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Bird flocking as a foraging strategy Thompson, William Andrew 1974

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BIRD FLOCKING AS A FORAGING STRATEGY  '  by  W I L L I A M ANDREW B.A.,  Pomona  C o l l e g e , 1967  A D I S S E R T A T I O N SUBMITTED THE  REQUIREMENTS  THOMPSON  IN P A R T I A L  F U L F I L M E N T OF  FOR THE DEGREE OF  DOCTOR OF P H I L O S O P H Y  Interdisciplinary  Studies  in Mathematical  We a c c e p t required  THE  this  Ecology  t h e s i s as c o n f o r m i n g  to  standsr-d  U N I V E R S I T Y OF B R I T I S H November, 1974  COLUMBIA  the  In p r e s e n t i n g  this  thesis  an advanced degree at the L i b r a r y  shall  I f u r t h e r agree  in p a r t i a l  fulfilment  the U n i v e r s i t y of  make it  freely  t h a t permission  British  available  for  of  the  requirements  Columbia, reference  for e x t e n s i v e copying o f  I agree  for  that  and study. this  thesis  for s c h o l a r l y purposes may be granted by the Head of my Department or by h i s of  this  written  representatives. thesis for  It  financial  is understood gain s h a l l  permission.  Department of The U n i v e r s i t y of B r i t i s h Vancouver 8, Canada  °a e t  /  3  Columbia  Jh*. I9?V  that  not  copying or p u b l i c a t i o n  be allowed without my  ABSTRACT  To a s s e s s model was primary and  the s u r v i v a l  constructed  source  data  of bird  upon  known  of information  on  small  restricted  to  flocks  Simulation  experiments  the  on f e e d i n g  success  parameters  (sensitivity  distribution, in  variation  behavioural  in  the  ratio  of  ways:  probability  of  under may  seriously  examine  as  mean  The l a t t e r  long-run  model  bird's  that (e.g.  a  two  parameter  mean d e n s i t y and  rate  may  prey  over  types'  Success 2)  more  of  birds  variation  as  i n twenty  methods  prey  and  the  appropriate feeding  survival.  were  was  simulated  day's inadequate  developed  To to  model.  distribution  characterization variance  in  and  be  likelihood  varying  behavioural  types.  no c a p t u r e s  behaviour,  experiments  prey  capture  a p p r o x i m a t e t h e s i m u l a t i o n w i t h a Markov  Simulation  i n the  prey  examine  i n number o f ' p r e y  c o n d i t i o n s when a s i n g l e diminish a  to  d i f f e r e n c e s between  two  measure  applicability  performed  populations),  for  a b i r d ' s making  ('risk') . winter  abundance 1)  size,  prey  confounding  variation  variation  Polymorphic  measured two  minutes  i n flock  the  theory  on low d e n s i t y  of v a r i a t i o n  analysis),  parameters,  (monomorphic v s .  were  The  published  avoiding  foraging  a simulation  behaviours.  and b r e e d i n g ,  populations. effects  bird  By  (on t h e b i r d s ) winter  flocking,  f o r t h e model was  passerines.  f a c t o r s of p r e d a t i o n is  based  value  of  mean  a prey  indicated population  density)  is  not  iii  adequate  to  importance until  to  of  the  e s t i m a t e the  'giving-up  bird  leaves  success.  Flocking  to  clumped.  highly  when  prey  population  were  (the  clumped  the  two  of  monomorphic,  flocking  or  time  immediate  'risk  1  Flocking also  means  rate.  the  reduced  necessarily  capture  time*  sensitivity  and  from l a s t  foraging  whenever  prey  enhanced  the  polymorphic  more  same s p e c i e s ) . neither  analysis  major  increased  prey  area) were  the  capture  to  feeding  moderately  mean c a p t u r e  rate  (a p o l y m o r p h i c  prey  types  When t h e  revealed  of  prey  prey,  not  population  was  nor decreased  the  mean  TABLE OF CONTENTS  CHAP.  1  INTRODUCTION  CHAP.  2  BIRD FLOCKING AND ITS EEHAVIGU RAI  CHAP.  3  SIMULATION  CHAP.  4  THE SIMULATION  CHAP.  CHAP.  5  6  1 COMPONENTS  METHODOLOGY  9 21  MODEL  30  Habitat  34  Movement  36  Feeding  43  THE SIMULATION  EXPERIMENTS  49  Sensitivity  analysis  49  Mcnomorphic  prey  populations  5.6  Popymorphic prey  populations  78  MARKOV MODELS  105  Methodology Identification E s t i m a t i o n of  Validation  105 of  states  transition  108 probabilities  111  122  V  Results CHAP.  7  CONCLUSIONS  133 145  REFERENCES CITED  153  OTHER REFERENCES  164  APPENDIX  A  166  APPENDIX B  177  APPENDIX C  183  APPENDIX D  187  APPENDIX  E  190  APPENDIX F  193  V1  TABLES  I  B e h a v i o u r a l components  of a search  II  Parameter values experiments  III  S e n s i t i v i t y a n a l y s i s to t e s t variation alters results  in  the  33  simulation  51  parameter  52  IV  Sensitivity analysis to test whether g i v e n parameter v a r i a t i o n produces g r e a t e r v a r i a t i o n in results  53  V  Prey c h a r a c t e r i s t i c s  58  VI  Mean c a p t u r e e n t i r e day's  r a t e per hour feeding  VII  Capture rate  given  VIII  Risk  IX  Designs  X  Mean capture rates polymorphic prey  XI  R e d u c t i o n i n c a p t u r e r a t e by t h e p a r t i t i o n a p r e y p o p u l a t i o n i n t o two morphs  XII  Mean capture rates for various f e e d i n g cn p o l y m o r p h i c p r e y  XIII  Risk for birds polymorphic prey  given  used  strategy  whether  averaged  over  an  individual variation  76  individual variation  77  of p o l y m o r p h i c p r e y e x p e r i m e n t s  feeding  cn  83  itononiorphic  on  73  fleck  moncmcphic  and  84  of  86  si2es  87  and  89  via  on  90  p r e d a t i o n s u f f e r e d by the l e s s morphs when the morphs a r e in  92  XVI  Apportionment o f p r e y c a p t r u e s between a b o v e and b e l o w - a v e r a g e b i r d s ( b i r d s d i f f e r i n Frey detection abilities)  93  XVII  Apportionment of risk between abovebelow-average birds (birds differ in detection abilities)  and prey  94  XVIII  Capture rate proportions 1:1 percentage of monomorphic p r e y  polymorphic prey in and 9:1 expressed as capture rate on similar  96  XIX  P e r c e n t a g e , o f a t t a c k s on prey dimorphic prey p o p u l a t i o n f o r and 9:1  type 1 in a prey r a t i o s 1:1  97  XX  C a p t u r e r a t e p e r i n d i v i d u a l p r e y on each mcrph in a dimorphic population expressed as percentage of capture rate on similar monomorphic p o p u l a t i o n  98  XXI  Risk prey  99  XXII  State space for  XXIII  Transitions models  XXIV  Chi-squared t e s t s that t r a n s i t i o n counts " q u a s i - i n d e p e n d e n t " f o r t h e Markov models  XIV  Risk for various polymorphic prey  XV  F r a c t i o n of the a t t a c k e d o f two e q u a l numbers  sizes  feeding  on  for d i f f e r e n t population  and  flock  prey r a t i o s  the  Markov  their  in a  dimorphic  110  models  bounds  for  the  Markov  are  112  124  viii  XXV  Validation of Markov models t r a n s i t i o n p r o b a b i l i t i e s with transitions  of of  126  XXVI  V a l i d a t i o n o f Markov models by a c o m p a r i s o n o f predicted s t a t e r e s i d e n c e t i m e s with a second set of s t a t e residence times  127  XXVII  Asymptotic capture Markov models  rates  as  140  XXVIII  Dominant  of  Markov  XXIX  Variance over mean t h e Markov models  XXX  Capture various Figures  eigenvalues  the  by c o m p a r i s o n a second s e t  given  by  the  141  models given  by  143  r a t e s and r i s k , monomorphic prey levels of clumping (corresponds 4-6)  at to  178  XXXI  Mean c a p t u r e r a t e s f o r various flock sizes, prey i n t h i r t y - s i x clumps of t w n e t y - f i v e prey e a c h ( c o r r e s p o n d s t o F i g u r e 7)  179  XXXII  Mean c a p t u r e r a t e s f o r various flock sizes, prey in twelve clumps of s e v e n t y - f i v e prey e a c h ( c o r r e s p o n d s t o F i g u r e 8)  180  XXXIII  Risk f o r v a r i o u s f l o c k s i z e s , prey i n thirtysix clumps of twenty-five piey each ( c o r r e s p o n d s t o F i g u r e 9)  181  XXXIV  Risk f o r v a r i o u s f l o c k s i z e s , prey in twelve clumps of s e v e n t y - f i v e p r e y each (corresponds t o F i g u r e 10)  182  XXXV  Estimated transition f l o c k i n g birds feeding  191  capture  rate  as  probabilities on monomorphic p r e y  for  ix  XXXVI  Estimated nonflocking  transition birds feeding  probabilities cn monomorphic  for prey  192  X  FIGURES 1A  Search paths minutes  IB  Area covered searching for  2  Probability density s e a r c h movements  3  Flock determination PFLOK  4  Mean capture clumping  5  V a r i a n c e o v e r mean c a p t u r e of prey clumping  6  Risk (probability of minutes) as a f u n c t i o n  of  three  by five  birds  searching  a flock minutes  rate  of  function  for  as  nine  governing  3 8  birds  3 9  random  4 0  of  4 4  prey  6 0  function  61  z e r o c a p t u r e s i n twenty of prey clumping  6 2  different  a  values  function  rate  as  a  cf  Mean c a p t r u e r a t e for various flock sizes, prey i n t h i r t y - s i x clumps of t w e n t y - f i v e prey each  6 5  Mean c a p t u r e r a t e prey in twelve each  6 7  for various flock sizes, clumps of s e v e n t y - f i v e p r e y  Risk f o r v a r i o u s f l o c k s i z e s , prey i n s i x clumps of t w e n t y - f i v e prey each  1 0  five  for  Bisk for c l u m p s of  v a r i o u s f l e c k s i z e s , prey i n s e v e n t y - f i v e p r e y each  thirty-  twelve  6 9  70  Mean c a p t u r e r a t e p e r b i r d a s a t i m e f o r v e r y h i g h l y clumped p r e y Mean capture rate per b i r d t i m e f o r h i g h l y clumped p r e y  as a  function ( C = 9 . 4 8 )  function  ( C = 5 . 2 4 )  Mean c a p t u r e r a t e p e r b i r d a s a function t i m e f o r r a n d o m l y d i s t r i b u t e d p r e y (C=0.07)  xii  DIAGRAMS  (APPENDIX A)  1  Subroutine  2  S u b r o u t i n e FLY  3  S u b r o u t i n e FLOGG  4  S u b r o u t i n e EAT  5  Subroutine  6  Main program  TMSTP  IMTATE  xiii  MATHEMATICAL NOTATION Owing document been are  to  has  the  been  adopted  produced,  with  explained  limitations  the  certain  r e g a r d to  below,  of  along  the  p r i n t e r cn  special  mathematical  with  some  which  this  conventions  have  notation..  These  possibly  unfamiliar  notations.  Summation i s  For  ***  i n d i c a t e d by n *** ®  example,  «  *** {Ai}  A 1+A2 + . . . + A n .  =  *** i=1  ***** Multiplication  is  indicated  * * * *  by  n For  example,  ***** * *  {Ai}  A 1*A 2 * . . . *A n,  =  * * i= 1  The  element i s  For  8  symbols or i s  example,  while,  Matrices  of  not  and  $  have  been  used t o  indicate  that  included in a set. ,  XS[0,1]  implies  t h a t 0<X<1,  X£[0,1]  implies  that  numbers a r e  s u r r o u n d e d by  either  |•s.  X>1  or  X<0.  an  xiv  Thus,  .2 .1 .5  .81 .6 .1|  represents  a 3x2  matrix.  1  CHAPTER QUE: The  INTRODUCTION  survival  value of f l o c k i n g  extensively  in  Goss-Custard  1972; P u l l i a m 1973; K r e b s 1973a).  have  proposed  existing  with  this  study  passerine  BS22£)  were  which  1970).  Typically,  season  and f e e d i n  latter  period  the b i r d s severe winters the  winter  case -.  flock  -  The  •  •  theory  (Hinde  through  in  sizes  1952).  the  of  from  tend  to  New  t i t  cold (Parus  England  (Morse  i n the breeding During  this  p r e d a t i o n p r e s s u r e on be n o t e d  larger,  nest s i t e s  in  in  1952) and t h e b l a c k -  i t should  be  T h i s appears  flocking  great  winter.  t o be l i t t l e  In a d d i t i o n ,  birds maintain  (Hinde  the  atricapillus)  flocks  food.  concepts  overwintering  are  i n England  there.appears  f o r examining  success.  has  some  and  these b i r d s are t e r r i t o r i a l  (Morse 1 9 7 0 ) .  winters  to i n c r e a s e the  distributed  value  while  R e p r e s e n t a t i v e examples  (Parus  authors  construct a simulation  survival  flock  which o v e r w i n t e r s  to  (e.g. Crook S  in birds.  f o c u s e s on t h e  capped c h i c k a d e e  patchily  information  discussed  Various  individuals  synthesized  behaviour  birds  climates.  enables  proposition,  model o f f l o c k i n g  i n recent years  which t h e y e x p l o i t  studies  The  literature  that flocking  effectiveness To examine  the  i n b i r d s has been  that i n  while  in  and t e r r i t o r i e s  to provide  the hypothesis that f l o c k i n g  an  mild  through  excellent  enhances  feeding  • J  of f o r a g i n g or predation s t r a t e g i e s  been a p p r o a c h e d i n s e v e r a l ways.  Here we w i l l  be  of  animals  concerned  2  with  two  of  these:  one  selection  of a p r e d a t o r  Schoener  1971;  other  1959a,b, the  H a c A r t h u r 1972;  parts 1965,  is  while  pursue  (or e a t )  rules  are  the  The  a given  combinations of d i e t  with  two  f o r a g i n g : 1) prey  predators attempt  t o maximize t h e  However,  objective  other  example,  maximizing  herbivores derived the  this  expected  yield  pursue  unit  handling  time)  for  not  that given  handling  1)  per  the  times  One  rate  of  whether  be  nutrient  that gain.  plausible; gain  effort prey  should  for  for large  (e.g.  which  effort  effort  .=.  major i m p l i c a t i o n  the  =  the average y i e l d  (e.g.  search  type  of  prey  and  2)  the  expected  p u r s u i t time per  search  of  are  where  i s g r e a t e s t ; and for  to  Decision  predator  effort  each  2)  caloric  +  unit  +  effort  pursuit  +  these'decision rules  knowledge o f t h e a v e r a g e s e a r c h , for  ("predation  assumption  net  may  that  the  (eat) any  yet l o c a t e d  assumes  1  and  the  (Holling  common s e n s e d e c i s i o n r u l e s  unit  further  upon  of  its  Two  i s g r e a t e r than  handling , times). is  theory:  yield  should  per  prey  rate  into  been l o c a t e d .  functions  (Westoby 1974).  from  predator  the  selection  (e.g. the  of t h e s e  where t o s e a r c h  based  diet  1974);  down  decisions  o n c e i t has  usually  Pulliam  process  theory  the  principles  1973;  predation  faced  derived,  Charnov  studying  1966).  predator  decisions")  and  involves predicting  u s i n g some o p t i m a l i t y  involves breaking  component  approach  pursuit  the average  and  caloric  i No i m p l i c a t i o n o f c o n s c i o u s d e c i s i o n making i s i n t e n d e d h e r e . A b i r d which i s "programmed" t o r e s p o n d t o s p e c i f i c stimuli in s p e c i f i c ways may n o n e t h e l e s s be c o n s i d e r e d a s a d e c i s i o n maker.  content  of  each  type of  prey  types  set';  C h a r n o v 1973)  always  into  try  predator  to  are  prey  and  predator's  level  of  much  more  the  this  shortcoming  of  the  optimal  hunger.  this  decisions.  uncertain  and o f t e n  foraging  theory  behaviours  optimal  operational  of  of  the  forager  the  this  it  provides  an  to  prediction  is  becomes  1971).  points  cut  the  must  a  In major  assumption  information  to  forage that  .idealized the  into the  appears  examine  the  with  predator  theory,  world,  should  which  change  (Schoener  necessary  'optimal  prey  latter  (1974)  the  breakdown  not  a satiated  choices  (the  predator  those  i n d i v i d u a l animals  which  theory  provides  should  do,  may  terms  does t h e  predation  predator, 1965,  with  functional  it  achieve  and s e l e c t  1959a,b, the  that  divide  make in  an  optimal  standard  actual  of  foraging  animals.  search  approach to  best  against  which an a n i m a l  which t o  this  changing  While f o r a g i n g an  However,  predation  Since  at  are  should  Pulliam  f o r m of  the  prey  its  failure,  category  which other  should  Futhermore,  known e v i d e n c e in  I n one  i n the  i n d i v i d u a l a n i m a l has  performance  predator  prey  ignored  selective  discussing  that  those  always i g n o r e .  preferred  to  the  two c a t e g o r i e s .  capture;  should  contrary  prey,  is  1966).  an i d e a l i z e d  does n o t  these  concept of  consider  ends.  the  That  the  theory,  prey  to  which  Boiling's  component been  a predator  the  and e a t ?  does c o n s i d e r  T h i s work has  response of  pursue  the  analysis concerned  to  prey  means by  is,  i n d i v i d u a l animal s e l e c t  what  how  in  area  in  A second behaviour (Holling primarily  density,  ie.  a  the  changes  density. are  in  The  capture  rate  components which i n f l u e n c e t h e  (for each prey Rate o f  successful  2.  Time o f  exposure  3.  Handling  4.  Hunger ,  5.  Learning  6.  I n h i b i t i o n by  7.  Exploitation  ( d e p l e t i o n of  8.  Interference  between  9.  Social  10.,  avoidance these  the  predator) prey prey)  predators  l e a r n i n g by  components, will  the  prey. ,  learning,  be  of  primary  a foraging  on  the  social  rate  of  time  and  hunger  explicitly,  they  are  held  I t i s acceptable  because the  result  short  simulation  to  of the  effects  become  assume fixed  satiated.  only  enough t o p r e v e n t  process. be  all  the  have  a  successful search.  Time  of  will fixed hunger  not  is  to cover a  In t h e as  be  i n the  considered  bird flocking constant  short  which t i m e t h e  Exploitation  local,  and  three  minutes), during  predation will  to  runs are  (twenty s i m u l a t e d  unlikely  its  hence  facilitation  interest in considering  strategy.  exposure, handling  time  decisions  facilitation  influence  model.  p^rey  search  the  o f f l o c k i n g as  major  predation  in  time  (by  interference value  changes  type):  1.  Of  accompanying  period  of  birds  are  i s an i n e v i t a b l e  simulation the  only  experiments  time h o r i z o n  s e r i o u s e x p l o i t a t i o n of the  i s kept  entire  prey  5  population. not  be  Finally,  determination  importance  be e s t i m a t e d  1.  Prey  density  2.  Prey  distribution  fixed  the estimated  perceptual  4.  Predator  learning  Predator  7.  exploitation prey  will  call  choice  prey - e . g .  1960; Dawkins  microhabitat  set  o f t h e prey  a  of  prey  -  search  1971a,b) e.g.  "prey  t u r n i n g r a t e and  predators. of encounter  r a t e s may  (ignoring other  Thus,  (we  predation  secondary  has  o f change i n  o f d e c i s i o n v a r i a b l e s and d e c i s i o n s ' t o make  these  predator  due  an  "secondary  the  sources  be b i a s e d  available  these  amongst  other  estimates  population).  additional  has  1970)  I n t e r f e r e n c e with  the  predator  time  In a d d i t i o n , p a s t to  that the  movement p a t t e r n - movement r a t e ,  giving-up  Factors  rates are:  l e a r n i n g t o d e t e c t camouflaged  (Royama  These  prey  (de R u i t e r 1952; T i n b e r g e n  niches"  the  field)  Conspicuousness of  Predator  type.  past experiences.  encounter  ( i t i s assumed  3.  image  r a t e s f o r . e a c h prey  from t h e p r e d a t o r ' s  alter  6.  will  i n making p r e d a t i o n d e c i s i o n s i s  of the encounter  which may  5.  and l e a r n i n g by t h e p r e y  considered.  Of c e n t r a l  must  inhibition  decisions").  And  predation decisions w i l l  the  have a  major i n f l u e n c e on t h e v e r y i n f o r m a t i o n used t o make t h e  primary  predation  decisions.  general  decision  process,  i t  Given  the  complexity  a p p e a r s most  profitable  cf  this  t o examine known  6 behaviours  of a  predator  uncertainty  a r e made and  in  may  flocks  (1973b) this  these  indicated  review  these  social  of  and  its  their  behaviours)  decisions  interactions  the  will  flocking  The  provide  pooled  decision  under  which  occur  making.  behavioural  Krebs  aspects  of  selected  must  prey  decisions  to  select.  may  be  of  influenced  by  valuable  (manifested  behaviours  experiences.  be  All  its.decisions  i n f l u e n c e d by  i t s own  make d e c i s i o n s  It  in  is  terms  i n f o r m a t i o n of the f l o c k  of at  near this  of f o r a g i n g  members i s  likely  a b e t t e r b a s i s f o r making p r e d a t i o n d e c i s i o n s .  from  interference. expect  birds  which  be  may  to the i n d i v i d u a l  realized  the  i s in a flock,  as w e l l as by  that  success.  costs  incumbent  When a b i r d  neighbors point  previously  where t o s e a r c h and  learning.  can  the  how  problem.  regarding  to  how  see  t o more e f f e c t i v e  p r e s e n t s a good  As  in  lead  to  of being  more  rapid  likely  exploitation  Thus, under d i f f e r e n t  environmental  for.  In t h i s  forms of s o c i a l ,  thesis  conditions  under  individual  foraging bird.  2  are  prey  that d i f f e r e n t  Chapter  In a f l o c k  which  I will  flocking  will . discuss in  a  of  general  context  reference  to  the  prevailing  i d e a s r e g a r d i n g the s u r v i v a l  following  net  behavioural  problem  topics:  an  be  conditions  we  may  be  to e l u c i d a t e those  the  a s p e c t s of the  to  greater  organization  attempt  is  and  The  benefit  and with  examination  to  the  ecological particular of  the  v a l u e o f f l o c k i n g ; and  a  7  discussion  of  other  components d i r e c t l y Chapter  3  investigations relevant to this  will  m e t h o d o l o g y and w i l l applied  including  the  built  to  in  subroutines  sources  examine  flocking  survival  the value  and  the  value  range  charts  of f l o c k i n g  third  of  and p a r a m e t e r s major  computer  great  tits  of  experiments w i l l  be  ones  when p r e y  investigate  populations are the  value  of value  a r e measured i n t e r m s o f t h e i n d i v i d u a l  bird,  these  t o p i c s , the study  d e n s i t i e s such a s those  passerines  wintering  overwintering  i n cold  features  and employed  transient  will  be  l i k e l y to climates  i n England).  