq.  with T =  1, a = 0.90  prey  1.3)  and  generated  random p r e d a t o r  and  h = 0.05.  d e n s i t y a t w h i c h number of p r e y  from  equation  Dotted eaten  the  (Eq.  lines i s 0.5  disc 1.20)  indicate Amax.  N U M B E R  OF  P R E Y  P R E S E N T  24  1.13)  giving aN PT 0  A(N ,P)P  {  1  2  2  )  0  (Beddington  1 + ahN  1975). E x p r e s s i o n s prey  density  0  + rt^P-1)  based  on  response  to  ( E q s . 1.5,  by making  the a p p r o p r i a t e s u b s t i t u t i o n  other  models  of  the  1.8 a n d 1.10) a r e o b t a i n e d for  a  in  Eq.  1.21  or  1.22. New  models  response  t o prey  component,  but  presented  using  a  modified  density,  and  omitting  the response  i n chapter  called  assumed of  equation  including  density  on  of t h a t  exploitation  to predator  density, are  ( E q . 1.3) h a s been g e n e r a l i z e d t o t h e s o equation"  Charnov  t h a t , as i n the d i s c  attacks  the  t o M u l t i s p e c i e s Models  "multispecies disc  authors  incorporating  2.  j_.6 E x t e n s i o n The d i s c  approach to d e s c r i b i n g the  a given  (1973)  independently and  equation,  Murdoch  by  (1973).  the i n s t a n t a n e o u s  species, A i , i s directly  the  search  difference  activities  to the  s p e c i e s , N i , by t h e r e l a t i o n s h i p (1.23)  s  effective  They number  related  Ai = a i T N i  where T s = t i m e  several  predator  spends  f o r prey  searching  and  s p e c i e s i . The t i m e  between  ( T ) and t h e t o t a l  the time  total spent  time  ai  spent spent  =  rate  of  searching i s in  feeding  handling prey, i . e .  25  T„ = T - .Z-hjAj  (1.24) or  T = T  + s  where s i s t h e rearranged the  number  and  solution  the  for Ai  of  Eq.  1.25,  the  type  used  generalize  to  multispecies  3 and  If  Eq.  i s substituted  into  3  3  Eq.  1.23,  3  each  species  4 responses, 1.5,  v e r s i o n of E q .  is  a^Ni  to  Eqs.  1.24  (1.25)  1+13=1  type  present.  by  response  represent  = 1  f o r Ts  i s given  .  f h,a.T_N. 3 3 s j  species  result  A  In  j  1.5  1.8  is  a similar and  1.10.  type  approach For  2.  To  can  be  example,  the  is  a.TN, i-J: 1 + Z a,h,N, J ni  Ai =  CI-26)  n  3=1  and  produces  these  kinds  difficult  of  type  2 and  involved  which  actual  instantaneous  numbers  number  3  3 responses.  is  types  that of  of  investigated of  prey  attacks)  One  between  in section  eaten can  problem  in p r a c t i c e ,  responses  in distinguishing  m u l t i - p r e y systems a r e The  3  type  formulations  to determine  difficulties in  both  3  i t may  be  operate.  (The  response  types  2.5).  (rather be  with  than  determined  the in  an  26  a n a l o g o u s manner t o equations over al.  as  time  a  using For  (1974).  single  set  of  methods Eqs.  species  differential  similar the  1.25,  a  1 +  Generalizations  NAi  integrating  by  Lawton  e_t  equations  (1.27)  N  f a^h-iNi 0=1 3  J  s = N (l - e - ^ - ^ / J ^ j ) )  A i  by  (1.28)  ±  incorporating  follow d i r e c t l y . i s found  the  give  N  also  and  employed  differential  J  to  expressing  ' i i  dt  solved  by  equations  to those  _  are  cases  the  response  I f B e d d i n g t o n ' s model  to predator for  A(P)  density  is  used,  solving s - a . ( P T - Z h-N .) A  N  Ai = N (l - e  1  ^w^-D  +  ±  Equations  1.29  predation natural 1.29  in a  of  that  species  the  environment. the  a predator  different in d i r e c t  faced  proportion  of  1.26  each  to  (1.29)  the  basic  Their  limited.  p o t e n t i a l prey  Equation  proportion  of  context.  i s , however,  each  between  all  multispecies  systems  a l l assume  number  incorporate  )  applicability  Equations  with  a  species  1.25, 1.28  choice will  its relative  assumes t h a t  species  in  components  the  between  simply  of to and a  capture  abundance  in  the r e l a t i o n s h i p diet  and  its  27  proportion  in  the environment  a b u n d a n c e . None of t h e active  selection  function null  i s one  hypothesis  are being subsequent  by  the  t h a t no  sections  f u n c t i o n of  allow  the  for  predator. Probably  of p r o v i d i n g a  expressed.  i n c o r p o r a t e prey  models  i s a simple  mathematical  possibility their  of t h i s  selection,  be  used  most  statement  p r e f e r e n c e or o t h e r complex  (They' w i l l  for  relative  this  of  useful of  the  behaviours purpose  in  t h e s i s ) . More complex m o d e l s , t h a t  are considered  in chapter  3.  28  CHAPTER 2: EVALUATION AND DEVELOPMENT OF MODELS  INCORPORATING  VARIABLE RATES OF E F F E C T I V E SEARCH  2.1_ I n t r o d u c t i o n Although  most o f t h e  m o d e l s have been b a s e d 1925,  Volterra  1965), many adopt  classic  predation  on t h e a s s u m p t i o n  specific  (non-random)  prey  themselves  Oaten  1975, E g g e r s  et  a l . 1977,  Milinski random  (Ivlev  Hassell  search  (e.g.  i s t h a t they  stability  in  For  this  reason,  describing  or  ability  is  factors  interactions.  1972, May  essential  explaining  Pyke  1979, H e l l e r and  major  by  This  has  been  arguments  incorporate into  prey  promoting  theoretical  non-random s e a r c h  non-  of h i g h  1973, 1978, Murdoch to  the  1977,  areas  e x p e r i m e n t a l l y and  i t  et a l .  to locate  are p o t e n t i a l l y  1958, S m i t h  of  c o n s e q u e n c e o f some forms o f  predator-prey  demonstrated both Huffaker  the  these  1974, Murdoch and  1978, A k r e and J o h n s o n  1979). The i m p o r t a n t  density)  (e.g.  1961, Werner and H a l l  often  and t h a t  by t h e b e h a v i o u r  1977, 1978, 1982, H a s s e l l  1959b,  predators  strategies,  s t r a t e g i e s may be e n h a n c e d o r i n h i b i t e d  (Lotka  1935, H o l l i n g  have shown t h a t search  parasitism)  o f random s e a r c h  1928, N i c h o l s o n and B a i l e y  recent experiments  (and  1977).  mechanisms  predator-prey  models. In single  terms of the i n s t a n t a n e o u s prey,  random s e a r c h  subsidiary  components  parameters  representing  namely, t h e d i s c  i s represented  (section  equation  response  the  1.2)  basic  t o the d e n s i t y of by m o d e l s  are  i n w h i c h no  operating  components  ( E q . 1.3) o r t h e random  a  are  and  the  constant;  parasite  and  29  random  predator  functional form  of  equations  response  (Eqs.  non-random  social  facilitation  causing  t h e b a s i c components  effective its  search,  Pr  (recognition!  Pr  (attack  prey  by  prey  inhibition)  exert  consist  of  a  as  follows:  a =  encounter).  Pr Pr  time  density and  of  a,  via  (and d i s t r i b u t i o n )  the  number  When  most  of prey  was  with  density  both  the  the hunting  of prey;  contact  per t i m e ) .  