d i s c u s s t h e a p p l i c a t i o n o f Markov  special  be d i s c u s s e d  results  Throughout, the terms  t o low a v e r a g e p r e y  and  behaviours  the simulation experiments,  I n examining  Chapter 6 w i l l  will  Flow  are polymorphic.  of the f l o c k .  analysis  be  ones i n v e s t i g a t e p a r a m e t e r s e n s i t i v i t y ; t h e  be a v a i l a b l e t o s m a l l (e.g.  may  models.  Three groups of s i m u l a t i o n  when p r e y  restricted  o f the v a r i o u s  present  the f i r s t  monomorphic;  net  problems of t h e s i m u l a t i o n  a d e t a i l e d d e s c r i p t i o n o f t h e model,  model.  5 will  discussion.  second  and  major  w i l l be i n c l u d e d .  Chapter  discussed:  predation  d i s c u s s some a n a l y t i c methods which  present  the  the  study.  to i n t e r p r e t a t i o n of simulation  Chapter 4 w i l l  and  review  concerning  feeding  of simulation to  examine  trends.  models. questions  Topics  will  models f o r The method of  long  include:  8  estimation derived Markov  of  transition  transition  probabilities;  matrix;  and  model t o o b t a i n i n s i g h t  analytic  into  long  validation  of  the  m a n i p u l a t i o n s of the run  and  short  term  behaviour. Finally, interesting  chapter results,  investigation, effectiveness  and  7  will  make provide  summarize  some a  some  of  suggestions short  fcr  commentary  o f t h e methods e m p l o y e d i n t h e s t u d y . ,  the  more  future on  the  9  CHAPTER  TWOj. BIRD FLOCKING AND I T S BEHAVIOURAL  This chapter flocking. will the  Emphasis  later  be d e v e l o p e d  birds  feeding  discuss  will  goal of t h i s study  enhance  two  will  the  i s the examination  to  winter  (Morse  1970).  a  review  be  little  prey  movement,  learning  The for  over  •flock  1  aggregation between  Crook  be of  1965), o n l y  and 2)  during the  birds  of  prey  has i n t e r e s t e d 1922).  capture time,  mean with  a  foraging  some  the  birds.'  of  the flock,  social  major  word an  attraction  are generally  movements.  by s e v e r a l a u t h o r s  a short discussion  ecologists  Here  In o p e r a t i o n a l terms f l o c k s  reviewed  a  the e f f e c t s of  predators, giving-up  Miller to  of  the  learning).  (eg.  foraging  will  1 as r e l e v a n t to  the e f f e c t s  flocking  restricted  been  on  concerning  These.include  between  of bird  by t h e c o h e s i o n  has  i n Chapter  rate,  (social  a century  individuals.  identified subject  interference  adaptiveness  will  justified  and f o r a g i n g s u c c e s s s a r e  investigations  flocking.  on c a p t u r e  and i m i t a t i o n  half  p r e d a t i o n on t h e  p r e d a t i o n on t h e b i r d s  previous  bird  distribution  of  may  Following the discussion cf flecking  of  of  which  flocking  T h i s e x c l u s i o n appears  b e h a v i o u r a l components i d e n t i f i e d consideration  hew  bird  Since  (with t h e e x c e p t i o n d i s c u s s e d b e l c w ) ;  appears  be  of  and e v i d e n c e  of  considerations  g r o u n d s : 1) p r e d a t i o n on t h e b i r d s independent  value  i n t o a model o f a f o r a g i n g f l o c k .  success,  there  on  survival  be on t h e c o n c e p t s  have been e x c l u d e d .  largely  COMPONENTS  (eg. ideas  As  the  Rand 1954; will  be  10  given  here.  reducing  Flocking  the  increasing  bird's  the  investigators 1971;  bird's  success  (Lack.1968;  i s selected  means  enhance f e e d i n g  in  predation,  foraging.  Goss-Custard  or  While  1970; V i n e  1972; P u l l i a m  o n l y i n so f a r  have  2)  by  various  1971; H a m i l t o n  1973) have a r g u e d  as  i t  been i d e n t i f i e d  that  directly  affects  watching  away from the  spent  sacrificing  in  Murton,  solitary  actively similar  a  a  p r e d a t o r s i s time  taken  f o r the b i r d s t o reduce activity  the  actions  "looking  Hence,  foraging.  Powell  (of  these noted  around"  the . b i r d s foraging  in time  (1974) d e m o n s t r a t e d  noting also that  a  t h e mean  a model hawk was s m a l l e r f o r t h e 10)  than  for  p r e d a t i o n on t h e b i r d s  t h e model. , H e n c e , t h i s h y p o t h e s i s w i l l  Copy  (1971) have  more t i m e  i n flocks.  without  for detecting  greater proportion of their  flock  earlier,  time  I s a a c s o n & Westwood  time t o . s i g h t i n g  indicated  may  Since  necessary  phenomenon f o r s t a r l i n g s ,  in  flocking  predators. ,  already present  those  engaged.in  reaction birds  this  wood-pigeons s p e n d  (head r a i s e d ) t h a n spent  for  i t i s desirable  the a b i l i t y  predators.  flocks  watch  out f o r approaching  foraging,  time  that  by which  efficiency.  Reduce t h e t i m e b i r d s spent  2.  to  t y 1)  success.  Several  in  value f c r a b i r d  f o r by p r e d a t i o n p r e s s u r e s on t h e b i r d s , i t  be d i s c u s s e d h e r e  feeding  1.  susceptibility  B u s k i r k 1972; L a z a r u s  flocking will  may have s u r v i v a l  of a successful  bird.  single is  birds.  not  As  included  n o t be e x a m i n e d . T h i s may  involve  11  the copying  o f prey  types  prey  'sites*  1972;  Krebs 1973a).  included It  is  again  (i.e. specific  discussed  Reduce t h e t i m e density  more t h o r o u g h l y  birds  of  effort:  due  to r e s u l t foraging  built  from  the  success  to  1971).  means  beneficial.  i n t h e c h a p t e r and  "The speed  rather precisely  individuals (Short  into  1961).  of  with  prey  which a an  w i t h a minimum o f  While  into this  model,  social  low  to provide  moving  the f l o c k i n g  effects  areas  hypothesis  i t is likely  interactions  and  on movement.  Increase  the  time  i n aggressive  lost  may be  f e e d i n g i n each a r e a  foraged."  not d i r e c t l y  that  i s the primary  later  of  (Krebs e t a l .  spend f o r a g i n g i n a r e a s o f  1922; Cody  period  previously  1974).  behaviour  copying  4.  (Miller  optimum  is  microhabitats)  Copying  moves may be a d j u s t e d  wasted  1971) c r t h e  i n t h e model by which f l o c k i n g  i n Chapter  flock  (Burton  time  available acts  As p r e v i o u s s t u d i e s aggression  f o r f o r a g i n g by r e d u c i n g t h e of  (eg.  solitary Hinde  was low i n w i n t e r  birds  (Earash  1952) had i n d i c a t e d  flocks,  this  hypothesis  was n o t c o n s i d e r e d . Flush  out i n s e c t s  hypothesis as  probably  1915; . E r o s s e t  1969).  This  only a p p l i c a b l e to t r o p i c a l  birds,  temperate-zone b i r d s g e n e r a l l y f l o c k  adult  The  i s  (Swynnerton  insect  availability  value of f l o c k i n g  i s low  (Yapp  i n the winter  when  1970).  i s presumably c l o s e l y  linked  with the  12  density prey  and  (e.g.  proposed  distribution Brown  that  1964;  (temporal  Brown S O r i a n s  territoriality  dipersed,  and  conditions  the d i s a d v a n t a g e  thought food  flocking  (more e y e s  may  be  by  spatial)  1970).  favoured  when f o o d  t o be o u t w e i g h e d  sources  and  It  is  s e a r c h i n g ) , the p o s s i b i l i t y  of time  i n marginal feeding areas f o r a b i r d  prey  distribution  predators  has  theoretical random,  attack  i s found.  suffer prey  A  decline  in  i n Murdie's  serious  greatly  deficiency  in  rate  the r e d u c t i o n in a  flock.  host  parasites  investigators. that  rate  than  prey  these  the importance  prey  studies  is  Since the of prey  is  the that  randomly cne  (1971)  became  of one  i n the  or  rate.  a r e c a p t u r e d once  as  or  On  when s e a r c h  the c a p t u r e  the capture  o n l y t o random s e a r c h . affects  imitating  o f random s e a r c h , M u r d i e  study  rich  1971a,b) d e m o n s t r a t e d  attack  capture  is  of prey s h o u l d reduce  (1967,  higher  of  affect  the c a p t u r e o f t h e r e m a i n i n g  conclusions apply search  that clumping  a  rate  several  not  and  of  g u e s t i c n o f how  when a l l p r e y i n a clump  However,  not e n s u r e  does  In a s i m u l a t i o n s t u d y  a slight  clumped.  by  birds,  (1972) s u g g e s t e d  while Paloheimo  prey  the  attack  examined  (1959) a r g u e s  distributed  found  been  the  distribution  rate,  clumped  affects  grounds Rogers  host  Rashevsky  f o r a g i n g animals  latter  tc locate  preference  For i n d i v i d u a l  the  when i t i s f o u n d  microhabitat spent  generally  Under  the i n c r e a s e d a b i l i t y  of s u c c e s s f u l  available  when f o o d i s e v e n l y  i s patchy.  of s h a r i n g food  of  more  prey d i d  clump.  that  their  pattern  of  distribution, i t  13  is  necessary  t o consider the  search  randomly.  pattern  after  Many p r e d a t o r s  the capture  shown t h a t i n s e c t s Dixon  1959;  Smith  prey of  of  a  capture,  1937;  item..  search  the predator  concentrates  ( area-restricted  1950;  lasts  random).  T h u s , whenever p r e y  i t s , former  be a d a p t i v e ,  i t s search  Holling Franck  higher  search are  1969;  which  abundances. large  mode  (often  aggregated,  1971),  6  birds  the  spend  many p r e d a t o r s hunting  Dawkins  time  1968, 1971).  t h a t s o many p o t e n t i a l  prey  (Tinbergen,  rate.  prey.  These  (Griffith  &  Impekoven S 1968).  different  the prey  disproportionately  i n the areas  1971 ; ..birds Such b e h a v i o u r populations  the  apparently  which compare of  a  This  which  1961; Eeukema  areas  spend  1970).  area  that the attack  on randcm  a r e those  in  the  area-restricted  were i n s e c t s  (Ivlev  of observations  predators In f a c t  than  the p r e d a t o r s  (insects - Hassell Smith  prey  1 9 7 4 ) , and f i s h  portion of t h e i r  density  given  clumped  Hassell  line  in  i n c r e a s i n g t h e mean c a p t u r e  s t u d i e s i n which  A related  1970;  on  1967; S m i t h  times  after  after  In s e v e r a l s t u d i e s i t h a s been d e m o n s t r a t e d  include  1957;  ( C r o z e 1970;  immediately  up t o a few m i n u t e s ,  resumes  is  search  Banks  search'; Crcze  1  predator  rate  their  not  Numerous s t u d i e s have  Fleschner  pattern  do  p r e y . . By i n c r e a s i n g t h e r a t e o f t u r n i n g a f t e r a  change i n b e h a v i o u r  will  predators  1966; H a s s e l l 1968) and b i r d s  their  recent success  search  that  (parasites) a l t e r  of a prey  (Laing  Bansch  1974) a l t e r  capture  evidence  of higher -  prey  Gcss-Custard i s no s u r p r i s e  (both  animal  and  14  vegetable)  have  clumped  1936;  1939;  Burnett  Smith  Gerard  1965;  Taylor  Wh.en.ever with  one  been  predators  1974;  Cheke  dees o c c u r given  a  Dybas S D a v i s  aggregate,  (Griffith  1974).,  choice of  S  Holling  I t has  at natural  densities  different  despite  stability  distribution were low,  distributed attack  the  the  foraging guestion  area than o f how  area  before  the  predation  i n areas  Berthet  of  o f low to  when  S  can  to stay  area  a  Rogers  the  l e a v i n g f o r another  the  high  Hassell  prey  are  density  density.  arises.  of  Thus  interaction. interference,  When  higher  densities  parasite  on  randomly  were  high,  hosts.  t h e r e comes a poin't  better to search  depleted  should  h a b i t s of c a r r i o n  &  to a t t a c k ; R o g e r s 6  was  depleted,  t o do  long a predator  parasites  predators  efficiency.  are  in  has  that i n t e r f e r e n c e  predator-prey  clumped  expect  interfere  insect  parasite  parasite  on  to  that p a r a s i t e s leave areas  attack  higher  i n one  which the p r e d a t o r  1969;  attack e f f i c i e n c y  But was  prey  Ullyett  predators  a r e l a t i o n s h i p . , between  and  hosts.  efficiency  As at  adds  areas  (1974) a l s o o b s e r v e d  densities  likely  using  densities  i n t e r f e r e n c e , but  d e n s i t y sooner than  prey  are  (provided  prey  occurs  Cheke  1962;  been d e m o n s t r a t e d  1974), t h a t a g g r e g a t i o n  interference  they  several researchers  Hassell  high  Parsons &  I n t e r f e r e n c e between s e a r c h i n g  i n v e s t i g a t e d by  as t h e  1958;  (eg.  1971).  predators  another.  distributions  one.  continue Croze  Thus,  searching  (1970),  crows, proposed  in a  new the  in  an  studying  t h a t the  crows  15  gave up and prey data  left  capture fairly  notion  that  this  foraging  theory  giying-up  predicts  mean p r e y  this  The  times  (1973)  has  that  s h o u l d be  interpreted  no  different  prey  in  s h o u l d be Gibb's  giving-up  time  may  also  vary  with  The  i s -one  the  of  1973). should  i s , when p r e y s h o r t ; while  comparitively  are when  long.  studies of t i t predation this  light.  Smith  times  d e n s i t y , while.Krebs,  minutes.  (Charnov  That  Tests  S Dawkins  between  Byan & Charnov  the p r e d i c t e d changes with changes i n  giving-up  o f 6.5  relatively  ( G i b b , 1958,1960,1966) i n  differences,  last  the g i v i n g - u p time  h y p o t h e s i s have been i n c o n c l u s i v e .  found  the  model f i t C r o z e ' s  theory.  density.  are scarce, giving-up times  Ernamonia  find  optimal  with the o v e r a l l  Charnov on  some- t h r e s h o l d .  since  p r e d a t o r s s h o u l d have a g i v i n g - u p t i m e  Furthermore,  common,  when t h e e l a p s e d . t i m e  w e l l w i t h a mean g i v i n g - u p t i m e  of  prey  area  exceeded  predictions  vary  an  prey  predator  of  (1971)  areas  of  (1974) d i d  density. density  The  (Hassell  1971).  type  While  i t i s well  eaten  by  abundance  that,  i s not a simple  i n density, size,  of c a p t u r e can  predators tend.to  attention seem  a predator  t h e number o f a p a r t i c u l a r function  o r d e n s i t y , n e i t h e r t h e above f a c t o r s  between p r e y difficulty  known t h a t  on  conspicuousness,  wholly account  forms of l e a r n i n g  c o n c e n t r a t e a t t a c k s on  nor  the  prey  differences  palatibility the  and  observations  c o n c e n t r a t e a d i s p r o p o r t i o n a t e amount of  t h e more a b u n d a n t p r e y  t o be s e v e r a l  for  of  prey  types  (Holling  1965).  underlying this  t h e more a b u n d a n t  prey  There  tendency  type (s).  to  These  16  include  1.  'search  1 9 7 1 a , b ) , and 1971).  'niche  Both  preferences, 1968)  2.  and  forms  hunting* of  i n contrast  i t s counterpart  Rabinowitch  1968)  Search detect  image f o r m a t i o n '  a  image f o r m a t i o n  difficult  type  to d e t e c t . ,  De  sticks,  (Garrulus  had  had  chanced  others.  great  Thus, . i t  distinguish  the  studies  provided  images.  Croze  on  appears  indirect  shell  on  that the  that  conspicuous  the  from  the  by  the  predator's and  (Irincjilla Once a  Burton  guarters  of  the,  a s i n g l e type Boer  bird found  searching  crows  to  preying  be  the  find  was  one laid,  of made out  stubble.  wood-pigeons.  feeding  f o r the  found  on  cryptic.  c l o v e r and  .(1971)  out  colour  (1971)  of f e e d  to other  When l a i d  on  (the i n i t i a l  t h a t about .three  Den  of  selectively  He  experiments.  appropriate  rapidly  proved  fields..of  m i n u t e s o f the  stick  Several  shells.  g r a i n on  on  when  the  i t  study  shell  .experimentor).  specialized  that  b i r d s . . , learned.,-,. how  cryptic  had  l e a r n i n g to  chaffinches  types  there  (e.g.  therefore . i n i t i a l l y  .with  several found  of  started  Clarke  prey  t h a t b i r d s form  mussell  c o l o u r s of  prey  Allen S  sticks.  evidence  b i r d s preyed  which t h e y .had  short-term  unfamiliar  however,  (1970) p e r f o r m e d a f i e l d  a s h i n g l e beach a l l t h r e e  Dawkins  i n f i n d i n g , t h e prey.  b a i t s hidden under p a i n t e d  Croze found  and  caterpillar,  caterpillars  Smith S  to  found  mixed  garrulus)  difficulty  upon one  (1952)  were  Dawkins  phenomena.  r e f e r s t o the  Ruiter  (Geometridae)  coelebs)  long-term  1960;  (e.g.  r e j e c t i o n , : of  which i s c r y p t i c ,  caterpillars jays  lead  'familiarity*  the  which a r e  prey  (Royama 1970;  learning  to  (Tinbergen  that  15-30 the  17  proportions  of  £iniarus taken background. mcrph than  two by  colour  captive  After  yellow larvae  tits  birds  (green l a r v a e on  morphs (Parus  had  a green  of  the  larvae  spp.)  varied  been t r a i n e d background),  when p r e s e n t e d  to take they  o f Bujgalus with the  took  cryptic  more  w i t h e q u a l numbers on  the  green  a  green  background.  The  best  direct  evidence  that  birds  i m a g e s comes f r o m  Dawkins'  domestic  Two.prey t y p e s , g r e e n  chicks.  backgrounds,  green  investigate presented  grains,  rapidly.  This  reinforcement, a few  minutes  Search (1960)  to  insects gave  to  a c c e p t a b l e prey portion forest.  on  was  within  young).  a prey  When  background,  cryptic  grains only  net  began  finding  grains increased  maintained  24.hours  initially  predation  . He to  of the b i r d ' s d i e t As  to  (and  without  within  just  conditions).  the  tended  used  single,  the c r y p t i c  type  be than  found  what  that  ignored, their  proposed  patterns  ( a c t u a l l y Tinbergen determined their  a  Once the c h i c k s  image  image f o r m a t i o n was explain  two  images.  of  with  and  searching  the a t t a c k r a t e  u n d e r some  rice,  form  minutes.  being forgotten  orange  were  types of g r a i n  searching  experiments  surfaces,  were t a k e n i m m e d i a t e l y ,  a l a g of s e v e r a l  cryptic  and  form s e a r c h i n g  pebbly  chicks  w i t h t h e two grains.  (1971a,b) l a b o r a t o r y  orange  whether  conspicuous after  and  do  by  Tinbergen  cf t i t s prey  on  forest  parent  birds  a t low  prey  making  up  relative  densities, a  smaller  p r o p o r t i o n i n the  became more common, t h e b i r d s  would a t  18  some p o i n t s w i t c h  to taking  particular  type  than  prey  i t i n great  image* was  was  that  f o r the prey  sufficiently  1960)  the  type,  high.  have g a t h e r e d  Thus,  would t h e n be t a k e n i n g r e a t e r  i t s p r o p o r t i o n amongst a l l p r e y  explanation  abundance.  birds  populations.  the  Other r e s e a r c h e r s  additional  proportion Tinbergen's  formed a ' s p e c i f i c  but only a f t e r  the  searching  contact  frequency  (Mcok, Meek S H e i k e n s  evidence  which  supports  this  view. An data  alternative  i s taken  similar  'niches' the  by Royama  to  hypothesis,  arguing  niche  For  Thus,  patterns  thresholds  the  f o r prey.  most  the bird  are  of  basically  searching  the b i r d s l e a r n  image  in  which  The b i r d  learns  i t s hunting  'niches'  time  to see i f t h e i r  w i l l , have t h e  option  to  whenever a p r e v i o u s l y u n p r o f i t a b l e  study  i twill  t h e two mechanisms - ' s e a r c h i n g  for  Exactly  of this  n o t be n e c e s s a r y  to  image f o r m a t i o n ' and  s i n c e b o t h p r o d u c e much.the same n e t e f f e c t , a  preference  preference'). formation  search  data  and s i m i l a r  profitable.  hunting',  s h o r t term  that  t o sample o t h e r  the purposes of t h i s  distinguish  his  rejects,  and s p e n d s  changes..  i t s feeding becomes  'niche  'niches'  continues  profitability  he  to  of Tinbergen's  While  i n i t s place  (microhabitats)  but  change  (1970).  Tinbergen's,  profitable  there,  interpretation  what  preference  such as those  a  p a r t i c u l a r . prey experiences a r e not c l e a r .  of R o l l i n g  type  are necessary  (a  'prey  f o r the  Models c f a t t e n t i o n  (1965) and  Dawkins(1969a,b)  19  appear be  quite  formed  with  o f h u n g e r may Holling if  here  affect  by  i n a flock,  1.  social  involving  mimicry  demonstrated  i t  In  Parus spp.  item.  might  nesting  be  synthesized  given  simulation  (eg.  This  to create  may  cases be  is  o r 2.  i t w i l l not  , (whereas  in  a  necessary  to  make t h e  copying  study  has  tits  been  -  Krebs,  (for,chickadees, suggest  as  well.  no more t h a n a minute o r two  distinct to  used  copy, a n o t h e r  natural conditions  for  birds  as  Burton's f i e l d observations  proposed  <4.  -  minutes  same b r a n c h )  ( f o r .great  (Horn 1968; Ward  i n Chapter  a few  learning  A bird  conditions  A d e t a i l e d d e s c r i p t i o n o f how were  formation  For our purposes  well  manifested  sharing  level  p o s s i b l e when  Social  place  phenomena o c c u r , u n d e r is  within  1972) and i n t e r s p e c i f i c a l l y  reinforced.  colonial  prey  - Krebs 1973a).  copying  is  phenomena.  intraspecifically  similar  information  learning.  laboratory  MacRoberts 6 C u l l e n  unless  learning  t o d i s t i n g u i s h these  distinction).  This  of  in a similar  for a similar  preference  1952;  can  1971a,b).  two i n t e r r e l a t e d  necessary  prey  (De R u i t e r  c a n be e x t i n g u i s h e d  form  searching  searching  that  the rate of  (Dawkins  additional  covers  i s that the preferences  a s few a s a s i n g l e c o n t a c t  not r e i n f o r c e d  forage  The e v i d e n c e  1 9 6 5 ) , and t h e y  An  be  likely.  from  explain  the  sort  the a d a p t i v e n e s s  of of  1965).  the  factors . discussed  a model o f f o r a g i n g  I t i s preceded  by. a  bird  flocks  discussion  m e t h o d o l o g y and some a n a l y t i c t e c h n i q u e s  of  here will the  for analysis  of s i m u l a t i o n  results.  21  CHAPTER THREE:. SIMULATION An  abstract  model  METHODOLOGY i s a s e t o f c o n c e p t s and  relationships  among  concepts  relationships  and  mathematical supply  a  them.  model as a  set  of r u l e s  1960).  a  are  model  mathematical  for  Commonly,  the  model may  propositions, theory  (Kowal  concerning  which  be  is  ( B a r t o n 1970).  case  of  manipulated  make the  various model  Deductions  is  of a  from  must of the  phenomena  mathematical the  propositions.  The  to  t h e s e back  yield  additional  i n the  mathematical  intc  prepositions  a s e t o f new  conclusions.  o f t h e model a r e  verified,  execution or manipulation of a  I f the the b a s i c  standard  mathematical  f o r the a n a l y s i s  reduced  one  regarding  assumptions to  this  a  workable  simpler  level  special  too  One  t h e model, cf  of  complex  techniques,  o f t h e model. regarding  a  propositions  ( t h e model's s t r u c t u r e ) a r e  simplifying  made  dynamic  models.  by  options are a v a i l a b l e  use a  real  Computer s i m u l a t i o n . m o d e l s a r e  mathematical  To  appropriate.  the  t h e m a t h e m a t i c a l model be  the  implicit  these predictions  model  to  manipulated fact  these  the p r o p o s i t i o n s  mathematical  phenomena y i e l d s  judged  Simulation  in  ones.  of p r o p o s i t i o n s  Translating  real  To t h e e x t e n t t h a t t h e model may  are  1971).  the  t h e n be  model  phenomena,  formulation  phenomena i n q u e s t i o n i n t o  mathematical  real,  p r o p o s i t i o n s about  model b e g i n s w i t h t h e t r a n s l a t i o n real  mathematical  for translating  mathematical theory i n t o (Parzen  In  the e x p r e s s e d  is  two to  until  complexity.  model t h e n r e p r e s e n t s p e c i a l  22  cases of the o r i g i n a l that  one  can  statements  often  about  abstraction  of  Alternatively, particular model.  the  model  simple,  model.  work  often out  (specific  feasible.  the  can  the  and  the  limits  make  high  their  values) the  advantage  often  broad  degree  cf  various  of the o r i g i n a l  manipulation  W h i l e t h i s method  of  applicability.  implications  parameter  simulation  has  concise  However,  models  one may  cases  T h i s approach  make  such,  Computer  complex  model.  frequently  of  a  fails  to  p e r m i t t h e b r o a d s t a t e m e n t s which a r i s e  upon  the  which a r e more l i k e l y  model, i t does p r o v i d e c o n c l u s i o n s  be r e p r e s e n t a t i v e predictions altering  which  are  not  t h e model, whereas  o b s e r v a t i o n s may  Several ecology  of the r e a l  phenomena.. verified  of  the  may  counterpart  Walters  been  models,,,  models  a  may  upon 1925;  Volterra  structure mortality  of  by  the to  predators  Leslie  have  a basis for  model  the  new  used i n  provides  population  single  Many o f . t h e s e  Lotka-Volterra  account  f o r such  (Bartlett  been  1945;  Frank  extended  to  1960;  to  i n t h e model.  a  good One  dynamics  population  c r of  models a r e based equations  (Lotka  phenomena as  random  1957)  t h e p o p u l a t i o n and i t s e f f e c t s  (e.g.  models  of  1931)  prey  of  populations.  generalizations  capture of  Such  be  of  model  use o f m a t h e m a t i c a l models i n e c o l o g y , models has  several interacting  provide  (1971)  common t y p e o f e c o l o g i c a l These  addition,  o f m a t h e m a t i c a l models a r e commonly  and a n i m a l b e h a v i o u r .  discussion  In  with the s i m p l i f i e d  have no d i r e c t  sorts  simplification  and  upon  the  fecundity  Pennycuick  account  for  age and  1969).  stochastic  23  elements terms  (Kerner  of  1973).  1957)  their A  stability  second  "compartment"  an  describing  the  which  flows  (compartments),  the  can  contain  examples  good  energy  and  Volterra  1957; Watt approach  1959; Rcyama is  (Holling  1931;  this  model  ecological  modelling  t h e components  t h e component bird  flocking  that  the p u b l i s h e d  passerines ascertain  By seen  is  the  model  S Bailey  cf  1935;  refined  and  a  contrast,  within (1969)  and  Holt  version  this  or  on t h e e c o l o g y  of  analysis" down an  "experimental experimentally,  finally  type.with  Lotka  Eeverton S  involves breaking  components  A  process.  (e*g.  components  subprocesses  model i s o f t h i s  parameter v a l u e s  levels  Dyne  and o f c o m p e t i t i o n  individually,  literature  By  a p p r o a c h t o model b u i l d i n g .  these  has s u b s t i t u t e d  little  Van  For  level.  integrating  models t o p r o d u c e a model o f t h e o r i g i n a l  The  the  flow.  o f energy through  1966)., T h i s  examining  is  _trophic  (1971).and  May  a number o f  trophic  between  "experimental  simpler  into  in  1973;  model  .materials by  1971). . The most  1965,.  process i n t o  components",  Nicholson  Holling's  1959a,b  split  Patten  Examples a r e models o f p r e d a t i o n 1925;  et a l .  i s divided  flow  extensively  ecological  energy  of  type of e c o l o g i c a l  Fredrickson  system  overall  be s t u d i e d .  analyzed  of  be  of  t h e s y s t e m may  third  type  model. ,• The  ecosystem  been  (e.g.  common  compartments t h r o u g h example,  and have  process.  the q u a l i f i c a t i o n and  behaviour  of  f o r the performance of experiments to f o r t h e components.  m a t h e m a t i c a l models i n a n i m a l b e h a v i o u r  u s e . . The major e x c e p t i o n  have  has been t h e use o f Markov  24  models t o d e s c r i b e Nelson 1971;  1964 ; Slater  1973).  A  sequences  Altmann  1968;  5 Ollason variant  o f b e h a v i o u r , (e.g. Chatfield  1972;  Straw  on t h i s  1961;  . S Lemon 1970 ; F e n t r e s s  1972;  approach  Hiepkema  Slater  will  1973;  be d i s c u s s e d  Eaker  later in  this chapter. The  decision  investigation First  one  model, not  of  use  some  a  real  simulation  and  second  construct  one  There  must d e m o n s t r a t e are  several  reasons  include:  2.  To p r o v i d e i n s i g h t  3.  To h e l p i d e n t i f y  respects  predictions  m a t h e m a t i c a l models necessary Kowal  into  how  real  which  »1)  decisions.  mathematical  an a n a l y t i c  model i s  one  wish  might  (Garfinkel  1965).  system o p e r a t e s ;  may  distinguish  among  are  world  model o f may  giving  quantitative  be more d i r e c t l y  not  always  of a mathematical  testable.  simple  measurements may set  potential  tc  test  be d i f f i c u l t t o of  guidelines  results. However, as  which  for  model:  real-world  the. mathematical phenomenon,  e.g.,  theory  represents  continuous  the  obtain.  naturalness  with  to  a workable t h e o r y ;  a m a t h e m a t i c a l model has t h e  (1971) p r e s e n t s a p r a c t i c a l  construction  into  the given  experiments  over a v e r b a l  Thus, , i t s  any  the  theories.  these  advantage  aid  .  To o r g a n i z e masses o f i n f o r m a t i o n  a l l  that  a m a t h e m a t i c a l model o f some s y s t e m  several  to  phenomena i n v o l v e s two  1.  In  model  must p r o v i d e r e a s o n s f o r c o n s t r u c t i n g  feasible.  These  to  the  versus  the  25  discrete  As  2)  ability  3)  comprehensiveness  4)  tractability  5)  c o n s i s t e n c y with  will  be  criteria  Let  the  have  of  the  then  which  other e x i s t i n g the the  bird  integral of  underestimated  flocking  to  flock  behaviours  individual  behaviours  that  the r e l a t i v e  no  key  impacts  of  solitary  birds  birds?  can  in  a.  flocking  these the  flocking not  model i n  asked  sysnthesis  behaviours,  of  using known  behaviours? yet  If  identified,  These  might  significance  generate  be  has  elicited  may  lead  how*these Third, flocks  Predictions  flock  been  when  behaviours  were i g n o r e d .  different  have some c o n t r o l  under which  q u e s t i o n s t c be  Identification  circumstances.  of  guiding  The  which a r e o n l y  elements  the  changing  in  of a.bird  whose  to  success?  regarding  last  the  t h e o t h e r hand,.a s u c c e s s f u l s y n t h e s i s  foraging  predictions,  and  use  behaviour.  individuals or  second  emphasis  generate  some key  models."  model.  First,  a r e i n g r o u p s . .. On  birds  major  behaviours  t h e r e must be  are  indicate  first,  above c r i t e r i a .  bird  behaviours  birds  mathematics,  later,  of the  esthetics,  us examine t h e p r o p o s e d  individual  of  of the  received  variables,  predictions,  and  model a r e t h r e e f o l d .  not,  versus several  to generate  seen  construction  light  f u n c t i o n , one  behavioural o f key  S e c o n d , what  are  components  on  behaviours  over  to experimentally behaviours  would  might  which  testable  change  with  what a r e t h e c o n d i t i o n s ( i f any) forage about  more  such  successfully  c o n d i t i o n s may  that  then  be  26  tested  experimentally.  For  a l l  three  appropriate. sort and  The  o f model.  of  the  also  relevant  experimental  and  here  which a r e flocking  the  as  In  model  in prey  t e r m s of these  handling ability responses  of  bird  This and  other  prey,  manipulate birds  each in  the  valuable  in identifying evaluating  emphasis  what may  be is  relevant  was  termed  b i r d s and  the  placed  a  characteristics. size  to  memory o f  past  (in  rate of  learn,  has  It  is  role  of  upon  the  and  the  "mechanistic"  represented  and  to  experiments  phenomena  b i r d s these i n c l u d e ability  amenable  prey p o p u l a t i o n .  for  model,  most  an are  major d i f f i c u l t y  i t s l o c a t i o n and  detect  While i t  case.a  l e d to  the  a model.  second  In t h i s  each prey item  For  The  difficulties  t r a n s l a t i o n between r e a l - w o r l d  time).  to  using  individual  include. conspicuousness  to  model.  one  strategy.  Each  i t seems  i s the  be  some  b i r d s ' behaviour,  question  a foraging  elements.  construct  physical  practical.  critical  the  the  selectively  manipulating  constructing  model.  the  we  is  relevant  approached  study.  and  of  simulation  t o be  directness of  the  possible to  model may  likely  a  third  laboratory  that  of  methodology  demands.that  behaviours  The  been i n a s s e s s i n g  simulation  complexity  best  setting,  overwhelming. field  be  theoretically  the  the  model be  can  a  question  e n v i r o n m e n t and  necessary that  might be  first  Owing t o  biological  question  questions  explicitly For  the  terms  of  movement,  repertoire  events.  The  of model  27  is  c o n s i s t e n t with  was  modelled  other  enter  into  behaviours,  the  model  mathematics  are  not  manipulated  primarily  comprehensiveness Chapter  2.  excluded  from  is  a  as  a  The t e s t  generates  the a d a p t i v e Chapter  Of  Hence,  computer  o f t h e model h a s  the  of comprehensiveness testable  models  are  step  the  size  simulate Siniff of  an  of  in  individual  independent  common  as  of both a  starting  more complex movement  animal  were a b l e  from  animal  such, the must  mcdel.  be The  discussed  i t may  in been  be t h a t  t o e x p l a i n the value will  be whether  the  p r e d i c t i o n s regarding be r a i s e d  point with  again  to simulate k l i n o k i n e t i c ,  one  ,  s i m u l a t i o n .modellers. movements key  (Kitching position,  patterns.  i t s home r a n g e ,  only  generally  variables. then  exponential  (1969) have s u c c e s s f u l l y over  model,  p o s i t i o n . a n d time,  negative  and t u r n i n g r a t e v a r y  B Jensen  the  and t u r n i n g r a t e a s t h e  distributed  around  birds'  e l e m e n t s have  This point w i l l  f  various, behaviours  consider step s i z e  be  simulation  comprehensive  of f l o c k i n g .  the  model  been  Since  7 when t h e u s e f u l n e s s o f t h e , m o d e l i s e x a m i n e d .  Stochastic  will  the  important  movement, h a s r e c e i v e d much a t t e n t i o n  these  component  As  (predation o f the b i r d s ) ,  experimentally  value  of  one.  already  potentially  t h e model  many  stochastic  tractable.  As c e r t a i n  of f l o c k i n g .  in  determining,  model i s n o t s u f f i c i e n t l y  model  models i n t h a t each  on t h e b a s i s o f p u b l i s h e d d a t a and t h e o r y .  chance elements  the  existing  1971).  If  animals  distribution Ey  letting  i t i s possible.to  Holgate  (1971)  and  s i m u l a t e d t h e movements  and R o h l f S D a v e n p o r t orthokinetic  and  (1969)  tropotaxic  28  behaviours.  Analytic  Analysis  The  Of A S i m u l a t i o n  derivation  systems i s o f t e n  of  frequently  for  economic o r p r a c t i c a l  is  often  chapter  data  measurements.cannot  reasons.  from  to  complex  be made  modelling  constrain  the f e a s i b i l i t y  is  applied  then  to  population  o f long  In  interpretation  of  such  in  i s d e v e l o p e d and  those  often  simulation  runs.  social  trace,  behaviour  and  as w e l l  the impacts of lagged  addition,  as  must  as t r a c e  account  at a given  c h a r a c t e r i s t i c s often  i n molding t h e behaviour  constraining  the s o c i a l  the  patterns  attributes  r e q u i r e m e n t s make s u f f i c i e n t l y  of  a r r a y of  moment  of  the  variables.  In  assume an i m p o r t a n t  o f t h e i n d i v i d u a l organism as well which a r e  f o r t h e l o c a t i o n o f each  i n time as well as  method  dynamics  f o r each a n i m a l , a long  simulation  as  The  as t o  prey s i t u a t i o n s . .  occurring  role  where  a degree o f d e t a i l  environmental f a c t o r values  spatial  cases  reexamine t h e b e n e f i t s o f f l o c k i n g i n the  of animal  must  the  data  model.  simulations  requires  monomorphic and p o l y m o r p h i c  Simulations  facilitate  i s most u s e f u l  approach  either  In such c a s e s a v a i l a b l e  biological  The method  biological  i s unavailable,  amenable t o t h e d e v e l o p m e n t o f a s i m u l a t i o n  demonstrated. the  models o f complex  The n e c e s s a r y  the necessary  6 a methodology  results  analytic  unfeasible.  and  Model  feasible,  i n d i v i d u a l a t each  of  each  location.  detailed simulations  models moment These  expensive to  29  run  and d i f f i c u l t , t o It  based and this to  the simulation  magnify  strategy  a  object  microscope image  of  illuminate  model p r o v i d e s  a different  a  more  with the  which  provides  interest.  One  of construction a h i e r a r c h y o f  constructing the  to develop second  level  models  model t o f u r t h e r s i m p l i f y t h e s y s t e m  s p e c i a l phenomena o f  perspectives Each  thoroughly.  i s p o s s i b l e i n such c a s e s upon  filter  analyse  a  real  series cf lenses world,  " c u t " and tool  consider  models as a n a l o g o u s  providing  certain features  effective  may  which for  o f the o b j e c t .  magnification,  in  answering  which  will  hence  particular  questions.  The (Chapter  particular analytic 6) i s a Markov  relationships  obtained  Markov p r e s e n t a t i o n  "lens"  process from  of the  model t h r o u g h the  model  powerful mathematical techniques long  r u n d y n a m i c s and  stability.  simulation makes  which  .it  which  be  developed  input/output  a r e viewed. amenable  to  The some  permit a n a l y s i s of system  30  CHAPTER FOUR:. THE SIMULATION This  chapter  simulation the  model.  literature  model  i s  presents  on f l o c k i n g  and  a  detailed  description  The model i n c o r p o r a t e s d a t a  structured  behaviours  MODEL  the  and  and f e e d i n g b e h a v i o u r  primarily  on  interactions  the. basis  between p a i r s  one  behaviour  i s i n c l u d e d i n which an i n d i v i d u a l  to  a  of  group  •behaves*  like  a  understanding the  birds.  of  real  Thus, flock  particular  should  of b i r d s . of  individual  cf birds. bird  The  Only  responds  us  how  well  cur  mechanisms c a n e x p l a i n  b e h a v i o u r a l components a r e b u i l t type  resulting  into  from  t h e model: experience,  social learning. Two t y p e s Short  o f movements a r e c o n s i d e r e d .  movements  of  an  individual  bird  These movements a r e i n f l u e n c e d n o t o n l y search from 2.  from  t o which t h e model  tell  behavioural  movements, p r e f e r e n c e f o r p r e y  1.  extent  ideas  the  phenomenon o f f l o c k i n g .  T h r e e main  and  the  of  f o r food,  neighboring  Integrated  The.  second  preference account  one a r e a  behavioural  effects  and  repulsion  b i r d s i n the flock.  fora particular  the  by t h e i n d i v i d u a l s  b u t a l s o by i t s a t t r a c t i o n  movements o f t h e whole f l o c k  b i r d s moving from  looking f o r food.  t o another  which r e s u l t  (Hinde  1952).,  component o f t h e mcdel,  prey  type,  short-term  i s i n c l u d e d t o take  o f v a r i o u s mechanisms s u c h  i n the  as s e a r c h  into image  31  formation  (Tinbergen  restricted  search  1970). , A l t h o u g h importance produce  (Croze  1970)  there i s  of these  the  1960; Dawkins  some  different  effect  of  1971a,b; and  'niche*  argument  mechanisms  short-term  Croze  as  1970),  hunting to  (Krebs  the  area  (Rcyama relative  1973b), t h e y a l l  p r e f e r e n c e f o r one t y p e o f  prey. The  third  learning. evidence  type  Murton  i n c l u d e d i n - t h e model i s s o c i a l  (1971), and Krebs  et a l .  which s u p p o r t s t h e i d e a t h a t  copy  one  study  the copying  copying  o f behaviour  another  was  in  learning  value  of  was o f f o o d t y p e s ; o r nature  supposedly  flocking,  birds  presented  in a fleck  tend  to  t e r m s o f f e e d i n g b e h a v i o u r . . In M u r t o n ' s  of l o c a t i o n  social  (1972) have  i n that  of  Krebs  of food h i d i n g  places.  provides the b a s i s f o r  s i n c e each  individual  may  et a l . This  the  survival  benefit  from t h e  linked  with the  cf  available  experience of others. The  value of f l o c k i n g  density  and d i s t r i b u t i o n  prey.  It  favoured is  is  (temporal  generally  when f o o d  patchy  i s presumably c l o s e l y  (e.g.  proposed  i s evenly Brown  thought food  t o be o u t w e i g h e d  sources  (more e y e s  spatial)  that  territoriality  d i s p e r s e d , and f l o c k i n g  S  c o n d i t i o n s the disadvantage  and  Orians  1970).  of s h a r i n g food  Under  by t h e i n c r e a s e d . a b i l i t y  of successful  birds,  of  i n marginal  feeding areas f o r a b i r d  food  the latter  to locate  s e a r c h i n g ) , the p o s s i b i l i t y  preference  spent  when  when i t i s f o u n d  microhabitat time  may be  i s rich  of i m i t a t i n g  and t h e r e d u c t i o n in a flock.  32  Of 1954;  the  ways f l o c k i n g  Crook  searching search  1965),  this  efficiency.  f o r prey  may  aggregate  action.  elements  of  a  may  The  be  enhance  study  will  From a v a r i e t y  affecting  In  a d d i t i o n . Table  of  these  observable may  of  d e f i n e the prey  b e i n g , found indicate  and  that  spatial  By  and  using  the complexity of  on  influence  captured.  a  the  behavioural These  categories:  prey  preference.  an i n c r e a s e i n in  basis  of  those  (cited  characteristics of  of  way  we prey  of a given  studies  each  terms  In a s i m i l a r  difficulty  the  two  the l i k e l i h o o d  prey  of  their  (Table I ) .  manifested  Previous  palatability,  temporal  (as  of  whenever p o s s i b l e ) .  relevant  conspicuousness,  the  2  down , i n t o  the e f f e c t  environment  which  studies  those,affecting  components  behaviours  characteristics  and  I indicates  behavioural  a s s o c i a t e d with  s t a t e g y were s e l e c t e d  movement  (Rand  t o d e f i n e a s e a r c h s t r a t e g y by  b e h a v i o u r a l components.were b r o k e n those  success  f o c u s on enhancement  s e t of behaviours  said  search  feeding  prey  above)  are  capture,  size, and  distribution.  