would  For  likely  to  affect  present  that  have  simplicity,  i s amalgamated  subsidiary  into  of prey i s the  two  at a given  time  (Nt)  successfully (At). patterns  alter  d e p e n d e n c e o f a on A t s u g g e s t s  the predator  one  components, a r e t h e  searching  success  The  in searching for  species  s t r a t e g y and i t s  amount o f p r e v i o u s  them.  previously attacked  o f a on Nt s u g g e s t s  that  spent  of  recognition).  one  of prey  Dependence  handling  i n terms  only  the  prey  on t h e r a t e o f  covered  r a t e , ER,  Pr  variables  and  the  (pursuit|  parameter,  (success).  (predator  (success| attack attempt).  i s the encounter  a l l the predator's  some  i n f l u e n c e by  of  (area  of c o n d i t i o n a l p r o b a b i l i t i e s  subcomponents  their  t o have t h e g r e a t e s t e f f e c t  the product  the  represent  d i g e s t i v e pause, s u b s i d i a r y  and n o t i n a t t a c k i n g and c o n s u m i n g  present,  A l l other  contacts  t o v a r y . As most  attempt| p u r s u i t ) .  if  to  a . T h i s v a r i a b l e c a n be e x p r e s s e d  subcomponent  resulted  and  are l i k e l y  subcomponents  first  or i n h i b i t i o n  usually  components  1.20).  s e a r c h . The s u b s i d i a r y components  facilitation  will  and  m o d e l s c a n be c o n s i d e r e d  hunger,  time  1.16  may  depend  on  that the  has a l r e a d y had w i t h i t s  prey. Although  the  hunger  level  of the predator  can a l t e r  the  30  search  s t r a t e g y , i t c a n o f t e n be t h o u g h t  the h a n d l i n g time per prey and  is  through  leading  2.1).  result  The  baseline  random  2,  a  involving Pathways  responses  3  type  is  sufficiently will  if  4 response.  will  t h e components  to result  v i a pathway  capable  of  into a modified  a l l type  t o type  3  the  pathways responses.  2 responses  that  that  have  would  4  2 and  4  responses  as  would  in  encounter  rates  a r e t h e most  8, p a r t i c u l a r l y  causing their  the  i s p r o n o u n c e d . Pathways  type  reductions  that  were n o t o p e r a t i n g , o r t y p e  effect  in  search ( F i g .  Provided that  produce  lead  different  is  strong,  probably  i n c r e a s e d . Type 4 r e s p o n s e s  produced  eight  t o the b a s e l i n e response  i f the i n h i b i t o r y  substantial  to give  i s transformed  a  pause)  subcomponents a c t  interactions  or  increasing  digestive  remaining  these  inhibition  relative  4 are unlikely  or  of  the  r a t e s of e f f e c t i v e  ( t y p e 2) r e s p o n s e  facilitation  obtained  density  of e a c h  involving  require  variables  effect  are depressed been  to  type  facilitative  The  Nt o r A t , b u t n o t b o t h ,  pathways  type  (by i n c r e a s i n g  not c o n s i d e r e d f u r t h e r . either  of a s s i m p l y  i f the prey  predators injury  this as  likely are  prey to  be  distasteful  d u r i n g an a t t a c k  attempt. Models d e r i v e d to d e s c r i b e d i f f e r e n t responses  (chapter  1)  have  distinction  between  mechanisms  dilemma  whether  to  is  realistic, models  develop  often incorporating  that  are . capable  but a r e p u r e l y d e s c r i p t i v e .  not  types  usually  producing models  large  of  considered  the  response.  The  the that  numbers o f  functional  are  biologically  parameters,  of d e t e c t i n g t h e shape o f a Biologically  realistic  or  response,  models, which  31  Figure  2.1.  