s i m u l a t i o n methodology,  of the prey environment  the search process.  The  chief  and  we  are a b l e to  handle  the s t o c h a s t i c  nature  drawback o f s u c h  an  approach  2 On great tits Betts (1955), G i b b (1954, 1960), K l u i j v e r ( 1 9 5 1 ) , K r e b s e t a l . , ( 1 9 7 2 ) , l a c k (1966), Owen (1954), Rcyama (1970), Smith & Dawkins (1971), L.Tinbergen (1960): North A m e r i c a n f i n c h e s - Cody (1971): c a r r i o n crows Croze (1970): domestic chickens Dawkins (1971a,b): Brewers b l a c k b i r d s Horn (1968): E u r o p e a n t h r u s h e s Smith (1971): wood-pigeons Murton (1971), M u r t o n , I s a a c s o n S Westwcod (1971).  33  (a)  Movement  Parameter  Besult  1a  Hopping  Increase search less  1b  L e n g t h o f hop  Increase area searched, search l e s s thorough  1c  Turning  Decrease area searched, s e a r c h more t h o r o u g h  rate  rate  Strength of i n t e r - b i r d attractions Individual Giving-up  Space within  to  join a  Maximum copying  within  birds farther flocks  Increase  flock  apart be tween  time  Beduce f l o c k  cohesion  threshold  ability distance  cohesion  preference  Minimum reinforcement level for prey preference Copying  crowding  flight  (b) P r e y contact  area searched, through  Increase flights  time  Integrated threshold  Prey  Increase flocks  distance  Propensity flight  of Increase  Beduce d e v e l o p m e n t prefence (reduce rate  of prey learning  Reduce memory span preference  of  Increase learning birds i n a flock  rate f o r  Increase learning birds i n a ficck  rate for  prey  for  T a b l e I_j_ B e h a v i o u r a l components o f a s e a r c h o r d e r e f f e c t s when i n c r e a s e d .  strategy  and  their  first-  34  is  the  possibility  and  ecology  danger  does  extensive are  of  to  appear of the  study  of  t o be  a  movements  and  tit)  simulated,  a  a  the  15  necessary  second  recent  feeding  h i s t o r y and  by  flock.  A d e t a i l e d d e s c r i p t i o n of  is  the  compensation  we  model  and  the  passerine  of  the  lasting  by  in  (e.g.,  i s determined  modified  tc  many  simulates  experiments  activities  the  and  enabling  behaviour  s t r a t e g y , whose e x p r e s s i o n  number  environment  o f .a. s m a l l  basis,  bird's  this  t o examine a p r o c e s s  The,  activities  A  In  behaviour  study  owing t o t h e  complex p r e y  role.  c f the  current  conditions,  major  minutes.  In  severe,  highly  feeding  on  aspects  available studies.  experiments  which c h a n c e p l a y s  search  animals.  p r e c i s e l y experimental  repetitions  great  ignoring v i t a l  modelled  not  nature  able  control  the  of  20  by i t s  the  bird's  members o f i t s  model f o l l o w s .  Habitat The birds  model i s d e s i g n e d  outside  associated  the  with  dimensional  breeding  nesting.  world,  incorporate forage.  in  typical  prey  morphs) on  the  Prior  each run  to  in  wcodland  which e l i m i n a t e s  problems  the the  birds  of  foliage  trees  model, the  in  a  t r e e s or  cf  trees  twoon  in  which 'prey  s i z e and  distribution  the  overlaps  deemed s u f f i c i e n t  divided into distinct  b a s i s of conspicuousness, of the  feed of  canopy  w o o d l a n d s , i t was  stands  are  f l o c k s of small  modelled  Since  continuous  The  season,  The  either  ground beneath t r e e s . considerably  to simulate  the  to  birds  types'  (or  palatability. and  abundance  35  of  each  prey  program type  type i s s e l e c t e d ,  each  commensurate  cell  with  requiring  of a grid  the  scale  as f i n e  be most  The  are  subroutines  the  introduced into read  to  options include as  particular producing  i n t h e computer  and  FINIT.  s t a n d s a r e assumed  was  with found  precise  a  random  of prey  First  wide  or within range  to highly  Following  of  prey  data  c f the prey  to s p e c i f i c  Program locations  (uniform,  normal,  the trees,  area  and  in  i n a stand, ranging  a  thus from  aggregated.  the  distribution  p r i n t s a map f o r e a c h FANAL  The p r e y  input data.  distributions  o f t h e p r e y , s u b r o u t i n e s FMAP  FANAL p r o v i d e i n f o r m a t i o n a b o u t  prey._  input data.  distribution  throughout  a restricted  TINIT  input  o f each  distributions  These c a n be a p p l i e d  from  t o be s g u a r e .  additional  assignment  model by t h e  Subroutine  the c h a r a c t e r i s t i c s  p r e y t y p e from  stand,  spaced-out  the  A scale  activities,  o f 3 by 3 f e e t  t h e model i n two s t e p s .  several  exponential).  and  A grid  each  was r e c o n c i l e d  Then s u b r o u t i n e F I N I T e s t a b l i s h e s . t h e  well  prey o f  searching  as p o s s i b l e ,  TINIT  establish  abundance o f each  as  birds'  l o c a t i o n , of t h e s t a n d s o f t r e e s  simplicity  types.  of  f o r e g o i n g was i m p l e m e n t e d  establishes,the  are  t h e computer  satisfactory.  initiation  For  limits  c o v e r i n g the t e s t a r e a .  a grid  computer s t o r a g e c o n s t r a i n t s . to  these  m a i n t a i n s a r e c o r d o f number o f i n d i v i d u a l  in  hence  within  stand of t r e e s  performs  t h e prey showing  a statistical  distribution. the  locations  analysis  FMAP of  c f the prey  36  distribution including also  in  an  each  index  referred  stand of  of  trees  clumping  to as C throughout  for  each  prey  type,  (*mean c r o w d i n g * , L l o y d  1967;  the remainder o f the t e x t ) .  Movements Movements  of  birds  are  categories:  individual  movements.  The f i r s t . c l a s s  factors for other  relocation  affecting a single  food,  b)  birds.  move. ,  the  bird.  result  change  which a r e  remaining  in  birds,  actively  birds  direction  o f movement f o r . e a c h  movements  As  and  15-second  period  factors  1.  Direction  2.  Cody  bird  al.  1972) .  3.  Location  the  same  direction  various  the  search from  do  not  is  stimulated  by  birds  move, Hence,  must be  a r e as  these the end  calculated vectors.  follow:  shew  preference  between  feeding  a  is  for  episodes  1971).  Location  another  this calculation  by  of the response  o f p r e v i o u s movement - b i r d s  in  integrated  prey  intensity.  The  (e.g.  handling  the  an a p p r o x i m a t i o n t o t h e i n t e g r a t i o n  continuing  a)  broad  and c) r e p u l s i o n  movement  as  which e n t e r  two  and  These i n c l u d e  e n v i r o n m e n t a l and s o c i a l s t i m u l i . stimuli  into  o f movements i s e x p l a i n e d  a t t r a c t i o n to other Birds  For  classified  of successful  birds  i f the latter  of a l l other when  their  -  bird  has j u s t c a p t u r e d  birds  another  bird  distance  (individual distance).  -  separation This  a  bird  attracted  a prey  is  distance  (Krebs e t  repelled  i s l e s s than  a  to  from  threshold  i s reduced  when a  37  bird's attention i s closely feeding  (Crook  as f o l l o w s .  1961).  The  focused  upon some a c t i v i t y  such  as  T h e s e f a c t o r s combine i n t h e c a l c u l a t i o n  movement  response  of  one  bird  to  another  ( ' s o c i a l movement') i s c a l c u l a t e d by t h e f o r m u l a : if  D>T and b o t h b i r d s s e a r c h i n g f o r prey, otherwise  MR where  MR = movement  T = threshold a = scaling  distance  the  attraction  the l i n e  move  closer  takes  place  addition  of  ('non-social') Variation  in  was m o d e l l e d  and v e l o c i t y  on a p r o b a b i l i s t i c direction  density. and  On t h i s  given  given  basis  was f o u n d  Thus,  to  during  total  MR  (D>T) to  movement  by t h e  be  vector-  stochastic.  a 15-second  period  preference  features of varying distribution  was  i n response  The.probability distribution  2 and e x a m p l e s o f t h e r e s u l t i n g 1(a,b).  with  response  incorporating  and s t a t i s t i c a l  The  birds./Searching  movement  used are  3  This function yields t h e Maxwell v e l o c i t y random movement i n two d i m e n s i o n s . 3  data.  used t o d e t e r m i n e a movement v e c t o r  i n Figure  i n Figures  to  The  basis a probability  to the environmental s t i m u l i . is  r e p u l s i o n , and  The. maximal  considered  the b i r d s ,  MB n e g a t i v e  each o f t h e o t h e r  was  direction  maintaining  calculated  to  and w i t h  D-T = 1/a.  to the flock  movement  from  between t h e two b i r d s .  when  responses  between  f i t t h e ..curve  together.  response of a s i n g l e . b i r d  foliage  to  attraction  (D<T) t h e b i r d s move a p a r t ,  birds  for  D = distance  separating  parameter used  movement i s a l o n g positive  response,  distribution for  A  : *'v-.  *  \  i.*o"...  t  / / K/ /  .  *  N 1  S  Figure la: A t y p i c a l s a m p l e o f t h r e e b i r d s w a s s e l e c t e d to i l l u s t r a t e m o v e m e n t a s the b i r d s s e a r c h f o r p r e y . T h e a r e a d e p i c t e d is 60 ft. x 6 0 ft. T h e t h r e e b i r d s m o v e 5 5 , 6 4 , a n d 4 7 f e e t ( r e a d i n g l e f t .to r i g h t ) i n a p e r i o d of 5 m i n u t e s . T h e a r e a s s e a r c h e d are a p p r o x i m a t e l y 42, 4 2 , a n d 30 s q u a r e f e e t r e s p e c t i v e l y .  ^  (  © — —  . — _  A3;  y  —17^7.  z  Figure 1 b T h i s figure i l l u s t r a t e s the a r e a c o v e r e d by a flock of 9 b i r d s s e a r c h ing f o r 5 minutes. . The a r e a depicted is 60 ft. x 90 ft. The shaded a r e a i n d i c a t e s the region s e a r c h e d by three b i r d s whose s e a r c h a r e a s ove r l a p p e d conside rably. In this flock the b i r d s s e a r c h 2/3 of the a r e a they would have s e a r c h e d separately, thus a l l o w i n g intensive s e a r c h of some a r e a s . A s s u c c e s s i n c r e a s e s flock density increases, r e s u l t ing in g r e a t e r o v e r l a p of s e a r c h a r e a s .  . 30  f(r)  ,20  = 2Sre  A  O =  1/2  V  0  < 2a)  =  rt  £: O  0,  P (,|r 10  . oo 0  F i g u r e 2:  -.ul-  16  .963  18  I l l u s t r a t e d i s the p r o b a b i l i t y d e n s i t y f u n c t i o n used to s i m u l a t e s e a r c h i n g movements, g i v e n f o r s e v e r a l v a l u e s of the parameter ( 3 . T h i s f u n c t i o n g i v e s the p r o b a b i l i t y that a b i r d w i l l move r f e e t i n 15 seconds. Thus the p r o b a b i l i t y t h a t a b i r d w i l l move no more than, x f e e t i s the i n t e g r a l o f f ( r ) from 0 to x. To the r i g h t o f the f i g u r e a r e given u., • the average d i s t a n c e moved; a, the standard d e v i a t i o n of movement d i s t a n c e s ; and the p r o b a b i l i t y that a b i r d move more than two standard d e v i a t i o n s more than the average. The v a l u e 6 = .06 was used i n the s i m u l a t i o n experiments ( a l s o i n F i g u r e s l a and b ) .  c  41  T h e s e movements subroutine  TMSTP  ( s e a r c h i n g and  (Diagram  movements were s i m u l a t e d normal and  distribution  velocity minute  2  bird  First  the  numbers a and  these  two  b i r d ' s movement, the where  2  iterations  we  A).  two  Using  were c a r r i e d  out  in  searching b  from  a  numbers as t h e  bird  moves  at  x a  ^  experiments). above,  drawing  2  \| ( a +b ) /T (4  by  Appendix  (0,s ).  y components o f t h e r-  1,  social)  To  per  s e t s =. 1/TX2B .  with  birds  in  the  After  1  the  i s t h e number o f i t e r a t i o n s  minute  match t h i s  i s calculated,  T  a l l  the  probability  the  are  simulation  f u n c t i o n given  movement r e s p o n s e all  per  moved  to  of  each  their  new  locations. So  far  we  have  movements made by must  consider  social  bring to  bird. i t to  take  Croze  flight (1970)  restricted exceeds bird  flies  flight, other  observed  limit  that f l i g h t  b i r d s i n the  be we  t h a t a crow  the  time  area. may  flock.  In  serve  flocks,  as a s t i m u l u s  calls  the  (Hinde  it is  in a  capture  which p o i n t bird  to i n d u c e  bird  example,  prey  one  Other  the  takes  flight  initiating  1952).,  a  likely  searching  last  once  only  activities  For  continue  we  n c n s o c i a l or  feeding  time'), after  Typically,  characteristic  and  the  Now  consider  of t r e e s ) ,  will  'hopping'  prey.  vicinity.  since  {'giving-up  for  need  (or s t a n d  t r e e i n the  small  t r i g g e r e d by  bird's searching  to another  to another  makes c e r t a i n  a  may  the  search  case  edge of a t r e e  area u n t i l  some  which  I n the f i r s t When  the  considered  the b i r d s i n t h e i r  flight,  stimuli.  single  just  in  flight birds  42  feeding  successfully  are  less  unsuccessful  ones.  flight,  the  level  of c a l l i n g  i n the  flock  join  birds may  take  up  The subroutine 1.  2.  b i r d s f l y with  last  prey capture)  N<G,  4.  The  of  the  success,  flock  follow, logical  Diagram  2,  parameters, simulation  To  at  or  f l y this  their  the  the  a l l the events  A.  edge  current  (N-G)/N i f N than  G  (the  (periods  giving-up  since time).  . b i r d s may  from  join  i t in this  (total  of  one  the  thresholds,  leader.  opportunity in detail  values Table  various will  period  minute).  birds f l y , a l l birds in  The  (see  the  the  fly.  first  steps:  them beyond  in  in  period.  i s greater  number o f  experiments  complete  following  which f l e w  distance  appendix  join and  f o l l o w i n g depends upon v a r i o u s  their  recently  incorporated  food  sequence i s given  and  is  period.  following periods  and  If a sufficient  birds  current  is initiated,  of  the  whole c o u r s e o f  performs the  probability  does.not  three  probability  feeding  This  bird  Once a f l i g h t  o r any  5.  the  than  of  movements c a r r y  handling not  This  flight  subroutine  do  number  flight.  f l y i n the  period  join  e x c e e d s some t h r e s h o l d  governs  Other  the If  will  which a r e  or previous 3.  This  to  time.  whose s e a r c h i n g  trees  Birds  the  which  FLY.  Birds  of the  If a sufficient  to a minute's  logic  likely  be  the  (integrated  i n the  flow  thresholds discussed  II).  d e s c r i p t i o n of f l i g h t ,  same  flight). chart  are  set  with  of as the  -  i t i s necessary  to  43  give  the o p e r a t i o n a l d e f i n i t i o n  A  flock  if  they  than  as used  in  the  model.  i s d e f i n e d as f o l l o w s : two b i r d s a r e i n t h e same a r e i n t h e same s t a n d  PFLOK  feet  estimated  PFLOK.  apart.  from  pictorially  the  of f l o c k  the  The  value  of  observation.  Figures  determination  cf flocks  Subroutine  sorting  o f t r e e s , and t h e y  FLOGG  (see Diagram  of birds into  3,  are  PFLOK  flock  not  more  (40 f e e t )  3(a,b,c)  was  present  f o r s e v e r a l values of Appendix  A)  performs  flocks.  Feeding In  modelling  necessary  to take  the into  feeding  account  behaviour  observations  which.show t h a t b i r d s t e n d  to concentrate  and  in  tend  to  Tinbergen,  take  prey  'runs*  1960; Royama, 1 9 7 0 ) .  and  maintained  given  some s u f f i c i e n t  prey  type,  a bird  frequency  will  of  direct  of  •specific  will  experiences presents search prey  image*  sufficient  success  a somewhat d i f f e r e n t is  directed  are l i k e l y  thesis,  towards l o c a t i n g  t o be f o u n d .  a t t r i b u t e s r a t h e r than  search  i n predation. feeding  type.  suggesting  by  on  one  on t h e b a s i s the  as t h e b i r d  Rcyama  (1970)  that the birds'  'prey-niches',  prey,  by  That i s ,  Once f o r m e d ,  feeding.  L.  these  formed  sites  T h u s , he e m p h a s i z e s s e a r c h for  types  (e.g.  images',  be r e t a i n e d so l o n g in its  prey  (1960) e x p l a i n s  i t s future searches  o f t h a t prey  i t is  various studies  type  successful  of various c h a r a c t e r i s t i c s search  one  search  by s u c c e s s  from  on common  Tinbergen  phenomena on t h e b a s i s o f • s p e c i f i c conditioning  c f the b i r d s ,  their  by  where site  attributes.  Figure 3 These three pictures illustrate flocks as dete r mined by subroutine F L O G G for three different values of the parameter P F L O K . The same 24 birds are depicted in each case on an area 180 ft. x 180 ft. The value 40' f t. vas used in the simulation ?:-:oer~L~ent s. •  „  45  However, (e.g.  t h e s e . two  Turner,  1965).  characteristics  Search (Diagram bird were  form  and  4,  with a p a r t i c u l a r  grid  size  o f 3 by  one  bird  an  distance  with  movement  and  could  search  background search  one  A bird  random  probability  Pi  probabilities conspicuousness, prey  is  consuming prey  has  o f the prey  in  cell  whose  chosen  w i t h i n each  an  and  i n the  where i reflect  grid  i s removed from  prey  foot.  .the  in  cell.  The and  in  speed  of  p e r i o d depending e x t e r n a l and  bird  and  internal  *enccunters'  capturing prey  a prey  characteristics,  difficulty  upon  capturing  such  system.  'direct-  direct-detection  type.  with  as  size, If  to s e a r c h , h a n d l i n g  Handling  time  may  The vary  * Capture a u t o m a t i c a l l y f o l l o w s d e t e c t i o n , i . e . the may not r e j e c t a p r e y i t e m , and t h e p r e y c a n n o t escape.  the  These  cf capture.  f o r a g i v e n number o f p e r i o d s . the  area  selection  the  ceases  The  the  represents  bird  contents  This  variation  o t h e r such  p a l a t a b i l i t y and  the prey  initial  15-second  prey  EAT  a s s o c i a t e s the  15-seconds g i v e n a  as  detecting  subroutine  location  opportunity of d e t e c t i n g  order,  captured,*  in  variation  well  habitat  preferences.  as a p p p r o x i m a t i n g  o f one  of  as  hunger,  mutually e x c l u s i v e  o f the s i m u l a t i o n r u n .  in  1970)  averaging  be  out  Each b i r d ' s  was  density of f o l i a g e , factors.  carried  3 feet  (Croze  net  the b a s i s f o r prey  at., the b e g i n n i n g  distance*  represents  need  model f o r m u l a t i o n p r e y and  are A).  detection  prey  jointly  Appendix  which  any  In t h e  capture  established  grid  hypotheses  a and  captured f o r the  predator  ,  46  different  prey  handling  periods,  activities. short  term  t y p e s , depending the  bird  In a d d i t i o n , (1 p e r i o d )  primarily does  not  when a b i r d  preference  on s i z e .  luring  engage  in  any  th^e ether  renews s e a r c h i n g , i t has a  for  prey  of  the  type  just  captured. If  a  bird  makes two s u c c e s s i v e c a p t u r e s  type, a preference f o r that Holling  1965),  detecting the  bird  with  type a  prey  for  success  (as s u g g e s t e d  preference  a r e changed.  according to c h a r a c t e r i s t i c s  bird  with prey  of  detecting  Additionally, remaining  i s formed  preference f o r prey and the  prey  capturing  probabilities  types  model we assumed  (Qij,i*j)  that  a prey  the capture o f a l t e r n a t i v e a)  prey.  of  prey  type such  and  success  on  are substantially  Thus,  i#j  Qij = probability  by  bird  preference f o r type  the  preference  so  j .  long  A bird  as  prey  to Q i i .  any  of  the  reduced.  In  success,  excludes  ( i , j = 1 , . . . ,n)  without  of capture  prey  c f type  i by a b i r d  are captured  then  with  If  the prey  of a  preference;  with prey p r e f e r e n c e  i . e . the p r e f e r e n c e i s r e i n f o r c e d . M p e r i o d s without  the  (i=1,...,n)  o f c a p t u r e o f a prey  prey  the  we have:  a  i  Pi  preference e f f e c t i v e l y  prey  type  habitat,  from  types; P i = p r o b a b i l i t y  more t h a n  about  Ey d i r e c t i n g  where n = number o f p r e y  frequency,  goes  by  i i n c r e a s e s i t s chances prey  for  0<Pi<Qii<1  b) .Qij=0_  of  prey  and c a p t u r i n g p r e y i n much t h e same way as b e f o r e , but  probabilities  search  the  A  prey  c f t h e same  a  with  retains  sufficient bird  goes  preference i s  47  forgotten. impact very  on  The the  feeding  l a r g e , we  type.  As  feeding  stategy  bird the  choice  M  becomes  have a b i r d  which  decreases,  the  with more  feeding  influenced (1971)  by  and  Krebs  allowing its  an  analysis  this  In  the  parameter  the  learning.  birds in  Findings  the  such  model  as  those  behaviour,  copying  the  model  by  with  preference  s u c c e s s f u l neighbor, copying  preference.'  IMTATE  birds within  distance  value  DIM  can  adequately  which  prey  provided bird  I f the  This  Appendix of  captured the  observe another  the  said  to  next p e r i o d ,  DIM  represents  is  that  'temporary  preference.  (Diagram 5,  neighbor  no  the  A)  lcgic  to  acquire  a  preference'  bird  acquires  i s executed  by  which s e l e c t s from a l l unsuccessful  a prey.in  the  maximum d i s t a n c e bird's feeding  bird  current at  which a  behaviour  the  period.  to  : The  copy  neighbor i s  prey  the  flock or  bird  prey  Murton  types  unsuccessful  The  of  also  prey  into  close.  is  (1972) i n d i c a t e t h a t b i r d s i n a  feeding  the  i s examined.  