effective  Eight  search  Pr(success). time t  (Nt)  (At). result  the  Vertical i n an  baseline  v i a the  Each or  pathways l e a d i n g  number of  increase  response  subcomponents ER  i s a f f e c t e d by  arrows or  to v a r i a b l e  either  successful  decrease  ( i n w h i c h no  (encounter the  the  rate)  and at  to time t  pathway  in prey consumption  subsidiary  of  prey d e n s i t y  attacks  i n d i c a t e whether  rates  components  will over  the  operate).  <"S  Direction of influence  S  ^ °  1  PREY SOCIAL FACILITATION  2 3  PREY SOCIAL INHIBITION  N.  A  t  Nt  A  t  ST  5 6 7 8  EXAMPLES Prey v e l o c i t y i n c r e a s e s w i t h increasing N  1 ER  2  Prey v e l o c i t y d e c r e a s e s w i t h increasing N  FACILITATION BY PREY CONTACTS  Increased a b i l i t y t o locate 3 p a r t i c u l a r prey types w i t h increasing experience  INHIBITION BY PREY CONTACTS  . Learn t o a v o i d areas con" t a i n i n g p a r t i c u l a r prey types  PREY SOCIAL FACILITATION  I n c r e a s i n g numbers o f weak ^ prey w i t h i n c r e a s i n g N  PREY SOCIAL INHIBITION  Group defense r e a c t i o n b e 5 comes p r o g r e s s i v e l y b e t t e r developed w i t h i n c r e a s i n g N  r  Pr (success) FACILITATION BY PREY CONTACTS  INHIBITION BY PREY CONTACTS  t E x p e r i e n c e l e a d s t o improved ' capture success 7  E x p e r i e n c e l e a d s t o a deg crease i n the p r o b a b i l i t y o f a t t e m p t i n g an a t t a c k  33  consider are  t h e u n d e r l y i n g mechanisms l e a d i n g t o  desirable  because  power. However, an one  mechanism  representation developed the  rate  than of  equation may  be  of a n o t h e r  to date  that  have  mechanism.  search  greater  is biologically  biologically  result,  predictive  realistic  unreasonable  For  example,  3 responses  is affected  end  for as  a l l  have assumed  by p r e y  (Eq.  1.5)  learning,  was  derived  in r e a l i t y  that  density, rather  f a c i l i t a t i o n ( F i g . 2.1),  equivalent may,  indirectly, If  provide  a s At  i s d e p e n d e n t on  the o b j e c t i v e  response  has  predation),  often  desirable  that  t h e p r e d a t o r has range  prey  so  that  i n c r e a s e s with prey  (Eqs.  1.5,  diversity  the 1.10 of  ability and sigmoid  to  use  type  any  3  is It  learning  which  type  of  3 the  risk  predation  these  is  respectively) (section  1977,  Over  positively average  Hassell  important, and  Hassell  to  generate  2.2).  i t is  i t s prey. is  of  suggest  of p r e d a t i o n t o t h e  responses  shapes  as  to s t a b i l i z e  curve,  capable  In p a r t i c u l a r ,  3 responses  of the R e a l , F u j i i 1.8,  them.  of  equation  types.  d e n s i t y (Murdoch  As d e t e c t i o n of t y p e evaluate  determine  the p o t e n t i a l  of t h e t y p e  density-dependent  to  response  to i d e n t i f y  with prey  subduing  representation  kind  ( i . e . , t o d e d u c e t h e c o n s e q u e n c e s of  i t i s legitimate between  represent  Nt.  simply  been p r o d u c e d  differentiating  lower  is  encounter  o f p u r s u i n g and  adequate  to  one  r e p r e s e n t some o t h e r  unless simple  to the experience  however,  specifically  i t must  a  models  t h e amount o f p r e v i o u s c o n t a c t w i t h t h e p r e y . A l t h o u g h these  the  usually  to d e s c r i b e type  of e f f e c t i v e  predator of  they  the  1978). I  first  equations  N e x t , new  a  wide models  34  based  on  pathways  predator-prey of  prey  ( F i g . 