incorporated  The  it.  unsuccessful  ,is  corresponding  of  r e p l a c e an  prey  sensitivity  s u c c e s s f u l l y i n the  nearest  single  opportunistic.  i s employed  the  a  and  prey  subroutine  to  a major  example, i f M i s  s p e c i a l i z e s on time  'temporary  the  For  Imitation  nearest  sufficiently  M can•have  decreases,  et a l .  i m i t a t e each o t h e r * s  parameter  a more s u c c e s s f u l one  behaviour of  social  sites..  the  behaviour e x h i b i t e d .  i n f l u e n c e of changing  The  prey  of  bird copy - •  remaining  p o r t i o n s of  the  computer program  handle  the  48  input  o f parameters  positions OUTS) . FSTAT and  of  the  Subroutine provides  (MAIN, birds BMAP  capture  OUTS p r o v i d e summary  distance  moved,  each  prey  the  flocks  type,  number prey  Diagram  6,  Appendix  (BINIT) and o u t p u t prints  statistics  initial  (EMAF, FSTAT, OUTR,  a map o f t h e  statistics  A),  bird's  locations.  and s e a r c h s t a t i s t i c s . f o r each  and l e n g t h o f f l i g h t s ,  p r e f e r e n c e and s u c c e s s  bird  OUTR  including  f e e d i n g s u c c e s s on  of b i r d s  which were f o r m e d d u r i n g t h e s i m u l a t i o n .  f o r each o f  49  CHAPTER FIVEj_ THE SIMULATION EXPERIMENTS Three here.  groups  The f i r s t  changes  in  discover  which  foraging birds  of  simulation  examines  the  behavioural  feeding  are  of  the  The  critical  be d i s c u s s e d  aim here i s t o in  will  (monomorphic  examine t h e e f f e c t s flock  (differences examines,  type  size  cf varying  and o f i n t r o d u c i n g  i n behaviour f o r d i f f e r e n t the  flocking  examine t h e same q u e s t i o n s as t h e s e c o n d  two o r more p r e y t y p e s  making  a  parameter the  the  (polymorphic p r e y ) .  The prey These  group,  but  available.  Analysis  order  sensitivity  types  the prey  individual  when  experiments  to  identify  analysis small  was  variation  key  parameters  conducted.  o u t p u t . . The s e t o f r e s u l t s in later  standard.  Thus,  corresponding  experiments  f o r each  t o two l e v e l s  hypotheses:  (A)  the  The  (usually  a t a t i m e and o b s e r v i n g how  v a l u e s used  two  prey  of  has  in  several  value  prey  birds).  population  In  determining  prey  single  of f i x i n g  Sensitivity.  to  a  on  distribution,  with  model  examines the v a l u e c f f l o c k i n g t o  These  group  parameters.  will  The second  population).  third  sensitivity  o f the behaviours  success.  variation  the  experiments  10?)  much v a r i a t i o n  model  a  involves  i n one i n p u t is  created  o b t a i n e d with t h e parameter be  referred  to  as  the  we have two s e t s o f r e s u l t s  o f t h e parameter. results  the  procedure  about  will  parameter  in  do  We t h e n t e s t  not d i f f e r ;  and  the  (B) t h e  50  percentage change  change i n r e s u l t s i s n o t  in  the  parameter.  The  p a r a m e t e r s whose v a r i a t i o n results  (A  and  r e g u i r e improved  2).  to  identify on  was  run  of the  v a r y i n g the  inter-bird  attraction,  integrated  preference, distance  and  the  individual  flight  minutes. for  of these  parameters  nor  be  possible.  hopping  B  The  rate,  prey the  maximum  variation of  do  the  to e s t a b l i s h by  .zero results  the  of  join.a  for  prey  inter-bird  i n mean  and  cf  higher  However, f o r s m a l l v a r i a t i o n s i n p a r a m e t e r s  Two  capture  captures  standard  varying a single  not r e v e a l s e c o n d  a  analysis  span  prey environment.and T a b l e  they  present  -  detection likelihood.  in probability  were o b t a i n e d  little  strength  clumped  results  has  and  rejected)  memory  copying,  were u s e d :  used  -  d i s t a n c e , propensity to  for and  variation  a moderately  time,  in  20  this.analysis II gives  values..  parameter  the  Since at  a  order  interactions.  these  interactions  generally small.  Mean rejected) and  simulated  IV  values  identify  the  I I I and  these  are  on  Tables  parameter  percentage  1) t o  A  threshold,  a. f l o c k ,  variation  then  following:  maximum d i s t a n c e  within  whose  (neither  model may  measures o f s e n s i t i v i t y rate  two-fold:  a major i m p a c t  parameters  results  by  flight,  was  the  measurements i n l a b o r a t o r y c r f i e l d  those  simulated  simplification  aim  B rejected) - determination  may  impact  has  g r e a t e r than  to  minutes  capture to prey  rate  (=success)  was  detection likelihood  giving-up ( = r i s k ) was  time.  The  sensitive  highly sensitive without  probability  of  prey  (A and  preference  zero c a p t u r e s  to giving-up time.  In  B  in  20  addition.  51  £§havioural  component  Flockincj  E5U^25£JSi£2  Comments  R a t e o f h o p p i n g and turning  SCI=3.0  Same  Strength of i n t e r - b i r d attraction  PEXP=15.0 f t .  -1. 0  Individual  P1TRB=1.0 f t . P2TRB=3.0 f t .  Same Same  P F L ¥ = 2 . 0 min.  Same  PFOL=1.0 min.  99.0  Nc flights joined by nonflocking birds  Integrated,flight threshold  INTF=4  99  No integrated flight for nonflocking birds  Prey c o n t a c t t h r e s h o l d t o form p r e f e r e n c e  PCT=2  Reinforcement r a t e to maintain preference  MHUN=2.0 min.  Same  Copying e f f e c t preference  Reduces PCT t o 1 a f t e r copying  None  DIM=15.0 f t .  -1.0  Giving-up Propensity  distance time to join  Maximum c o p y i n g  flight  on p r e y  distance  Table I I I B e h a v i o u r a l p a r a m e t e r s used  birds  items  I f PEXP<0, nc attractions take p l a c e  Same  No c o p y i n g f cr nonflockin g birds  i n the s i m u l a t i o n . e x p e r i m e n t s .  52  V  2  Z  A1sig  510  chi  1.58  5. 13  ~  —  58  —  1.46  6.54  -.35  NS  64  0.75  NS  +20*  2. 74  9. 95  2.97  ..003  45  3. 37  ..07  +10*  3.72  17.64  4.46  <..001  37  8.80  .003  +10*  1.74  6.71  0.46  NS  54  0.32  MS  -11*  1.58  5.13  0.00  NS  58  0.00  NS  +25*  2. 39  10.55  2.04  ..021.  50  1.28  NS  Memory span +12* Maximum c o p y i n g  1.41  6.54  -.50  NS  67  1.72  NS  dist.  1.52  5.21  0.19  NS  61  0.19  NS  1.94  6.46  1.05  NS  51  0.98  NS  1.40  5.?2  0.54  NS  55  0.18  NS  Stand.  fl  —  Prey with  detection pref. +6.1%  Prey  detection  without  pref.  Giving-up  time  Propensity to join flight Propensity to ignore f l i g h t if  feeding  Integrated  2  A2sig  flight  threshold  +10*  Strength  S  inter-bird  attraction Individual  +10* dist.,  (searching) Individual  +10* dist.  (feeding) +10* 1.64 7.38 0.17 NS 64 0.75 NS T a b l e III: S pe rn es ai dt i vwiittyh i n a n a l y s i s t o t e s t the hypotheses t h a t v a r i a t i o n i n t h e p a r a m e t e r p r o d u c e d no a t i o n 1.31 i n mean c a p t u57 r e s 0.02 (A1) NS and i n flock +10* 2.06 v a r i8.27 ..097 p e r c e n t a g e z e r o c a p t u r e s (A2) . R e s u l t s a r e f o r twenty s i m u l a t e d minutes. N o t a t i o n : M = mean captures, S = estimated variance i n captures, Z=[ N / ( S ( p ) + S ( s ) ) ]* * [ M (p)-M (s) ] where (s) and (p) d e n o t e t h e s t a n d a r d and v a r i e d p a r a m e t e r results respectively, N=100 (number of b i r d s i n each sample), A1sig = p r o b a b i l i t y of s u c h a h i g h (low) v a l u e o f Z under hypothesis A1 (standard n o r m a l v a r i a t e , o n e - s i d e d t e s t ) , *0 •= p e r c e n t a g e c f b i r d s making no c a p t u r e s , c h i = [ 2 N - 1 ]*[%0 ( p ) - * 0 (s) ] / [ (*0 (p) +*0 (s)) * (200-*0 (p)-*0 (s) ] , A2sig = probability o f such a h i g h v a l u e o f c h i (1 d e g r e e o f f r e e d o m ) u n d e r h y p o t h e s i s A2. 2  2  2  2  2  2  2  53  H  Stand.  S2  1.58  5.13  B1sig  %0  chi  2  E2sig  58  Prey detection without pref. Alt. Stand.  + 20% +20*  2.74 1. 90  9.95 7.39  2.02  i022  45 48  0.22  NS  G i v i n g - u p time A l t . , Stand.  +10% +1051  3.72 1.74  17.64 6.21  4.04  .0001  37 53  4.96  .026  Spread w i t h i n fleck Alt. Stand. ,  + 10% +10%  2.06 1.74  8.27 6.21  0.84  NS  Integrated f l i g h t threshold, . . +25% Alt. Stand. +25%  2. 39 1.98  10.55 8.02  0.96  NS  ;  T a b l e IVj. S e n s i t i v i t y a n a l y s i s to t e s t the hypotheses that variation in t h e p a r a m e t e r p r o d u c e d no more t h a n t h e c o r r e s p o n d i n g p e r c e n t a g e change i n mean captures (B1) and i n p e r c e n t a g e z e r c c a p t u r e s (B2). R e s u l t s a r e f o r twenty simulated minutes; the ' a l t . S t a n d . ' r e s u l t s a r e t h e e x p e c t e d r e s u l t s under h y p o t h e s e s B1 and B 2. Notation: as i n T a b l e I I I e x c e p t t h a t t h e ' a l t e r e d s t a n d a r d ' (as) r e p l a c e s t h e s t a n d a r d f o r e a c h t e s t where M (as)= (1+a)M(s), S z (as)= (1+a) S 2 ( ) , %0 (as)= (1+a)-*%0 ( s ) , a=V/100, and v = p e r c e n t a g e change i n the parameter. 2  S  54  significant the  changes  parameters  integrated  flight  six  increased  range  of,  to  flocking  be  both  should  further  Ryan  and  necessary  model.  The  a behaviour risk  and  try  to  note  only apply  success  .&  Charnov  (Parus a t r i c a j j i l l u s  reduce  greatly  their  increased.  giving-up In the  are  in  modify. time  the be  an  examined  as  times  to the  detection  b i r d s and  high  suggests  one  that  birds  different with  the  i t s giving-up  hour).  predicted that a t one  may  sensitivity  entirely  when t h e i r  of  characteristic  able to a l t e r  L.). t r a i n e d  A  before  rates  same.conclusion,  (e.g.; under  to  nearly  model.  feeding  The  these  regarding  very  the  primarily  Using  should  found  parameter  In c o n t r a s t t o p r e y  can  (1973) r e a c h e d  (1974)  threshold.  sensitivity  must be  i s specific  i t .  i n a s h o r t p e r i o d of t i m e  high  absolute  to g i v i n g - u p  that a bird  such  the  where p r e y  model a r e time  optimize  while  t o c o n s i d e r how  considered  regarding  which t h e y  for  that conclusions  likelihoods  which i n t h e  in  found  risk.  as t h o s e  giving-up  were  within flock,  increase  nonflocking birds.  prey,  chickadees will  an  is  of the  detection  approach, Charnov  time  it  statements  probabilities, the  detect  prey  generalizing  of  spread  decreased  use  rejected)  the changes i n  only f o r integrated f l i g h t  feeding rate w i l l  as d i f f i c u l t  well  the  g r e a t e r than  detection probabilities,implies  absolute  of  and  no  not  cases,  proceeding  results affect prey  these  success  B  t h r e s h o l d and  were f o u n d  of  Before  success  (A r e j e c t e d ,  changes i n r i s k In a l l  in  Krebs,  black-capped food  food  density  density i s  s i m u l a t i o n e x p e r i m e n t s which f o l l o w a  55  small  g i v i n g - u p time  value  (as d e m o n s t r a t e d  giving-up less  time  pronounced  was used  (2 minutes)  by t h e s e n s i t i v i t y  increases,  i t s effects  t h a n on f l o c k i n g  found  for  flocking  longer  g i v i n g - u p time optimal  birds  should  (at l e a s t  value).  cases  where t h e g i v i n g - u p t i m e i s l e s s  t i m e s : 6.5 m i n u t e s  for carrion  1970);  11.5  L .) ( K r e b s  minutes  indicated  the  to  besides to  even  crows  great  advantage  greater no  with a greater  c a n be.made f o r  t h a n t h e one we u s e ,  light,  the  o f measured g i v i n g - u p  (Corrus  blue  n o t e on v a l i d a t i o n  corone  herons  concerns  i n t h e model d e s c r i p t i o n ,  individual and  any  no s u c h c l a i m  in  as  L.) (Croze  (Ardea h e r o d i a s  1974).  A final  (with  for  Thus,  f o r giving-up times  While  reasonable  because  on n o n f l c c k i n g b i r d s a r e  become  the  appears  i t s suboptimal  analysis)  birds.  than  selection  despite  exception of integrated  flock  fleck  movement.  behaviour i s generated  flight)  by t h e c o l l e c t i o n o f  b i r d s r e s p o n d i n g t o e n v i r o n m e n t a l and  the  other  individual  the i n t e g r a t e d  (awareness  flight  of) the f l o c k .  birds.  That  internal  the observed  reaction  consistency of  t h e f l o c k movements g e n e r a t e d i n t h e s i m u l a t i o n  experiments  flock  a  movements o b s e r v e d i n t h e f i e l d  independent  substantiation  provides  o f t h e model's  cues  i s , no b e h a v i o u r  i s g e n e r a t e d by a b i r d ' s Thus,  As  validity.  positive  with and  56  Monomorphic P r e y P o p u l a t i o n s In with  the experiments  the advantages  which  birds  type.  sets of simulation  first  through  compares  third  flocking  flocking  measures o f s u c c e s s o v e r the second  we w i l l  experiments  and  fourth  differences  i n behaviour  considered  to  first  sets  two  vary  examine  of  the  several  are  a r e examined.  impact  abilities  experiments  on  The  o f prey d i s t r i b u t i o n s .  between d i f f e r e n t  i n their  birds  sizes  to  are o f a s i n g l e  were p e r f o r m e d .  nonflocking  a wide r a n g e  sets  which a c c r u e  when t h e p r e y  the success of various f l o c k  and  concern o u r s e l v e s  i n terms o f f e e d i n g s u c c e s s  individual Four  follow,  cf  In The  introducing  birds.  The b i r d s a r e  t o d e t e c t p r e y , and t h e repeated  under  these  conditions.  The  effect  In  of  flocking;  the f i r s t  foraging  attraction..  normal  flocking  group  there  subsequent in  an  random  I n one group  (social  group).  no  f o r each  experiments  we  examine  120  bird  refer by  values  120 y a r d s .  p_rey.  how  involving  birds  show  i n the ether  ( t h i s c o n s t i t u t e s the used  back t o T a b l e I I .  and i n d e p e n d e n t  the  copying), while  behaviour  F o r t h e parameter  of  a monomorrihic  of experiments  attraction,  flocking  experiments  area  with  c h a n g e s when we v a r y t h e b e h a v i o u r s  social  control  feeding,  s e t of s i m u l a t i o n  success  is  on  Initial  in  these  The b i r d s site  of the other  and  forage  selection i s  birds.  Thus,  57  the the  birds  have  no  initial  f o r a g i n g area are s c a t t e r e d  type  (prey  variable  characteristics in  the  distribution. thirty-six  prey  equal  kept  fixed  covered  by  a clump, from  Lloyd  clump  was  To  directed  toward,  (Owen 1 9 5 4 ) .  insufficient  f e e d i n g may  mortality  feeding risk  prey  as a s e c o n d  of  this  These  400  density  t o 2.5  within ranged (within-  crowding,  square.yards  t h e number o f p r e y . i t e m s  losses  flocking  to  birds  criteria  rather  success  of  -  during  f c r evaluation  than  a  growth may  10%  ratio  of  not  single  increase a bird's i n one  c o n s i d e r the r i s k  the  per  experiments.  significantly  one  area  period, feeding a c t i v i t i e s  feeding  used,  are  The  w i t h i n clumps  measure o f f e e d i n g s u c c e s s .  initially  into  locations  Densities tested  from  maintenance  - we  spatial  each) whose c e n t r e s  the average  yard  The  divided  of experiments.  to e s t a b l i s h  weight  1952)  are  prey  V).  the p a t t e r n of  As t h e mean c a p t u r e r a t e  of  since  (Kluijver  were  that  s e t of  During  measure  uncommon  to areas  noted  Table  L l o y d ' s i n d e x o f mean  the importance  sufficient  of  square  with  i t i s necessary  reproduction  series  per  in this  feeding success.  primarily  900  Within  a l l of a s i n g l e in  foraging area.  estimated  varied  determine  mid-winter, of  prey  I t s h o u l d be not  is  consequently  1967), c o r r e s p o n d i n g  clump.  cut  between e x p e r i m e n t s .  9.48  clump d e n s i t y was  1  set  the  both  a clump, and  to  'prey ,  ( t w e n t y - f i v e prey  i n the  throughout  0.07  are  case  clumps  i s varied  900  environment  In each  are randomly l o c a t e d  per  preference within the area.  Twc  or be a  day's chance  night are  of  are  not  insufficient measures of  v a r i a n c e t o mean i n  ££§2 c h a r a c t e r i s t i c s  i n model  0.5 grams  S ize Handling  Value  time  P r o b a b i l i t y of d e t e c t i o n and c a p t u r e by b i r d : a) w i t h o u t p r e y p r e f . b) w i t h p r e y p r e f .  T a b l e V_: Prey c h a r a c t e r i s t i c s .  Comments a p p r o x . . 20 gram/day f o r maintenance  30 s e c o n d s  0.25 0.75  Accounts f o r conspicucusn e s s , p a l a t a b i l i t y , and d i f f i c u l t y of capture  59  feeding  r a t e , the other  a fixed  period  the p r o b a b i l i t y  of obtaining  no f o o d  in  of time.  Results Results representing 4-6  simulation  twenty  (see A p p e n d i x In  with  Figure  B f o r these 4  Griffiths  drop  at  we  see t h a t  show  clumping  values  the  same  show  that  with  of food  the  higher suffer  clumping.  and n o n f l o c k i n g  slope  birds.s  flocking  for  lower  rate  risk  increase  birds differ birds  be  expected,  high  correlation  when  feeding  cn  and  linear for nonflocking  (the two r e g r e s s i o n Figures  linearly  5 and  significantly 1.?5  6  i n C* , t h e  the r e g r e s s i o n  t h e model, f l o c k i n g than n o n f l o c k i n g  increases  o f f o r even  p>.10).  In both cases  nonflocking  Thus,in  mean c a p t u r e a  different,  Figures  to l e v e l  the  of increase  each)  form).  (as might  Both f l o c k i n g  b o t h measures o f ' r i s k *  flocking  in  the mean.capture r a t e  (although  pattern  are not s i g n i f i c a n t l y  fcr  i n tabular  o f t h e prey  tested).  lines  logarithm  runs  suggest t h a t the r e l a t i o n s h i p i s h i g h l y  range o f C  birds  results  (five  are s e t out  & H o l l i n g 1969), appearing  extreme  coefficients  experiments  minutes o f a c t i v i t y  the degree of clumping  e.g.  the  of  times  lines  (p<.001) that for  b i r d s do n o t have a  birds,  but  clumped  food.  they  do  And a s  S i n c e t h e two measures o f r i s k g i v e s u c h s i m i l a r r e s u l t s , o n l y one i s r e p o r t e d - p e r c e n t a g e o f b i r d s making z e r o c a p t u r e s i n 20 minutes - i n other experiments. 5  5.5.  MEAN C A P T U R E S I N 2 0  MIN  5-aL  Flocking:  r = .75  Nonflocking:  r = .84  <  o-o Figure 4:  0-5  5*5  1-0  C  6-0  ~ LN(lG*C)  The graph p l o t s mean capture rate against C' = In (16C) where C i s Lloyd's index of mean crowding, an indicator of within clump density. Points marked "x" are r e s u l t s with f l o c k i n g , those marked " t " are without f l o c k i n g . The regression l i n e s plotted through these points are not s i g n i f i c a n t l y d i f f e r e n t . The.goodness of f i t i s indicated by the c o r r e l a t i o n c o e f f i c i e n t s r = .75 (flocking) and r = .84 (nonflocking).  .cr> o  Figure  5:  The g r a p h p l o t s t h e r a t i o o f v a r i a n c e t o mean o f c a p t u r e s i n a t w e n t y m i n u t e period against C ( a s i n F i g u r e 4). X's r e p r e s e n t f l o c k i n g r e s u l t s , t ' s represent nonflocking r e s u l t s . The r e g r e s s i o n l i n e s d i f f e r i n s l o p e s i g n i f i c a n t l y (p < .001).  o-sl  PROBABILITY- OF ZERO CAPTURES I N 20  MIN  Nonfloc k. i n s z  o-al  r  = .96  y = .26 + .037.x  0-71  •0-6.  0-51  o-aL 0<  0-1-  O-Ql  0-0  0-5  1-0  1-5  2-0  2-5  C Figure  6:  3-0  3-S-  4-0  4-5  = LNC16*€)'  The g r a p h p l o t s t h e p r o b a b i l i t y o f m a k i n g no c a p t u r e s i n t w e n t y m i n u t e s a g a i n s t C' ( a s i n F i g u r e 4 ) . X's r e p r e s e n t f l o c k i n g r e s u l t s , t's represent nonflocking r e s u l t s . The r e g r e s s i o n l i n e s d i f f e r i n s l o p e s i g n i f i c a n t l y ( p < .001).  5-0  5-5  S-O cn  63  clumping  of  food  increases. along  The  with  the  increases,  implications  this  difference  of these  s e t examining  the  results  success  in  will  of  risk  alsc  be d i s c u s s e d  different  flock  experiments,  when  sizes.  In each  the  of c l u m p i n g  clump o c c u p i e s a s i n g l e  predicts t o be In  extreme case  the  0.57  for flocking  this  case,  nonflocking samples  probability  of  we  until  success  succeeds  (makes a c a p t u r e ) .  grid  period square  square  with  (20  simulated  i s t h e sum  of  sampling  the  eighty  draws  The  probability  the  probability  of  of  probability [(14364  which i s i n e x c e l l e n t equation.  are  the  birds. of  the  a  bird  or the  failure  are  any.  bird in  an  cf picking  detecting  samples  completed  of picking  not  have  feeding  minutes of searching)  but  minutes  scheme i n which  t w e n t y - f i v e prey Assuming  regression  model,  i n 20  ncnflocking  probability  z 5  =  for  (14364/14400) p l u s t h e  (0.75 *36/14400). the  can  the r e g r e s s i o n f o r m u l a  zero captures  0.78  either  without  given  and  as a B e r n o u l l i  (searches)  square,  making  birds  however,  birds  grid  f o r these  a  empty a  of  grid them  independent,  we  failure  + 0.75 *36)/14400]80 = 2 5  agreement  with  the  0.82.  