2.1)  and p a r a s i t o i d - h o s t  i s p r e s e n t . Pathways  (section  2.3)  pathways  B-D  problem than  A-D  and  briefly  of d i s t i n g u i s h i n g  one p r e y  2.2  type  outlined  Hassell  new  models  (section  curves accelerate  i s slight,  better  equations,  detail  representing  2.4). F i n a l l y , the when t h e r e i s  more  3_ R e s p o n s e s  e q u a t i o n s c a n be u s e d  n o t o t h e r s . To d e t e r m i n e perform  both  i s discussed (section 2.5).  response  When a c c e l e r a t i o n  usage  for  C o m p a r i s o n o f M o d e l s D e s c r i b i n q Type  functional  for  s y s t e m s when o n l y one s p e c i e s  types of responses  A number o f d i f f e r e n t  but  developed  A a r e c o n s i d e r e d i n t h e most  suggestions  are  are  than  over  to test  low p r e y  i t may be d e t e c t e d by  the  others,  the  densities.  some  w h e t h e r any o f t h e m o d e l s  whether  models  i n common  Real, F u j i i  and  respectively:  A =  bTN,'0n  (2.1)  n 1 + bhN, '0  ( f r o m E q . 1.5)  be^o™ A =  ( f r o m E q . 1.10), and  'o  1 + be^OhN, 'o  (2.2)  35  g + N + bhN Q  (from  E q . 1.8) were compared  describing defined  a  by  i n terms of  range of s i g m o i d the  feasible  2  Q  flexibility  in  ( t y p e 3) c u r v e s . F l e x i b i l i t y  was  range  of  their  values  for  two  model  attributes: (i)  the  value  possible (ii)  The  value  the r a t i o of  of A a t the i n f l e c t i o n  of the v a l u e s  of these  inflection  is  point  a  relative  second  i s a measure o f t h e  a l l three  calculated  equation d > bh  Eq.  5  of the  the  positioning  rest  2.2,  at  was  the by  o f o n e . Thus Amax  expressions imposed 3  sigmoid  which  ranges for by  response  Ainf  the  (larger  shapes); the asymptote  assigning  was g i v e n  of both  the  of  of the curve  simplified  Feasible  limits  produce a type in  t o 90% a n d 50%  9  speed  value  cases.  the  to  analysis  by d e r i v i n g  determining  corresponding  p r o d u c e more p r o n o u n c e d  p a r a m e t e r , T, a f i x e d in  0  (N o/N o).  measure  o f Ainf/Amax  The  of N  of A  values  approached.  t o t h e maximum  of A (Ainf/Amax) and  t h e maximum v a l u e  first  relative  is the  by  1/h  attributes  were  and  N  9 0  constraint  (i.e. n > 1  in  /N  5 0  and  that  each  Eq.  2.1,  g > 0  i n E q . 2.3, a n d b a n d h > 0 i n a l l  found  to  cases). Equation (Table  2.1 was  2.1). Equation  inflection  p o i n t s above  2.3  is  be  the  unable  most  flexible  to generate  0.25Amax, o r c u r v e s  that  rise  curves  model with  rapidly to  36  T a b l e 2.1. Range of p o s s i b l e v a l u e s of the i n f l e c t i o n p o i n t and t h e r a t i o o f t h e v a l u e s o f N c o r r e s p o n d i n g t o 90% and 50% o f t h e maximum v a l u e o f A (= 1/h) f o r t h e R e a l (Eq. 2.1), Fujii ( E q . 2.2) and H a s s e l l ( E q . 2.3) e q u a t i o n s .  Equation  F e a s i b l e range of v a l u e s of A at inflection  F e a s i b l e range of x = N / N 9 0  Real  Ainf  < 0.50/h  1 < x < 9  Fuj i i  Ainf  < 0.50/h  1 < x <  Hassell  Ainf  < 0.25/h  3 < x < 9  5 0  2.96  37  an  asymptote  (relative  2.2  cannot  an  asymptote.  