value p r e d i c t e d  by  the  \  64  Flock s i z e , The  p_rey. d i s t r i b u t i o n ,  second  between p r e y flock  for  each  single  experiment  flock  tested  variation flock  over  join the  experiments  in  that  prey  in  the  repeated,  ( INTF = 0 ) .  with twelve total  food  the flock  takes  in  the  thirty-six  i f  large  above  size in  a  f l i g h t , the from  1 t o 16  as b e f o r e .  an  was n e c e s s a r y .  clumps c f p r e y ,  remains  their  maintained  that  additional  While there  a large are  flocks  may  experiments  clumps o f t w e n t y - f i v e  clumps o f s e v e n t y - f i v e prey  supply  with  of food dumpings  aggregation  the  birds  Flock sizes  i t became a p p a r e n t  different  flocking  accordance  whenever a b i r d  Consequently,  replacing  in  same r a n g e  for  experiments  The b i r d s a r e  distribution  clump,  success  s e t of experiments  i s preset.  must  disadvantageous.  the  i n this  success  examines the r e l a t i o n s h i p  feeding  'naturally*  may e x c e l a t l o c a t i n g  prey  each  and  by r e q u i r i n g  birds  During.these  experiments  i n the f i r s t  flocks  behaviours,  remaining were  Whereas  their  modelled  of  distribution  sizes.  formed  set  and f e e d i n g  each.  Note  few be were prey that  constant.  Results  The 7-10  (see A p p e n d i x  Figure 7 each) 1.  results  of these experiments  B f o r t h e same r e s u l t s  (mean c a p t u r e s ;  thirty-six  r e v e a l s the f o l l o w i n g For a g i v e n f l o c k clumping,  are presented  size  main  i n tabular  clumps,  i n Figures form).  twenty-five  prey  points.  t h e mean c a p t u r e  rate  r e a c h e s a maximal v a l u e , and t h e n  i n c r e a s e s with decreases  (as  65  Flock Size  F i g u r e 7:  Mean c a p t u r e s In 20 minutes w i t h 36 clumps oE 25 prey  each.  The diagrams are a three d i m e n s i o n a l p l o t In p e r s p e c t i v e of moan c a p t u r e * (heif.ht of peaks) and a contour man of t h e sane p l o t . The x - a x l s g i v e s f l o c k s i z e , the y - a x l s fitves the decree of clump inp, i n C ("in 16-C, where C i s Lloyd's Index of mean crowding). The v a l u e s on the contour map arc numher of prey capture Jn 20 p i nu t e s .  66  also  indicated  in  Figure 4).  size,  there i s a  'best*  prey  Thus, f o r a f l o c k o f  environment  i n terms  clumping. 2.  For  prey clumping  most s u c c e s s f u l . of  the  size 3.  size  i s otherwise  flocks  are  Except size  a s m a l l range  for  -  flock sizes  clumping, to  form  are  flocks  (when  flock  clumping  larger  are  more  degrees  flocks  of prey  clumping  smaller  successful.  extreme  which  expected  prey  unconstrained).  while f o r high more  of  f o r a given prey  might be  v a l u e s of prey  successful,  4.  Thus,  'best*  F o r low  of •,  a given  fixed  yields  clumping, greater  there i s always  success  than  some  does  flock  solitary  foraging. 5.  The  highest  moderately  Figure  8  shows  that  several 1.  The  capture  clumped  r a t e i s f o r p a i r s of  feeding  on  food.  (mean c a p t u r e s ; t w e l v e increasing  birds  the  clumps,seventy-five  number o f f o o d i t e m s  prey  each)  per clump  alters  o f the p a t t e r n s . pattern f o r fixed  for  larger  flocks  flock have  size  still  h o l d s , but t h e  shifted  to  higher  clumping  still  peaks  values  of  clumping. 2.  The  pattern for fixed  stongly. 3.  fl  simple  flock  size  prey  h o l d s , but  net  as  , . relationship  between  i s no l o n g e r  apparent.  prey  clumping  and  'best*  67  F l o c k Size F i g u r e 8:. Mean c a p t u r e s i n 20 minutes with 12 ciumpii of 75  preyench.  The diagrams are a three dimensional p l o t i n p e r s p e c t i v e o f r.ean c a p t u r e s (h^tfiht of peaks) and a contour man of the same p l o t . The x - n x l ? F.lves f l o c k size, the .y-axla j?,lvos the decree of clumpi n g l h C' ( l n 16'C, where C l a Lloyd's index of mean crowding).' The v a l u e s on the contour map a r c number of ptey capture i n 20 minutes. a  68  4.  Again then  5.  t h e r e i s g e n e r a l l y some f l o c k the s o l i t a r y  Again  the  The of  larger  first for  clumped  major  test  these  the  results  difficulty reduced  i s f o r pairs  seems t o be t h e  flock  i n t h e second  with  about  a patch  compared  1.5 p r e y  success In t h e  per  s e t of tests  bird  a patch  per b i r d .  Thus,  of sixteen, the increased  when t h e number with  greater  4.5 p r e y  that f o r a flock  of locating  c f b i r d s on  o f . p r e y clumping.  p r o v i d e d o n l y about  r  small  rate  higher degrees  (16) . w h i l e  suggest  was  here  patch  same  better  food.  at  a single  a large flock  provided  capture  difference  flocks  which does  bird,  highest  moderately  size  the  of  patches  was  i n c r e a s e d value of the  patch. The  patterns of r i s k  success. clumping with come (C  From  for single  with  large  = 2-5).  ;  birds.  In a d d i t i o n  risk  generally  flocks  (12-16) and moderate  flocks  reduction  Figure  10), we f i n d  risk  decreases and  f o r large then  i n t h e high clumping  declines  s i z e . . The g r e a t e s t r e d u c t i o n s i n r i s k  (see  =2-4)  prey  i n c r e a s e s with  When t h e f o o d  (C  for  risk  = 2-4).  1  those  we s e e t h a t  Smaller risk  from  9  an i n c r e a s e i n f l o c k  noteworthy (C  Figure  are rather different  (2-6)  for  exhibit  moderate  a  supply i s divided  into  much t h e same p a t t e r n .  range  clumping  smaller  degrees  f l o c k s i n t h e moderate  risk  t c high  of clumping fewer In both  clumping  increases sharply f o r a l l flock , ( C .= ,4-6). 1  The  but  major  clumps cases range sizes  conclusion  69  Flock Size  : P r o b a b i l i t y of no .captures In 20 minutes with 1?. clumps of 75 prey e a c h . The d l n i ; r a r a 3 a r e a t h r e e dimensional p l o t In p e r s p e c t i v e and a cont o u r map o f the sane p l o t . The x-axts R I V C S f l o c k s i z e s , the y - a x l s g i v e s t h e decree, of clumping In C (--In l f , c , wlicre C Is L l o y d ' s Index o f mean crowding). The values on the contour r.ap are perc e n t a g e o f times a b i r d can expect to make no captures In 20 n t n u t c s .  71  here  is  risk  of  that  large flocks  obtaining  no  food  of  the  (up t o  16)  over  a  substantially short  time  reduce  the  horizon  (20  minutes).  £l§cussion The  results  simulation  experiments  flocking  does i n c r e a s e t h e e f f e c t i v e n e s s w i t h  exploit  food  stressed  that flocking  of  food  resources.  capture,  important  doing, badly. may  be  I n an  result  results  consequence of  which  i n an  suggest  flocking  than  a more a p p r o p r i a t e measure  of  workers  increase in may  reducing  the  fitness  have  the  that t h i s  u n p r e d i c t a b l e environment,  that  individuals  However, w h i l e p r e v i o u s  might  these  show  be  rate a  less  risk  minimizing than  of risk  maximizing  efficiency. In  the  yielding those  yielding  must  and  the  look  at  a day.  risk  period capture  account is  to  consider will  (the t i m e rate  unit  that of  the the  throughout  were  and  prey  the  not  be below  day,  to e v a l u a t e  rate.  that  an  can  work  level a  20  representative we  way  hours  rate for a  we  One  some c r i t i c a l  f e e d f o r 10  is  sizes  generally  probability  mean c a p t u r e  model)  flock  distribution,  capture  the  birds  these  Thus i n o r d e r  given  for risk  foraging success  assume  rate  risk.  F o r e x a m p l e , i f the  we  experiments,  capture  smallest  simultaneously  i f  size  size i s "best" for a  individuals for  flock  the g r e a t e s t a v e r a g e  which f l o c k  to  fixed  day  minute of  the  out  the  72  probabilities feeding for  of  critical  16  to  of prey  be a r e a s o n a b l e be a r o u n d  In  be  (neither  the s i z e  rate  when p a t c h e s  2.  select  In in  per patch  flock  between  the " r i g h t "  addition size,  expect  while f o r birds  are  per hour).  For  would seem  capture  to  r a t e has  to  are l i k e l y  i n locating  comes  to  copy  from,  be most  food  t o have  (because o f  neighbors).  interference In t h i s  apparent  feed the f l e c k  b e t t e r success of b i g f l o c k s  success  Thus,  when  The  between  system, the clumps  of  and dense enough t o  searching  in  the  patch.  o u t i n t h e s i m u l a t i o n r e s u l t s by:  The i n c r e a s e w i t h in  prey  The a v e r a g e  the f e e d i n g success of b i r d s  more p r e y  we  birds,  solitary  o f food are found.  should  enough  These t r e n d s a r e born The  hour  a t around.4  tendency  flocking  advantage o f f l o c k i n g  1.  single  t h e advantages of f l o c k i n g  s e a r c h i n g and a  increase  per  increased efficiency  large  (Table V I ) .  o v e r a 10 h o u r day.  more e y e s  are  over  prey  threshold  2.5 p r e y p e r hour,  for survival.  from  of  various  i n t h e m o d e l , 2.5 p r e y / h o u r  4 prey/hour  disadvantage  below  and when i n a f l o c k  i s favoured  resulted  food  fall  favoured 5.0  used  t h e model,  individuals  will  f e e d i n g r a t e s around  r a t e s above  favoured  to  a bird  r a t e s when a l o n e  critical  flocks  that  ( a l t h o u g h fewer  flocking flock  will  16) when t h e r e a r e  patches) ;  i n c r e a s e d prey clumping and s o l i t a r y  of the difference birds  ( p r o v i d e d we  size).  we would e x p e c t Copying  (e.g.  tend  risk to  r e d u c t i o n with be  more  an i n c r e a s e  beneficial,  when  73  Feeding  Flock s i z e Solitary F l o c k o f 16  0.0/h 0.00 0.00  T a b l e VI]. Probabilities that a bird's less then various levels c a p t u r e r a t e f o r 20 minute distributed).  rate 2.5/h 0.05 0.01  5.0/h 0.26 0.48  7.5/h 0.61 0.98  10.0/h 0.89 1.00  average d a i l y f e e d i n g r a t e will be (the assumption is made t h a t the periods is approximately normally  74  there  are  together. this  more  should  i n c r e a s e s and  b o r n e out  except  interference The cover  small  less  have  that  birds  as,prey for  that  and  Variation  be  may  likely  variation  with birds  the  was  prey  prey  degrees  are  prey  capture rate  birds  feeding  =  was  bird, flock  trends  are  where  4 -  in  The  results  when  patches have  interference  r e g a r d l e s s of the  only  suggests  the a r e a i t flock,  some  captures.  flock  and  birds  into  cn the  prey  capture  model.  The  probability  of prey d e t e c t i o n  probability  of p r e y d e t e c t i o n  These  found  should  alone.  hold  patches  flocks  5.when p a t c h e s  to  between t h e  to d i f f e r  when  patch i s s m a l l .  C* = 5 - 6  a patch  parameters:  earlier  per  examine t h e e f f e c t s  p r e f e r e n c e , ±10%;  likely  close  clumping  number o f  i s too s m a l l  p r e f e r e n c e , ±20%.  of  most s e v e r e  t o make no  i n two  These  s h o u l d be  those  will  are  role.  exhaust  experiments  birds  unsuccessful  increases.  high  between C*  of admitting d i f f e r e n c e s  without  clumping  between i n d i v i d u a l s i n t h e  These  the  A c o n s i d e r a t i o n of p h y s i c a l  when t h e p a t c h  will  benefits  p o i n t comes between  each  each.  when the  values of prey clumping,  this  prey  and  t r e n d s i n r i s k r e d u c t i o n as  very  area,and  while a f l o c k  covers,  produce  s u c c e s s f u l . than  75  25 p r e y  mainly  c o s t of s h a r i n g food  a  suggest  copy  p l a y s a dominant  Thus, f o r high be  to  Since copying  effect  size  birds  two  were  chosen  i n perceptual a b i l i t y , t o be  sensitive  to  since and  b)  changes  a) prey in  75  probability should has  s e e some s i g n i f i c a n t a  performed five  of prey d e t e c t i o n  major  impact).  f o r both  prey  prey p r e f e r e n c e  changes i f v a r i a t i o n These  flocking  values of prey  without  simulation  (hence  between  birds  experiments  and n o n f l o c k i n g b i r d s  we  were  f o r a range of  clumping.  Results Tables  VII  experiments. 1.  had no e f f e c t  flocking  prey prey 2.  the  results  of  these  upon mean c a p t u r e r a t e o r mean  or n o n f l o c k i n g b i r d s over  the e n t i r e  risk  range o f  clumping.  No s i g n i f i c a n t was  present  The major c o n c l u s i o n s a r e :  Variation for  and. V I I I  found  difference  between  the  i n mean c a p t u r e above-average  rate  c r mean  and  risk  below-average  birds. 3.  Given  individual  capture r a t e able birds  variation,  of the better  nor v i c e  versa.  flocking  did  not  b i r d s a t the expense N e i t h e r was r i s k  enhance  c f the l e s s  significantly  reapportioned. 4.  When  flock  size  s u c c e s s f u l than  was  smaller  fixed, flocks.  flocks  of  8-16  were more  76  C  F-Iv  flf-Iv  F-Niv  0.07 0. 13 0.25 0. 33 0.46 0.62 0.74 0. 88 1 .01 1. 23 1.48 1.86 2.44 3.64 5.24 9. 48  1 .28 1.30 1 .34 2.04 2.18 1.90 2.44 2.61 2.59 2.91 2.73 1.85 4.00 4.09 3.02 2.49  1.42 1.52 1.67 1.52 2.32 2.07 1.92 2. 19 2.06 3.11 2.34 2.54 2.81 3.31 2.40 2.57  1.30 1.55 1.27 2.12 2.04 1.40 2.26 2.45 2.23 3.23 2.20 3.02 4.28 3.34 3.44 2.35  T a b l e VIIj. Capture rate for flocking birds nonflocking birds with individual without individual variation, and individual variation.  1.08 1.66 1.76 1.46 1.85 2.20 1.88 2.16 2. 16 2.93 2.63 2.72 2.78 3.36 3.58 2.31  with individual variation, variation, flocking birds nonflocking birds without  77  C  F-Iv  Jf-Iv  F^Niv  Nf-Niv  0.07 0. 13 0.25 0. 33 0.46 0.62 0.74 0. 88 1 .01 1.23 1,48 1. 86 2.44 3. 64 5.24 9. 48  30 36 35 27 40 41 29 29 41 42 34 65 29 36 52 49  31 31 35 43 36 48 47 51 53 47 57 56 61 63 68 73  29 25 34 36 38 46 33 43 38 37 47 49 38 46 45 56  32 30 37 39 44 40 54 44 49 53 53 52 63 62 59 73  Table V i l l i R i s k o f making no c a p t u r e s i n t w e n t y minutes expressed as a percentage. The four columns respectively are f o r f l o c k i n g birds with individual variation, nonflocking birds with individual variation, flocking birds without individual v a r i a t i o n , and n o n f l o c k i n g b i r d s w i t h o u t i n d i v i d u a l v a r i a t i o n .  78  It  i s a somewhat s u r p r i s i n g r e s u l t  detection The  ability  made l i t t l e  by  variation  comparison process. small the  with The  i n prey  the  ones s u g g e s t s  prey  detection  effectively  success  i n success of  t h a t as  variation  for  than  inherent  of  the  in  a  average  small  in  search  comparison  to  the  average  of  flock  is  large  of  success  i n the  copying,  birds  prey  success. in  was  large flocks in  a result  abilities  greater  i s t h a t the  birds'  detection a b i l i t i e s  variation  greater  differences in  d i f f e r e n c e i n the  most p l a u s i b l e e x p l a n a t i o n  created  that  the  individual  abilities.  Pcl^mcr^hic  Prejj  Before of  Populations  presenting  prevailing  populations.  ideas In r e c e n t  may  select  form  of s e l e c t i o n ,  occurs  when  proportion morphs This  for visual  in  the  than  lower  of  5  1954; on  on  the  years  role i t has  polymorphism  of  predator  been a r g u e d in their  takes  than  been i n v o k e d ,  common  in  the  they  artificial  baits  Owen has  i n the to  in natural ^predators  in  This 1962),  greater and  rare  population. explain  cf s n a i l s  1965). been  given  (Clarke  population  occur  is  species.  morphs  f o r example,  1960,69;  that  prey  polymorphisms i n s e v e r a l s p e c i e s Clarke  review  polymorphism  known a s a p o s t a t i c s e l e c t i o n  frequency  maintenance  selection  experiments, a b r i e f  their.occurrence  phenomenon has  Sheppard  the  the (Cain  Apostatic  demonstrated  in  79  laboratory 1968;  studies  Allen  prey  for various  1972;  Manly,  Kettlewell  predation  on  Murton's  that  they  the  species 1965)  1972  apparent at high  and  1969).  densities  possibly we  combined  attack  rate  on  of  any  being  kept  the  are such  note  three  an  that  the  the  commoner  the.  this  Greenwood  d e n s i t i e s as  a trimorphic  little  difficulties of  Croze's  low  gave  polymorphism  in  population  to  predator  the  1969;  prey  Holling  well  was  of  difficulty)  (Greenwood  (1970)  monomorphic p o p u l a t i o n s  polymorphism  avian  predator,  a v a i l a b l e to the possibility,  increase  on  low  that  favored  handling  ( A l l e n 1972; very  natural  study  the  smaller  than  (overall  density  constant).  Accepting against  at  Using  b e t u l a r i a , while  preferences  of  Clarke  pigeons  are  a method o f  disadvantage  Finally,  with  situation  present  1972).  &  disproportionately  disproportionately .the  (Allen  moth B i s t e n  experiments  for density-independent  (Manly e t . a l . and  6 Cook  demonstrated  field  Confounding  accounting  Miller  species  a c r y p t i c morph o f t h e  (1971)  indication prey.  (1955)  bird  the  birds.  one In  feeding  be  may  this  examining the  birds'  may  an  ask  effective  what  section  value  success  we  ccunterstrategies investigate  o f . f l o c k i n g as on  a  strategy  one  a means  polymorphic  to  prey  80  population.  6  Since  preferences, p r e d a t o r s on one  modelled  can. expect  a polymorphic  required  encounter and  to  rate  polymorphism  prey.  maintain  is  copy  prey  and  prey  the  another  a  nearby site  unlikely  1.  extent  population mean  into  capture  flocking  and  two  birds)  and  individual  and  When  rarely  in  one  flock  learning  in  prey  the  flocks morph  to  birds. ,  types  were  intended  cf a  (morphs)  i s this effect  prey  to  a  more r a p i d l y  attacking  the  formed,  advantage  Therefore, birds  variation  do  mcnomorphic  reinforcement  a bird  partitioning  how  i n t o two how  thus  prey  effective  preference.  which f e l l o w  distinct  rate  cn a  the  t o copy,  solitary  the  term  be l e s s  c o n f e r any  actions,  does  Does t h e p a r t i t i o n the  to  s i m u l a t i o n experiments  what  prey  ability  t h a n do  questions.,  will  exceeds  t o . s w i t c h from  answer s e v e r a l To  rate  characteristics.  more r a p i d l y  The  they  short  p o p u l a t i o n than  the  bird's  potential  form  prey p r e f e r e n c e s are  However, g i v e n t h e i r  may  2.  prey  i s too low,  birds  that  whenever t h e p r e y e n c o u n t e r  rate  have  we  the  single  decrease  to  prey ;  influenced  the by  (as d e f i n e d p r e v i o u s l y ) ? types  flocking  and  increase individual  risk  (to  variation  Since the d i s t i n c t i o n i n t h e model i s between p r e y ' t y p e s ' , t h e s e t y p e s may r e p r e s e n t d i f f e r e n t s i m i l a r p r e y s p e c i e s rather than morphs of. a single species. Hence, this study a l s o p e r t a i n s to the c o e x i s t e n c e of s e p a r a t e prey s p e c i e s as a r e s u l t of density dependent predation (e.g. Murdoch 1969). In addition t h e morph d i s t i n c t i o n may be based upon d i f f e r e n c e s i n m i c r o h a b i t a t r a t h e r t h a n on v i s u a l d i f f e r e n c e s . Throughout the thesis the term polymorphism w i l l be used t o r e f e r t o any of these cases. 6  81  alter 3.  this  effect?  Does f l o c k i n g  alter  the r a t i o  o f a t t a c k s upcn t h e two  prey  types? 4.  To what e x t e n t does i n d i v i d u a l in  capture  rate  below-average  and r i s k  ones,  variation  create  differences  between a b o v e - a v e r a g e  and does f l o c k i n g  mitigate  birds cr  and  enhance  the d i f f e r e n c e ? . 5.  Do  the  birds  attack  disproportionately  The to  following  by  the a c t u a l  the  same  distributions one-prey  size  were i d e n t i c a l w i t h  prey.  At  random number s e q u e n c e s ) twenty  birds  combinations: without  individual  variation;  second  with  foraging  to  total used  variation;  in  clumps  highly  of  clumped (different  testing  flocking,  or i n d i v i d u a l experiments  the  clumping  variation;  without  prey  experiments  minutes,  and i n d i v i d u a l  types  determined  The  simulation  f o r twenty  simulation  performed  Five values of  The  without f l o c k i n g of  was  thirty-six  (C=0.07)  were r u n .  individual  set  types  distribution  of these, f i v e r e p l i c a t e s  with f l o c k i n g  flocking,  the  random  each  and  prey type  (900 t o t a l p r e y ) . from  were  the d i s t r i b u t i o n s  experiments,  ranging  four  (same h a n d l i n g time)  distribution).  were u s e d ,  prey  s e t t h e two p r e y  probability  prey each  modelled  In the f i r s t  of each  twenty-five  two  experiments  distribution  simulation  (C=9.48)  of  often?  examine t h e s e q u e s t i o n s .  (i.e.,  commoner  sets of simulation  were e q u a l i n number,  In  the  the with with  variation.  o n l y the p r e y  82  density  was c h a n g e d ; t h e two p r e y  previous densities experiments  variation  identical fourth  (1800 t o t a l p r e y ) .  examined,  individual  to  types  that  the  effect  present. in  the  a  9: 1 r a t i o  of  three types  two  sets  fixed  with  used  was  s e t (900 t o t a l p r e y ) .  