sigmoid that  generate  responses  If  response  produced  Amax and  produced by  below 0.25Amax, E q . curves.  As  t o the v a l u e  the  9 o /  N  are  5 o  Non-linear determine from  the  how  interactive computer  for  as  of  Similar  fixed,  the  pronounced  When t h e  decreases,  conclusions apply  than  inflection  distinct  to  is  sigmoid  a l l responses  when  Amax  and  ( F i g . 2.3).  non-linear  estimation  algorithms  could describe  parameter  designed  the  by  estimation  by  accessing  a Best  minimizing  the  lettered  curves  in  Figs.  an the  number  purpose.  p r e d i c t e d v a l u e s and  to  generated  monitor,  together  for this  obtained  were u s e d  data  m o d e l s were f i t t e d  d e v i a t i o n s between t h e by  techniques  NLMON, w h i c h l i n k s  of  estimates sums  the  2.2  of  "data", and  2.3  2.2). sigmoid  (by d e f i n i n g  a high  tended  smoothed out  t o be  When  and  adjustment routines  shapes g e n e r a t e d  inflection  the H a s s e l l  parameter v a l u e s , cases  p o i n t are  most  Equation  slow a p p r o a c h  more  the  inflection  parameter  Pronounced  c).  is  ( F i g . 2.2).  e q u a t i o n s . The  represented  (Table  2.1  generates  t h e p a r a m e t e r s were  squared  Eq.  2.3  p r o g r a m UBC  different  extremely  inflection  w e l l each e q u a t i o n  other  a t 0.5Amax).  0  the  2.2  point  fixed  w i t h an  Eq.  become more e l o n g a t e d . N  by  of N  by  other was  never  i n parameter v a l u e s , each resulted  N  equations  9 0  /N  equation  bad  (Figs.  2.4a  fit  and  positive in  attained. Following iteration  ratio)  5 0  c o n s t r a i n e d t o have  i t provided a p a r t i c u l a r l y  t r u e c o n v e r g e n c e was  the Real  p o i n t or a s m a l l  the  equation  from  these initial  of t h e o p t i m i z i n g  i n a l a r g e i n c r e a s e i n the e s t i m a t e s  of b  and  38  Figure (Eq.  2.2.  Curves  2.2) a n d H a s s e l l  inflection  point  generated  from  the Real  ( E q . 2.3) e q u a t i o n s when Amax a n d t h e  are fixed.  I n a l l c a s e s Amax = 10 a n d N i n f  (number o f p r e y p r e s e n t a t i n f l e c t i o n ) generated  f o r four  (Eq. 2.1), F u j i i  different  30%, 20% a n d 10% o f Amax).  = 30.  Curves  were  v a l u e s of A a t i n f l e c t i o n (40%,  Real Fujii Hassell  NUMBER OF PREY P R E S E N T to  40  Figure  2.3.  Curves generated  ( E q . 2.2) a n d H a s s e l l ratio  N  9 0  /N  5 0  figures  2.3) e q u a t i o n s  (prey d e n s i t i e s  Amax) a r e f i x e d . two  (Eq.  corresponding  I n a l l c a s e s Amax = 10.  with the lowest  the c u r v e s w i t h  from t h e R e a l  the highest  N  9 0  /N  5 0  ratios.  ratios  (Eq. 2.1),  Fujii  when Amax a n d t h e t o 90% a n d 50% o f N  5 0  and N  = 50 f o r t h e 5 0  = 30 f o r  N U M B E R  O F P R E Y  E A T E N  P E R  P R E D A T O R  42  Table 2.2. P a r a m e t e r e s t i m a t e s a n d sums o f s q u a r e d d e v i a t i o n s about p r e d i c t e d values (sse) for Real (Eq. 2 . 1 ) , F u j i i (Eq. 2.2) and H a s s e l l ( E q . 2 . 3 ) e q u a t i o n s . C r e p r e s e n t s the facilitation c o n s t a n t s : r e s p e c t i v e l y , n , d and g i n each of the equations. Estimates for E q . 2.3 were obtained s e p a r a t e l y for the case where the p a r a m e t e r s were c o n s t r a i n e d t o be positive (+) and w h e r e t h e y w e r e a l l o w e d t o become n e g a t i v e ( - ) . Generated data c o r r e s p o n d t o t h e l e t t e r e d c u r v e s i n F i g s . 