In a  prey  flock density  the prey  case  with i n d i v i d u a l  (900 t o t a l  of s i m u l a t i o n experiments variation  summarizes t h e s e e x p e r i m e n t a l  their  set cf simulation  were p r e s e n t e d  (900 t o t a l p r e y ) , and i n the. l a s t i n a 1:1:1 r a t i o  twice  size  The  s e t o f s i m u l a t i o n experiments  at  A third  of  first  were  s e t the prey  prey).  between  the  were  These  were c o n d u c t e d  only  birds.  in  last  f o r the  Table  IX  sets  of  designs.  Results Mean  prey  simulation  experiments  corresponding case.  The  reduction all  partition  distributed  prey,  nonflocking,  of  the  the r e d u c t i o n s t i l l  .002  t o .006).  two  i n Table X along  with  the  .caused  was  find  first  into  two  types  quite four  a  significant  (p<.0001; u n l e s s o t h e r w i s e with  reduction,  individual  the  case and the t h r e e - p r e y  were, e v a l u a t e d This  each  for  f o r t h e one-prey  i n mean c a p t u r e r a t e  1956).  Examining  rates,  are presented  results  hypotheses  Siegel  capture  the  while  large  variation, significant  small  for  cases  randomization fcr  nc i n d i v i d u a l (significance  test,  .randomly  h i g h l y clumped  separately  noted,  prey.  (flocking,  variation), levels  we  from  83  Experiment set 1 2 3 4 5  # prey types 2 2 2 2 3  prey ratio  total prey  flock size fixed  1 :1 1:1 1:1 9:1 1 :1:1  900 1800 900 900 900  No NO Yes No Wo  c a s e s examined F Nf N f F Iv Niv Iv Niv  * * * * *  * *  * *  * *  *  T a b l e IX_: Design of the f i v e sets of experiments. Under cases, examined the a b b r e v i a t i o n s a r e as follows: F flocking; Nf nonflocking; Iv with individual variation; Niv - without individual variation.  84  Clumping 0.07  0.74  1.48  5.24  9.48  2 prey types t o t a l c f 900  F -Iv F -Niv Nf-Iv Nf-Niv  1.12 1.08 1 .06 1.13  2.56 2.20 2.15 1.93  2.22 2.13 1.86 1.75  2.42 2.52 1.83 1.81  2.21 1.39 1. 14 1 .34  1 prey type t o t a l o f 900  F -Iv F -Niv Nv-I v Nf-Niv  1 .28 1.30 1 .42 1.08  2.44 2.23 1.92 2.16  2.73 2.20 2.34 2.63  3.02 3.44 2.40 3.58  2.49 2.35 2.57 2.31  2 prey t y p e s t o t a l o f 1800  F -Iv F -Niv Nf-Iv Nf-»iv  2. 16 1 .99 2. 14 2.06  3.79 3.26 3.84 3.14  4.15 4.51 3.33 2.91  4.99 3.41 4.28 3.63  3.26 2.82 2.31 3.08  3 prey t y p e s t o t a l c f 900  F -Iv Nf-Iv  0.95 1.25  2.34 2.04  2.40 „ 2.21  1.50 1.29  1 .54 1.30  T a b l e X_: Hean c a p t u r e r a t e s a t f i v e v a l u e s o f p r e y c l u m p i n g and f o r e a c h combination of flocking and nonflocking, with and without i n d i v i d u a l v a r i a t i o n , where F = flocking Nf = nonflocking Iv = individual variation Niv = no i n d i v i d u a l v a r i a t i o n . The m u l t i p l e p r e y results are f o r equal numbers, size and d i s t r i b u t i o n o f the prey t y p e s .  85  Whereas was  found  X) ,  i n the s i n g l e  t o have no  with  two  prey  mean c a p t u r e r a t e Additionally, prey  than  birds those  effect  upon mean c a p t u r e  individual  variation  prey  individual  i n the o t h e r  variation  t h e monomorphic c a s e  successfully  appears  smaller  that  the  the a d d i t i o n  of  a  greater  distributed  way  At  marginal  morph  protection  impact  set  for  with is  flocks  in  one.  rates  feeding used  here  protection  is  than i t i s f o r  Also  clumped  than  trimorphic  the d e n s i t i e s gain  flocking  with c a p t u r e in  twc most  better  of  morph t o twc  to  clumped.  (C=9.48), where  birds  alone.  of a t h i r d  second  to  magnitude  of  the  partition  into  two  Again  and  increase  are  did substantially  Table  polymorphism  prey  than  randomly  prey.  Another  smaller  .prey's  (see  distinction  t h e above r e s u l t s ,  than,those  f o r the a d d i t i o n  provides  This  t h r e e c a s e s . . The , s m a l l  reduced  from  prey  has.greater  case.  substantiate  it  when  a t extreme v a l u e s of clumping  with  rate  flocking  does s i g n i f i c a n t l y  (p<.00T); e s p e c i a l l y  experiments  more  s i m u l a t i o n experiments  types, flocking  i n the s i n g l e  prominent  prey  for birds  view  these  reduction prey  types  in;flocks  individual variation  results  in  capture  (see T a b l e  than  -is... t c rate  XI).  for solitary  appears  desirable  examine caused  the  by  the  This reduction i s foragers (net  (p=.066).  significant,  .10<p<.20).  When flocking  flock paying  sizes  are f i x e d  (see T a b l e  o f f r a t h e r more t h a n  i n the  X I I ) , we single  again prey  find case.  86  Clumping  F -Iv F Niv Nf-Iv Nf-Niv  0.07  0.74  1.48  5.24  9.48  .16 .22 .36 -.05  -.12 .03 -.23 .23  .51 .07 .48 .88  .60 .92 .57 1.77  .28 .96 1.43 .97  Table X l i The reduction of capture rate by the p a r t i t i o n o f the prey p o p u l a t i o n i n t o two d i s t i n c t t y p e s . Values a r e the d i f f e r e n c e in mean c a p t u r e r a t e between t h e one p r e y and two p r e y r e s u l t s when b o t h a r e a t t h e same t o t a l  87  Clumping Flock 1 2 4 8 16  size  0 .07  0. 74  1. 48  5 .24  9.48  1 .06 .90 1 .20 .90 1 .18  2. 15 2. 00 1. 70 1. 46 2. 11  1. 86 1. 25 3. 13 2. 54 1. 94  1 .83 .95 2 .20 1 .41 1 .95  1.14 1.65 1 .95 1.35 2.37  T a b l e XII,: Mean c a p t u r e r a t e s f o r d i f f e r e n t f l o c k s i z e s . The p r e y are of two t y p e s w i t h t o t a l d e n s i t y o f one p e r s i x t e e n s g u a r e y a r d s .  88  I n p a r t i c u l a r we f i n d ..large f l o c k s above t h e s o l i t a r y Risk,  the  apparent  one  minutes,  that  population prey  birds.  percentage  twenty s i m u l a t e d  p r o b a b i l i t y o f making no c a p t u r e i n  i s presented  risk-increases  with  (p<.00005); and t h a t case,  (16) c o n s i s t e n t l y e g u a l t o o r  to  be  an  in  Table  the p a r t i t i o n  effective  there  for  v a r i a t i o n between t h e b i r d s .  the trimorphic  third an  is  morph g e n e r a l l y  advantage.  single  prey  reduction, the  case,  and r i s k  large  increases  prey  are  feeding  equal  and  real  birds  foraging  equally  area  before  of  held the  b i r d s remain a t  the  as c l u m p i n g  greatest  risk  increases  for  regarding  the d i s t r i b u t i o n of  R:S, where R<S and R*S=1.  move  t h e prey a r e s u f f i c i e n t l y been  the a d d i t i o n  l e t us s u p p o s e t h e two t y p e s  limited  .will  clumped  ( f l o c k s of 16).  distributed exhausting  be n e a r l y  rare  to  ensure  ( b i r d s ..are the  food  period of  equal.  on t o new f e e d i n g  supply  But  grounds that  known  are  Since the  i n number, o v e r a l o n g  i n a s i n g l e a r e a . R.and S w i l l  general before  proportions  highly  Flocking  the  reduction  These r e s u l t s  produce  little  the t h i r d , q u e s t i o n  in.the  in  (see T a b l e X I V ) , as w i t h t h e  flocks  on t h e two p r e y t y p e s ,  captured  risk.  For f i x e d f l o c k s i z e  answer  attacks  as w e l l , with  increasing  largest flock size tested  To  have  population  as  means o f r i s k are  It i s  o f t h e prey  f l o c k i n g continues,  (p<.006) and p<.001), e s p e c i a l l y when p r e y and  XIII. .  in well  attacks  to l e a v e ,e.g.  a  Gibb  89  Clumping 0.07  0.74  1.48  5.24  9.48  2 prey t y p e s t o t a l c f 900  F -Iv F -Niv Nf-Iv Nf-Niv  39 41 43 40  46 48 46 49  44 49 64 63  47 51 73 68  54 75 82 83  1 prey t y p e t o t a l o f 900  F -Iv F -Niv Nf-Iv Nf-Niv  30 29 31 32  29 38 47 45  34 47 57 53  52 45 68 59  49 56 73 73  2 prey t y p e s t o t a l o f 1800  F -Iv F -Niv Nf-Iv Nf-Niv  18 15 16 17  21 24 39 33  31 32 41 44  20 42 55 61  37 46 67 63  3 prey types t o t a l o f 900  F -Iv Nf-Iv  43 34  42 52  42 67  72 80  65 86  Table  XIIIj. "the percentage probability m i n u t e s w i t h o u t c a p t u r i n g any p r e y . i n T a b l e X..  RiskT  that a bird The n o t a t i o n  w i l l go t w e n t y i s the.same as  90  Clumping Flock 1 2 4 8 16  size  0.07  0.74  1.48  5.24  9.48  43 45 33 36 29  46 65 45 46 32  64 60 45 52 26  73 70 55 59 35  82 80 60 61 31  Table XIVi Risk, the percentage probability cf a bird's gcing twenty minutes without c a p t u r i n g any p r e y , f o r b i r d s i n v a r i o u s f l o c k s i zes .  91  (1962)).  Thus,  7  i t is.of  interest  attacks  a r e c o n c e n t r a t e d on one  single  feeding  attack  ratio  simulation  episode  clumping in  of  Table  XV.  decreases are  how  this  the prey Two  a  series  term  types.  That  i s , birds  than  do  prey  detection  results  capture  abilities.  has  prey case  r a t e f o r above-average  variation  case  those  birds  (p=.010); but t h e  than  i n t h e no  variation  total  c a p t u r e s f o r both  XVI  w i t h 900  only ..a.small e f f e c t  one  of  prey  of  the  (p=.031).  more  evenly birds  prey c a p t u r e s  below and  total  the XVII  in  summarize  the  Individual  rate.  i s g r e a t e r than  belcw-average  and  average  prey.  on c a p t u r e  of b i r d s  R  ratio.  case. . Furthermore,  groups  cf  Thus, t h e f l e c k i n g prey  Tables  prey  presented  flock  attacks  examine t h e a p p o r t i o n m e n t  f o r the two  variation  their,  m a i n t a i n the i n i t i a l  between t h o s e b i r d s above and  the  T h u s , when  c o n c e n t r a t e d on  f e e d i n g by t h e m s e l v e s .  Next, l e t us  and  the value  when the b i r d s  in flocks distribute  more e f f e c t i v e l y  risk  first,  i n c r e a s e s (p<.00U).  Second, R i s l a r g e r  birds  by f l o c k i n g  Values f o r R are  c o n c e n t r a t e d , a t t a c k s t e n d t o be  two  1:1  the  mechanism f o r d e v e l o p i n g l o n g  distribution.  of  a  the  i s affected  clumping  which  since  f e a t u r e s are e v i d e n t .  as p r e y  to  of f e e d i n g b o u t s  to the prey r a t i o  model p r o v i d e s no  p r e f e r e n c e s ) , and  the degree  prey type or the o t h e r d u r i n g  (over  must c o n v e r g e  to observe  in  b i r d s dc  Thus,  the  the  nc  nc  worse  flocking increases  (p=.003 and  p=.013  for  D e p l e t i o n r e p r e s e n t e d no p r o b l e m i n t h e s i m u l a t i o n e x p e r i m e n t s since never more t h a n 11% o f t h e p r e y were t a k e n i n t h e t w e n t y simulated minutes. , 7  92  Clumping  flecking nonflocking  0.07  0.74  1.48  5.24  9.48  .435 .432  .407 .398  .357 .290  .347 .267  .291 .256  Table XVi The a v e r a g e f r a c t i o n of attacks on the less attacked population for t w e n t y minute f o r a g i n g e p i s o d e s ; two p r e y a t t o t a l d e n s i t y o f one p r e y p e r s i x t e e n s q u a r e yards.  F^ey types  93  Clumping 0.07  0.74  1.48  5.24  9.48  above below  65 47  58 42  126 130  49 S1  104 118  47 53  121 121  50 50  127 94  57 43  above below  54 54  50 SO  100 120  45 55  94 119  44 56  128 124  51 49  65 74  47 S3  Nf - I v  above below  45 61  42 58  102 113  47 53  108 78  58 42  91 92  50 50  79 35  69 31  Nf - N i v  above below  50 63  44 56  88 105  46 54  89 86  51 49  89 92  49 51  58 76  43 57  F -Iv F  -Niv  T a b l e XVJ.I Each p a i r of numbers represents the apportionment of prey captures between above-average and belcw-average b i r d s (the cases without i n d i v i d u a l v a r i a t i o n a r e the c o n t r o l group). The f i r s t number i n t h e p a i r i s an a b s o l u t e measure c f c a p t u r e r a t e , t h e s e c o n d number i s the p e r c e n t a g e o f t h e c a p t u r e s g o i n g t o t h e g i v e n group.  94  Clumping 0.07  0.74  1.48  5.24  9.48  F  -Iv  above below  32 4 1 46 59  42 4 6 50 54  38 43 50 5 7  44 4 7 50 5 3  46 62  43 57  F  -Niv  above below  40 49 42 5 1  46 4 8 50 5 2  40 53 36 4 7  46 4 S 56 5 S  70 80  47 53  Nf -Iv  above below  44 5 1 42 4 9  50 54 42 46  62 4 8 66 52  72 4 9 74 5 1  80 4 9 84 5 1  Nf - N i v  above below  38 4 8 42 52  54 55 44 4 5  64 5 1 62 4 9  68 5 0 68 5 0  80 86  48 52  Table Mill Each pair of numbers represents the apportionment of r i s k between above a v e r a g e and below a v e r a g e b i r d s (the c a s e s w i t h o u t i n d i v i d u a l v a r i a t i o n a r e t h e c o n t r o l g r o u p ) . . The first number i n t h e p a i r i s an a b s o l u t e measure o f r i s k , t h e s e c o n d number i s t h e p e r c e n t a g e o f t h e r i s k assumed by t h e g i v e n g r o u p .  95  above-average  and  gains a c c r u i n g to greater  than  the p o r t i o n average) the  affect  of the  nor  risk  is  with  with  the  variation by  was  .125  the  again  greater  percentage  of r i s k  in  (or b e l o w -  clumping  reduces  risk  n e t f o r the  below  and  this reduction  prey  risk  birds.  capture  rates,  amongst the  each  birds reducing their  of does  above-average  with  does r e a p p o r t i o n t h e  would.in  fact  performed  group  risk  are  this  protection  were  separately.  On in  prey a 1:1  more in  birds  (p=.065),  relative  to  9:1  effective  ratio,  test  tc  the  the  whether  selection  in in  a  i n T a b l e s XVIII  to  distributed  Polymorphism  prey  as  flocks  though the e f f e c t  with  1:0.  ratio.  proportions  the cn  (reduction i n capture  than  equal  were  1: 1, 9:1,and  to provide p r o t e c t i o n  and  birds  presented  p r e y each  between morphs b e i n g  seen  to  exert, apostatic  prey ' population  when t h e p r e y a r e 1:1  generally  case  Also,  variation  Flocking  respectively),  E i g h t e e n clumps of f i f t y ratios  with the  (p=.031) f o r t h e to  birds..  t o the above-average  population.  the  significantly  However, i n d i v i d u a l  o f , experiments  population,  the  the  birds  polymorphic,  net  with  birds.  Results  the  greater  contrast  below-average  XXI.  going  flocking.  above-average  simulated  birds  vary s i g n i f i c a n t l y  (p=.031 and  in  as measured  respectively),  f o r t h e above-average b i r d s but  significantly  individual  capture  on a p o l y m o r p h i c  birds  Finally,  above-average  does not  significantly average  the  birds  those a c c r u i n g to the below-average  birds  prey  below-average  the of  a  total  rate)  was  Furthermore, than  foraging  attacks prey  were  preference  96  Clumping 0.07  0.74  1.48  5.24  9.48  flocking  1: 1 9:1  88 91  81 94  70 90  83 89  89 98  nonflocking  1: 1 9:1  75 78  68 86  48 72  76 80  51 80  Table XVIIIi Total prey c a p t u r e r a t e as a percentage o f the c a p t u r e an i d e n t i c a l l y d i s t r i b u t e d monomorphic p r e y p o p u l a t i o n . morphs a r e i n t h e p r o p o r t i o n s 1:1 and 9:1.  r a t e cn The two  97  Clumping  flocking  nonflocking  0.07  0.74  1.48  5.24  9.4.8  1:1 sig.  54.5 NS  56.6 .034  59.0 .008  51. 2 . NS  42. 1 .02 0  9:1 sig.  92.2 NS  99.0 <10~  96.4 <10~*  97.0 .0002  1:1 sig.  49.1 SS  45. 1 NS  56.5 .076  51.4 NS  43. 9 NS  9: 1 sig.  90.1 SS  95.3 .002  96.8 <10"*  97.9 .0001  93. 2 .074  6  98.8 . <10-s  T a b l e XIX:. The p e r c e n t a g e o f a t t a c k s on prey type 1 in a dimorphic population. I f a t t a c k s a r e i n t h e same p r o p o r t i o n a s p r e y , t h e n f o r p r e y r a t i o 1:1 we e x p e c t 50% a t t a c k s on p r e y t y p e 1, and f o r prey ratio 9:1 we e x p e c t 90% a t t a c k s on t y p e 1. Significance v a l u e s d e r i v e d f r o m t h e normaLl a p p r o x i m a t i o n t o the binomial d i s t r i b u t i o n (minimum number o f s a m p l e s was 1 0 6 ) .  98  Clumping flocking  0.07  0.74  1.48  5.24  9.48  type 1 type 2 c o H o n morph r a r e morph  1:1 1:1 9:1 9:1  96 80 93 71  92 70 103 9  83 57 96 32  85 81 96 27  75 103 108 12  nonflocking type 1 type 2 common morph r a r e morph  1:1 1:1 9:1 9: 1  74 76 78 77  61 75 91 40  54 42 77 23  78 74 87 17  45 57 83 54  T a b l e XXj. Capture rate per i n d i v i d u a l p r e y on e a c h morph i n a' d i m o r p h i c p o p u l a t i o n , e x p r e s s e d as p e r c e n t a g e o f t h e c a p t u r e rate on an eguivalently distributed monomorphic population. The e n t r i e s a r e p a r t i t i o n s by morph o f t h e v a l u e s i n T a b l e X V I I I .  99  Clumping 0.07  0.74  1.48  5.24  9.48  flocking  1:: 1 9:: 1 1 :0 j  39 31 30  46 35 29  44 37 34  47 68 52  54 66 49  nonflocking  1:; 1 9::1 1::0  43 34 31  46 44 47  64 54 57  73 80 68  82 86 73  T a b l e XXIj. Risk, the percentage probability of a bird's gcing twenty m i n u t e s w i t h o u t making a c a p t u r e , f o r d i f f e r e n t prey r a t i o s i n a dimorphic prey p o p u l a t i o n .  100  formation  ' (causing  long  r u n s on  single  mcrph)  on  ratio,  t h e more common p r e y d i d r e c e i v e d i s p r o p o r t i o n a t e l y  case  except  the  when t h e p r e y  attacks  preference  In  was  the high  results  are i l l u s t r a t e d  Table  XX,  individual attacks  frequency  which,  of that  gives  the  predation  population,  rate  on  the  presence o f the second or e l i m i n a t e solitary  morph  morph  Thus,  the l e s s i n predation  for  cn e a c h  received  birds was  polymorphic prey  the  ratio  was  by  flocks  to  the  by t h e reduce  experienced  by  o c c a s i o n a l attempts to Finally,  risk  i n f l o c k s r e g a r d l e s s of  (p<.002) and  (p<.0003), e s p e c i a l l y when p r e y  fewer  t h a n on  not reduced  on common p r e y  lpwer f o r b i r d s  between t h e p r e y  morph p e r  Predation  in  on t h e r a r e and u n p r o f i t a b l e morph.  on  These  far  f l e c k i n g appears  b i r d s ; a l o s s w h i c h a r i s e s from  concentrate  strongly  morph i s a t a l o w e r r a t e  common  morph.  apcstatic  birds.  d i d t h e common mcrph.  whereas  prey  by t h e p r e s e n t a t i o n  attack.rate  morph., The r a r e  b i r d s on the,common  a monomorphic  produce  do s o l i t a r y  more  term  t o s e l e c t more  p e r h a p s more c l e a r l y  per i n d i v i d u a l than  nonflocking  risk  prey than  to  9:1  ( i n which  Thus, s h o r t  found s u f f i c i e n t  a d d i t i o n , f l o c k s appear  against  in  ratio).  the  were i n a  were d i s t r i b u t e d r a n d o m l y  were i n a 9:1  formation  selection.  However, when.the p r e y  shifted  ratio  attacks  some o c c a s i o n s .  a  polymorphism  were  increases  clumped.  Discussion There in  prey  may  i s some e v i d e n c e constitute  an  i n the l i t e r a t u r e effective  that  strategy  polymorphism  against  avian  101  predation In t h i s  (e.g.  C l a r k e 1969;  section  we  the i n v e s t i g a t i o n the a t t a c k r a t e The  distributed process prey  are  i n c r e a s e with  depends  upon  the  clumped  prey.  portions, attacked  small  to reduce  prey  two  of  Such  for  a  prey  clumping  the  when  apostatic  found  significantly  improving  improvement  images).  to  The  randomly  s i n c e the  learning  preference. prey  polymorphic mean  prey capture  i s n o t , however, s u f f i c i e n t  and  presented  These  in  for  unequal the  birds  i t s occurence i s to  be  prey  preferences  being  rare.  the  with  large  revealed that  enhance  of  are  selection  p r o t e c t e d by  in  effectiveness  prey  prey  experiments  1973b), r a r e morphs b e i n g  efficiencies  in  prey  .search  were  short-term  was  reduction  randomly  clumping,  whenever a p r e d a t o r f o r m s  Flocking  the  for  t h e common morph more f r e q u e n t l y than  population.  to  chances of f r e q u e n t encounters  morphs  simulation  support  small  f o r randomly d i s t r i b u t e d  When  the  is  small  to provide reinforcement are  extended  birds increases  a significant  formation  learning  but  chances  the  seems  (e.g.  by  and  additional  reduction  Polymorphism  of  assertion  i t becomes more s i z a b l e . a s  processes  benefits  lent  causes  .this  e t a l . . 1972).  prey.  experiments  polymorphism  prey,  increases. learning  polymorphic  While  distributed  Manly  t o e x p l o r e whether f l o c k i n g  simulation  predation.  1970;  have r e e x a m i n e d t h i s  on  hypothesis that  Croze  in  expected (Krebs  birds'  learning  environments,  hence  rates  for birds.  to o f f s e t  the  This  advantages  102  of  polymorphism  rates  by  birds  population abundant in  to the prey. in flocks  were  less  copying.  mcrph  than  the  capture  to l i e i n the  Thus, the b i r d s  to  another  predation). indicated  when  Additional that  flocks  feeding  combined  The  capture  the  ;  cn  advantage  would first  of  egually flocking for  s e a r c h on  the  include  s w i t c h i n g from  one  became  rare  focus t h e i r  using  through  fixed  f l o c k s a r e a t more o f on p o l y m o r p h i c  an  prey  provides  more r a p i d l y (this  rate  opportunity i t  experiments,  larger  the  on a l l t h e morphs o f a p o l y m o r p h i c  more common morph  smaller  is,  monomorphic p r e y p o p u l a t i o n .  t h e model a p p e a r s  locally  That  heavy  fleck  sizes,  an a d v a n t a g e  prey than  en  over  monomorphic  prey.  