2.1 and 2 . 2 .  Equation data g e n e r a t e d from  1. R e a l b = 2.743X10" C = 5.000 h = 0.100 sse  3.  Fujii  5.260x10" 0.123 0.101 0.461  7  2. Fujii b = 0.065 C = 0.041 h = 0.100 sse  b  Real  2.244X10"  Hassell (+)  3  2.591X10 2.843X10  1  Hassell (-)  1  1 3  0.088 22.877  2.674x10'*  3  2.362 0.095 0.848  3.506X10  1 6  0.091 1 .689  • 0.147 •42.801 0 . 137 5.464 • 0.446 •85.837 0 . 106 0.397  Real =  6.212X10"  C = 5.419 h = 0.100 sse  9  2.040X10"  0.090 0.101 0.476  3  2.463X10 3.069X10  1 f l  0.074 31.267  1 6  • 0.053 •52.867 0.187 5.495 cont.  43  Table  2.2  cont.  Equation data g e n e r a t e d from  4. F u j i i b = 5.556x10C = 0.0717 h = 0.100 sse  Real  2.649X10" 4.476 0.098 0.645  5. R e a l b = 0.1 54 C = 1 .226 h = 0.100 sse 6. H a s s e l l b = 0.556 C = 20.000 h = 0.100 sse  Fujii  Hassell (-)  1.612xl0 «  7  1  4.738X10  0.077 19.811 0.254 0.018 0.111 0.076  0.061 1.514 0. 106 0.042  Hassell (+)  0.194 0.021 0.114 0.512  0.370 4.539 0.095 0.033  1 6  • 0.062 •54.256 0. 174 1 .564  44  Figure Hassell  2.4.  Best  curves  indicate in Figs.  indicates and or the  (-)  by R e a l  ( E q . 2.3) e q u a t i o n s  f r o m one o f t h e s e figures  fits  that  equations that  solid  2.2 and 2.3.  (Eq. 2.1), F u j i i  (broken (solid  l i n e s ) to data lines).  l i n e s correspond  generated  Letters  on  to l e t t e r e d  For the H a s s e l l  equation,  (+)  t h e p a r a m e t e r s were c o n s t r a i n e d t o be p o s i t i v e  indicates  that  when t h i s c o n s t r a i n t  more p a r a m e t e r s became n e g a t i v e . fitted  ( E q . 2.2) a n d  curves are given  i n Table  was removed, one  Parameter e s t i m a t e s f o r 2.2.  45  0  50  100  NUMBER OF P R E Y P R E S E N T  150  46  NUMBER OF P R E Y  PRESENT  10  NUMBER OF P R E Y  PRESENT  48  g f o r a minute r e d u c t i o n i n the the  parameters  fit  was  would by  were  obtained, predict  the data)  allowed  but  the  sums of  squared  deviations.  t o become n e g a t i v e , a much  resulting  response  was  "type  a continuing decline in predation  i f prey  densities  were  (not  extrapolated  If  closer 4"  and  indicated  beyond  the  f i t to  data  point  was  with, a  low  data. The  Hassell  generated high  from  (Fig.  inflection produced  2.4d)  fitted.  to  responses  although  ignoring  The  and  the  Fujii  time  fitted  each o t h e r ' s data  flexible  By  suggests  although  type  stability  (e.g. H a s s e l l  is  the  of  r a n g e of  sigmoid  ignored  when  much the  equation  the data  towards  responses higher Real  Hassell  least  i t has  and  the this  and  than  to which  by  asymptote by  totally  estimating  i t s actual  Hassell  equation  distinct  value  equations  f r e q u e n t l y been  and  for  these  the  objective  least  shapes.  This  recommended f o r population  e q u a t i o n may Although  not  the  i n terms of d e s c r i b i n g a  other is  the  investigating  purposes.  flexible  t o be  sigmoid  1978), t h e H a s s e l l  responses,  the  shapes than  Real  a p p a r e n t l y p r o v i d e d a good f i t  the  the  for  most  the  well.  3 responses  t h e most u s e f u l model equation  be  shows  describing  well  However, i t a c c o m p l i s h e d  to produce the  that  bad  inflection  cases,  increases  comparison,  analysis and  equation  to  2.2).  a  performed  sigmoid  nature  (Table  The  when t h e  it  slow  f).  sigmoid  handling  provided  ( F i g . 2 . 4 b ) . In b o t h  with very  2.4e  also  equation  more a c c e n t u a t e d  was  the  the F u j i i  point  it  (Figs.  equation  to  equations detect  should  Real wide  not  the presence  be  be of  a  49  sigmoid  shape. C o n v i n c i n g e v i d e n c e  obtained  if  resulting  estimates  found  to  for  a  type  a l l three equations are f i t t e d  be  baseline  levels.  slight,  i t may  significantly  I f the tendency be  detected  ( n , d and g)  different  towards a type by  one  response  is  t o t h e data and the  f o r the sigmoid parameters  statistically  3  3  equation  from  are their  response  is  b u t n o t by t h e  others.  2.3 S i m p l e  Formulations and  This  section  effective  search  descriptions prey  i s concerned produced  need  i n space  predator  to and  response  take those  that  allow  in  descriptions  of that  (parasitoids) have  lead  into  (Hassell  particular binomial  and  probability to  the  represent  Murdoch and O a t e n  of  assignment May  1973),  to  have  rates  of  prey  prey  rates  the  and  prey are  behaviour  to  locate  represent  usually  of  and them.  spatial  tractable  way.  consisted  of  ( h o s t s ) and o f t h e  aggregation  of  predators  h i g h p r e y d e n s i t y . Some m o d e l s of  the  while  distribution the  prey  yet analytically  distribution to  how  predators  dynamics  regions  involved arbitrary  areas  1974,  the  of  difficult  meaningful  t o model s p a t i a l  encounter  account  aspects  heterogeneity  a  between  into  i t i s extremely  mechanisms  with modelling variable  v i a pathways A ( F i g . 2 . 1 ) . R e a l i s t i c  Unfortunately,  Attempts  i n Predator-Prey  P a r a s i t o i d - H o s t Systems  of the r e l a t i o n s h i p  density  dispersed  f o r Non-Random S e a r c h  such  prey  others as  distribution  1975). The p r o b l e m  here  to  different  have  used  a  the  negative  (Hassell  a n d May  i s that  when t h e  50  prey d i s t r i b u t i o n result many  is  is overlaid  often  an  overly  distribution  to  distribution  and  example,  complex and  the  predator,  intractable  in  the  the  overall  outcome  of  predator  response and  Holling  situations  where  is overdispersed.  grounds  that  they  mechanisms i n v o l v e d The  do  so  relating  predator-prey  and  simplicity,  the  model  with  hosts  the  parasitoids,  i d e n t i c a l to  prey  a  similar  to  derive  probability  for  of  the  hosts  on  the  underlying  (Royama  the  For  parasitoid  t y p e s of  1971).  from  first  components  of  Criteria  of  tractability d i s t r i b u t i o n of  demonstrated binomial  p r o p o s e d by  formulation  use  criticised  models  to  it.  number o f  interactions.  negative  that  the  analytical  is a  be  prey  that  prey  certain  formulation  Griffiths  predators  were  and  which  for  Holling accounts  depletion.  method o f  Methods variations  lead  it  to the  descriptive  parameters  and  assumed,  can  ( 1 9 6 9 ) , and  to  particular  assumptions  equation"  purely  the  both  d i s t r i b u t i o n of  distinguish  are  flexibility  was  represent  probability of  (1969) p r o p o s e d  parasitoid-host  a d o p t e d . A l t h o u g h no  standard  Such m o d e l s can not  and  a  the  the  a p p r o a c h u s e d h e r e was  principles,