The net  introduction  alter  s i g n i f i c a n t l y , the :  however, seem t o average birds. with  birds  reapportion reducing  variation  results risk  their  cited  among  risk  into  a polymorphic birds  is  average  birds  still  To  obtain  of i n t r o d u c t i o n experiments  prey more  birds,  relative  reduced  their  some i n d i c a t i o n s of f i n e r  further  prey  to  of  appeared  that  a g a i n s t . t h e below-  joining  prey  above-  below-average  the  a  below-  flock.  as t o the e f f e c t s  en p r e d a t i o n  (adding  morphs),  p o p u l a t i o n s were compared  dimorphic cases. reduced  by  It did,  with  Nonetheless,  risk  partitions  with t r i m o r p h i c  t h e monomorphic and morph  population selection pronounced.  t h e model d i d  above.  Compared t o t h e monomorphic p r e y c a s e i t  average  third  of i n d i v i d u a l  As e x p e c t e d ,  predation  addition  (primarily  when  with of a prey  103  clumping (at  was  least  fairly  under  the experimental  Considerations have  l e d to  unprofitable its  the  prey  search.on  Charnov  morph  with  are  Proper  less  diet  apostatic  hypothesis  population.  f a r as  the capture  selection  than  found  the  of s o l i t a r y  foraging birds  increase  capture rate,  should  ignore  Tullcck  on  a  are  loss  birds;  prey on  1971;  rare  prey  of a  ones c f a common  foraging")  should  polymorphism  morph rare morph.  lead  in  the  i n the s i m u l a t i o n experiments  in  predation  a loss arising  to occasionally  r a r e and u n p r o f i t a b l e morph. o f t h e commonest  effect  i t s diet  o f a p o s t a t i c s e l e c t i o n and t o  by s o l i t a r y  favor  (e.g.  predation  maintaining  experienced  the  select  the. p r e d a t o r  ones  ("optimal  Flocking.was  eliminate  should  o f more common o n e s ,  profitable  increase the i n t e n s i t y sometimes  that  valuable  so  selection,  marginal  (such a s r a r e morphs) and t o c o n c e n t r a t e  more  In  decreasing  c o n d i t i c n s modelled).  o f how a p r e d a t o r  items  the  1973).  interferes  h i g h ) , but with  on from  a l l but  common the  focus t h e i r  By s e l e c t i n g  type, f l o c k i n g  reduce  r e g a r d l e s s o f t h e abundance r a t i o s among p r e y  prey to or prey  tendency s e a r c h on  mere s t r o n g l y i n  b i r d s reduce  randomly  to  r i s k and  distributed types.  prey  101  S ujmary. Finally, intended and  one  must  note  that  simulation  t o examine p l a u s i b l e i m p l i c a t i o n s o f p r i m a r y  t o suggest  potentially profitable directions for  laboratory  e x p e r i m e n t a t i o n . , The r e s u l t s g i v e n  it  be  would  profitable  conditions  several  1.  adjust  Birds  feeding 2.  experiments are  Birds  test  giving-up  and  here s u g g e s t  that  under n a t u r a l  hypotheses regarding their  field  cr laboratory  flocking.  t i m e s on t h e b a s i s  of  recent  success. in  average  to  hypotheses  flocks  capture  t h a n do s o l i t a r y  neither  birds  .more n o r f e w e r p r e y on  when t h e r e  i s a single  prey  type. 3.  Birds  in  flocks  polymorphic prey  capture  population  more  prey  when  attacking  t h a n do t h e same b i r d s  a  foraging  alone. 4.  Birds for  i n f l o c k s have a s m a l l e r  a fixed foraging  period  variation  t h a n do b i r d s  in  capture  foraging  rate  alone.  105  CHAPTER S I X ! MARKOV  MODELS  M§ikodoiogy There procedure  are four basic  elements  described i n this  to the m u l t i - l e v e l  chapter.  They a r e :  1.  Construction of a detailed  2.  Choice  3.  D e r i v a t i o n o f a " b l a c k box" a s s o c i a t i v e  4.  T r a n s f o r m a t i o n and m a n i p u l a t i o n  of a mathematical  alternative In stated  foci  the  first  and t h e r e l e v a n t s y s t e m  behaviour  to  approach). holistic  of  structure;. mathematical  t h e . model  an  of  i s defined.  tentative  to identify  to the r e a l  identify  corresponding  in  both  the  real  provide  the  validity  of  model  possible  stage  world,  world  structural choices  cf  i.e.i n  (mechanistic  to real  o f t h e model.  basic  mechanisms o f  components o f b e h a v i o u r  In  as well as world the  attributes analytic  mathematical  data  second must be  structure  s i m u l a t i o n and t h e , d e s i r e d f o c u s .  determination o f a compatible  made,  parameters  t o ensure  with  to  In t h i s  by a n a l o g y  found  analysis  compatible  model;  a r e s e a r c h problem i s  p a t t e r n s i n terms of c o r r e s p o n d e n c e s  conducted  is  those  Tests  are necessary  been  underlying  stage of the procedure  model one can d i r e c t l y  stage,  mechanistic simulation;  and p e r s p e c t i v e s o f t h e p r o c e s s .  r e l a t i o n s h i p s a r e modelled the  modelling  When a  model  has  one f a c e s an i n f e r e n c e p r o b l e m ; how t o e s t i m a t e t h e  o f t h e mathematical  ( b l a c k box) model.  o f t e n compounded by t h e t e c h n i c a l  The  problem  and e c o n o m i c r e s t r i c t i o n on  106  sample  size  estimation  of  o f parameters.  relationships second  level  analytic  simulation  exist  runs  One  must  between  are a v a i l a b l e  note  the ease  model v a l i d a t i o n ,  manipulation  which  and  that  of parameter  the  permissible  also  richness  with,  f o r the  trade-off estimation,  of  possible  t h e model. . The  s t a g e employs v a r i o u s m a t h e m a t i c a l  techniques t o manipulate  second  model)  level  model  (black  m a g n i f i c a t i o n s and f o c i  In t h i s simulation  focussed and  In model,can  questions  the  are a v a i l a b l e  short-term  s t r u c t u r e chosen  behaviour  number  o f a Markov  sections  be o b t a i n e d from and  of  this  of  long  alternative  to c a s t  c h o i c e was  run  of  problem  stability  both  the  dictated  the r e s e a r c h  powerful a n a l y t i c  f o r an e x a m i n a t i o n  the f o l l o w i n g  are developed  8  nature of the s i m u l a t i o n , upon  to obtain  the  studied.  model i s a Markov p r o c e s s .  convergence, which  of the p r o c e s s  study the mathematical  by t h e s t o c h a s t i c which  box  final  and  procedures  long-term  and  model.  the procedures  a simulation  (or t h e  by w h i c h "real  a Markov system")  applied.  A Markov process i s any parametric stochastic process { X(T),T>0 } s u c h t h a t f o r any s e t o f n t i m e p o i n t s TKT2<...<Tn i n t h e index s e t o f the p r o c e s s , the c o n d i t i o n a l d i s t r i b u t i o n of X (Tn) , f o r g i v e n v a l u e s o f p r i o r X's, depends o n l y cn X ( T n - 1 ) , t h e most r e c e n t known v a l u e ; i . e . f o r a n y , r e a l numbers X1,...,Xn b e l o n g i n g t o the s t a t e space, P[ X (Tn) <Xn| X (T1)=X1 , ., . ,X (Tn-1) =Xn-1 ] = P[X (Tn)<Xn|X (Tn 1)=Xn-1 ] . I n t u i t i v e l y , t h i s means t h a t f u t u r e v a l u e s depend o n l y upon the current state (independent o f t h e past h i s t o r y l e a d i n g t o the current state). F o r a thorough treatment of Markov processes see F e l l e r (1957) o r P a r z e n (1962). 8  T  107  The  Develo£ment  In  order  model) as 1.  Of  to  a Markov  of  states  fact  i t may  a  system  process  we  proceed  models  by  might  be  i n other 2.  to  may  theoretical  structure  a later  3. the  that  be  based  or  upon  remainder are  the  one  system.  In  cf  Markov  results.  few the  theory  In  states  is  system  while  Pij.  These  or  data.  transitions  O t h e r s may  several  estimated  than  different  very  some  (p=1) .  system, or  the  probabilities  either  reasons  way.  inadequate.  transition  may  from  be  be  may  be  dictated  by  required  d a t a by  For  means  to  be  discussed  section.,, of  the  model.  t r a n s i t i o n counts are is  independent  model  is  necessary.  transition  counts the  sufficient. . no  be  simulation  more  comparing  e s s e n t i a l points  certain  which  that  several  using  a  following  representing  model  model may  the  of the  Validation  preceding  as  case  In g e n e r a l  create  a  the  of  (p=0)  The  to  (structural)  impossible  equal.  be  cases a simple  estimates  i n the  states S i .  selected  capture  Estimation  the  be  Models ,  (in t h i s  d i f f e r e n t s t a t e s and  some c a s e s i t may sufficient  the  valuable  using  Markov  model  I d e n t i f i c a t i o n of  set  in  Bird Flocking  current In  probabilities  of  the  Then are  must f i r s t  generated current one  -  our  must  i.e.  state  in  a  demonstrate random  of that  the a  require  estimation  exists -  us of  i.e. that  Markov  that  system Markov  that  process  i . e . that a  demonstrate  purposes w i l l  bias  i n t o a given  by  state -  independent  state  addition  consistent  not  One  the  history model  to  show  is a)  transition the  model  108  correctly  predicts  transition  4.  state residence  probabilities  homogeneity) simulated  mean  do  change  f o r the.time horizon i n v o l v e d  and b) t h a t t h e  ever  time.  (time  i n the simulation  (20  minutes).  A n a l y s i s and i n t e r p r e t a t i o n . . The r e s u l t s o f t h e v a l i d a t i o n  tests  may  which  be u s e d t o d e c i d e  is  possible  establishment  upon t h e  long-term  of transition  residence  probabilities.  probabilities  I^sntification Our success not  15  simplest the  bird  captured are 1.  .adequate  analytic results,  including  the bias  introduced  basis,  residence  a n a l y s i s on  without  use t h a t  time  success; the  feeding  model  c f the b i r d ' s  model c o n s i d e r s  In the s i m u l a t i o n  the  so t h e Markov  consideration  we w i l l  about  state  will  movements  activities  interval  Markov model t h e n i s a, two  searched a prey.  intc  parameters.  prey d i s t r i b u t i o n s ,  As t h e s i m u l a t i o n  possible  Given  s t a t e s , and s e n s i t i v i t y  and o t h e r  any e x p l i c i t  second  simplification  States  f o r various  locations.  of  an  major i n t e r e s t i s i n l e a r n i n g more  include  and  Of  various  times,  t i m e s by p a r t i c u l a r s t a r t i n g transition  degree  i n terms of s t a t e s e l e c t i o n , a n d t h e o r e t i c a l  Markov model, one c a n o b t a i n  a  not  time;  on  here.  The  model:  S1,  S2 t h e b i r d  s e a r c h e d and  following  distinctions  made: Learning  -  a  bird  which  characteristics  o f t h e prey  such  a  a  prey,  "prey  has  learned  has an i m p r o v e d  preference"  (e.g.,  something o f the  ability search  at  locating  image) - two  109  successive captures 2.  Memory s p a n - a p r e y  sufficient  frequency  periods.if 3.  Copying  while  - a bird  preference  must  be  t o be r e t a i n e d - p r e y  Handling  in a flock  necessary  copying  handling  preference; reinforced  preference  with  lost  in 8  reducing  the  not r e i n f o r c e d ;  reinforcement  4.  t o form a p r e y  - a bird  i t before  another,  a prey  preference  t o form  i s sufficient time  may c c p y  t o form  a prey  the  search  capture  preference;  which c a p t u r e s . a  resuming  -one  prey  for  must spend  mere  prey  time  -  two  periods. Taking  a l l four  space given  i n Table  In t h e c a s e Smaller 4.  distinctions  into  account,  XXII.  o f birds< n o t f l o c k i n g  state  importance  4-state  E  is  eliminated.  model model  i n helping tc establish  o f the d i s t i n c t i o n s .  and t h e 1 5 - s t a t e  4-state  state  models c a n be c o n s t r u c t e d by i g n o r i n g some o f p o i n t s 1-  These models may be i n s t r u c t i v e  relative  we o b t a i n t h e s t a t e  Thus, b e s i d e s  models a l r e a d y g i v e n ,  ( 1 ) , 5 r s t a t e model ( 4 ) , 8-state  (1,3),  model  we c o u l d  11-state  (1,4),  and  the  t h e 2-  develop  mcdel  a  (1,2,3),  9-state  model  (1,3,4).  In t h i s  paper  we.will consider just  (15  states) f o r flocking  one,  the 5-state  considerations leaving  model of  birds (4  handling  s t a t e s A,B,C,D,E.  two m o d e l s , a l a r g e one  (14 f o r n o n f l o c k i n g )  for time  nonflocking  birds)  and memory s p a n  Comparing  these  and a s m a l l in  which  a r e dropped -  twc models may  offer  1 10  State A  Bird searching without prey f a i l e d t o make a c a p t u r e .  B1,...,B7  B i r d i n s t a t e Bn was s e a r c h i n g w i t h PP and f a i l e d to c a p t u r e a p r e y f o r n s u c c e s s i v e p e r i o d s . The e i g h t h f a i l u r e r e s u l t s i n l o s s o f PP and r e t u r n t o s t a t e A.  C0,C1,C2  Bird searching w i t h o u t PP c a p t u r e d a p r e y (CO), handled i t the f i r s t period ( C 1 ) , t h e second p e r i o d (C2) .  D0,D1,D2  B i r d s e a r c h i n g w i t h PP c a p t u r e d i t (as above) .  E  Bird s e a r c h i n g w i t h o u t PP made no c a p t u r e c o p y i n g a n o t h e r b i r d f o r the next p e r i o d .  Table State  XXII.: s p a c e f o r t h e Markov  model.  preference  a prey  (PP)  c r handled  but i s  111  some i n s i g h t i n t o t h e handling  t i m e and  Table  XXIII  model and  any  table  be  may  memory  span.  gives  the  derived  from  the  gives  possible  thirty  remaining  output.  The  Estimating  their  this  one  distinctions  section  f o r the  small  A  estimated with  that  are  transition However,  a  probabilities more  of  realistic  process  approach  inference  point  the  of  early  (Silver 1965;  to 1963;  Vickson  Markov  1950»s.  (1961).  small  studied  approach  estimates  concerning  Billingsley basis  simulation  The  p r o c e s s has  and  be  to  consider  utilize  l e d to the  1967;  vertinsky  of Lee,  has  prior  form  the  of  process  Statistical  been  investigated is  given  by  made on  the  information  application cf.the  J u d g e and  exactly express  the  decisions  parameters  1974).  are  obtained.  processes  need t o  estimation  Martin  can  in  A review of e a r l y s t u d i e s  s a m p l e s and  the  probabilities  assumed  should  d i s t r i b u t i o n s o v e r unknown p a r a m e t e r s of  the  table  certain.  have  probability which  transition  the  date  about  since  simply  problem.  knowledge  from  large  similar  The  from  the  Transition Probabilities  the  known.  by  model by  four  H o s t a p p l i c a t i o n s of Markov p r o c e s s e s t o that  created  s t a t e s B,C,D. which  be  deal  lags  t r a n s i t i o n s f o r the  within  must will  time  probabilities.  t r a n s i t i o n s of  twenty-six  next  of the  possible  known bounds and  eliminating  The  importance  of  Markov  Zellner  1968;  about  Bayesian chains Roussas  112 Transition  Probability  Comments  A —>A  P1  A —>C0  P2  A —>E  P3  P 1 ° = p r o b a b i l i t y o f no capture i n 80 p e r i o d s s i m u l a t i o n r u n . P 2 C 0 1 6 based upon c h a r a c t e r i s t i c s of the simulation and t h e p r e y densities used. P3 s h o u l d v a r y considerably with the degree of prey clumpedness.  B1—>B2  PU  B1—>D0  P5  B2~>B3  P6  B2—>D0  P7  B3—>BU  P8  B3—>D0  P9  BU—>B5  P10  B4 — >D0  P11  B5~>B6  P12  B5—>D0  P13  B6—>B7  P l l  B6~>D0  P15  B7~>A B7—>D0 B7—>E CO—>C1 C1—>C2 C2~>A C2—>D0 C2-->E Do—>D1 D1—>D2 D2~>B1 D2~>D0  E —>A E ~>D0 E ~>E  '  P16 P17 P18 P19=1 P20=1 P21 P22 P23 P24=1 P25=1 P26 P27  P28 P29 P30  8  We e x p e c t  P5>P2  We e x p e c t  P5>P7>P2  We e x p e c t  P7>P9>P2  We  P9>P11>P2  expect  We e x p e c t  P11>P13>P2  We e x p e c t  P13>P15>P2  We e x p e c t P15>P17>P2 We e x p e c t P18>P3 Only p o s s i b l e t r a n s i t i o n . Only p o s s i b l e t r a n s i t i o n . We e x p e c t P22>P5 We e x p e c t P23>P18 Only p o s s i b l e t r a n s i t i o n . Only p o s s i b l e t r a n s i t i o n . We e x p e c t P27>P22; P27 s h c u l d considerably with the distribution. We e x p e c t We e x p e c t  T a b l e XXIII.: P o s s i b l e t r a n s i t i o n s i n one p e r i o d the t r a n s i t i o n p r o b a b i l i t i e s .  vary prey  P29>P5 P30>P23  (15 s e c o n d s )  and  bounds  on  113  For the c r e a t i o n complex  simulation,  appropriate. knowledge  as  random  analysis  of  simulation  model,  inferred  the  mathematical  simulation  distribution be  inferred  the  loss  is  and a s p e c i f i e d  function.  The c h o i c e  problem.  For  deviation  criterion,  one  minimizes the l i k e l i h o o d  some bounds bounds  side. used  compensate  prior  o f Eayes' r u l e a  Using  the  point  posterior  e s t i m a t o r s can  sc as to minimize  function  choosing  depends on t h e  the  least  an u n b i a s e d e s t i m a t o r  deviations  for  may be  (in cur case a  frcm the  I n c o n t r a s t , some a s y m m e t r i c a l T o s s to  on  through t h e use o f Eayes'  of a l o s s  of large  of  i n the  These  probabilities  obtains  are  basis  embedded  Similarly,  function,  example,  prior  which  the  to.obtain  obtained.  f o r the .transition  particular  suspected  P (A | B)=P (B | A) *P (A)/P (B) where P(A|B) probability of A given the occurrence o f 9  on  As a c o n s e q u e n c e  loss  seme  The b a s e s f o r  considerations.„  runs).  distribution  either  1962).  expressions  probabilities.  a  especially  parameters,  example,  possible  biological  assumes  cf  e x p r e s s e d i n t h e form o f  (Savage  For  model is  o b s e r v a t i o n s o f t h e p r o c e s s a r e made  of  can be be  one  these  variables  Markov  approach  d i s t r i b u t i o n s are modified  when  posterior  over  i t i s often  from  probability  on  Bayesian  a r e many.  transition  sample  level  In t h e Bayesian approach  such.distributions  9  second  the  distributions  considered  various  a  a b o u t t h e unknown p a r a m e t e r s  probability  rule  of  is B.  over  the  square which  estimator functions or  under  conditional  114  representations sequence  of  of  a  particular  transitions.  In  from t h e s i m u l a t i o n  distribution  was  matrix.  assumed  probabilities The  intervals  depended on bounds model  or  prior  incorporated  prior  now  in  general  from  estimate  V =  [N] =  function  (L1,...,Lr), r *** 8 *** i=1  From  these  {Li} =  transition  the  order  used i n o u r m o d e l l i n g  be  We  expressed We  defined  simulation we  relationships  examples of d e r i v i n g  (P1,.., ,Pr) .  transition  addition,  matrix.  make  have among  We  shall  least  square  procedure.  as f o l l o w s .  have by  beta  over the  was  o f the  In  transition  F(P). where  cf  estimators are derived  e x p e c t a t i o n s f o r P w h i c h may density  row  the a n a l y s i s  on  the  observed  distribution  knowledge.  which were  Bayesian  the  which t h e d i s t r i b u t i o n  i n detail several  Bayesian estimators  prior  single  knowledge  consider  to  over  an  model, a m u l t i d i m e n s i o n a l  a  a  biological  probabilities  wish  of  inferred  certain  In  as  in  estimating  probabilities  transition  transition  certain the  N.  We prior  probability observations,  I i i s t h e number o f o b s e r v a t i o n s P i  N.  we d e r i v e a p o s t e r i o r  density  H(P|[N]) = K ( [ N ] ) G ( L 1 , . . . , L r | P ) F { P )  function  115  r *** g = 6 *•* i=1  where  and  Finally, the  we  loss  Bayesian  K ([N ])  i s a normalization  must s e l e c t of  estimate  our  example:  a)  the  Pi's  will  b)  the  loss  function  be  P  constant.  function  for  a  L(P,P)  true  which  value  c f P.  expresses Then  the  minimizing  L (P,P)H ( P | [ N ] ) d F .  the  transition  will  be  the  sum  probabilities; of  the  sguared  estimation  i.e.,  L (P,P)  minimize  positive,  =  V(P;[N])  av (P; [N ] ) / 3 P i = be  loss  P i s f o u n d by  In  To  a  obtaining  V(P;[N]) = J  errors,  {Pi**LiJ  0 and  where  r *** «  *** i=1 it  that  { (Pi -  is  Pi) }. z  necessary  a l l the  roots of  and the  sufficient  that  determinant  D (x)  116  V11-X V21 D(x)  V12 V22-X  =  |vr1  and  vr2  ...  Vrr-x  V i j = a*V (F;[ N ] ) / a P i 3 P j .  ... J* {2 ( P i - P i ) H (P | [ N ]) d p i . . . dPr },  V(P;[N])/aPi = J  Since  V1r V2r  Vij  = 0 for i#j = 2 f o r i= j .  T h u s , D (x)  =  (2-x)**r,  Now  a v (P;[N ] ) / 3 P i = 0  implies  Pi =  ...  whose o n l y  problem  distribution Pi  and  are elements  interrelations  The  Pi's  independent of  t h o s e P's  i s positive.  the e x p e c t e d v a l u e of P i f c r  posterior only  (x=2)  J * {PiH(P|[N]) dPl...dPr}  = E (Pi),  The  root  which  density remains  function is  H(P|[N]).  to  select  perform the n e c e s s a r y i n t e g r a t i o n s .  i n a Markov between them  may  be  o f t h o s e P»s i n i t s own  matrix, there (row  split  sums =  into  Then  be  a  prior  Since  the  number  of  1).  groups  i n o t h e r groups group.  will  the  and  such  t h a t each  i s not  Pi i s  independent  117  K  ***** F (P1,.., ,Pr)  =  * *  { F i [ P 1 ( i ) , . . . , P K i ( i ) ]},  * * i=1 where i  K  i s t h e number.of g r o u p s , t h e r e  (i=1,...,K)  density  labelled  a r e K i members i n group  P1 ( i ) , . . . , P K i (i)  with  F i  estimates Choices  o f each  P i depend o n l y  group.  1.  F i = 1.  2.  F i = C f o r (P1 ( i ) , . .. , P K i ( i ) } & R i £ [0,1 ] * * K i  This  f o r F i were r e s t r i c t e d  upon r e s u l t s  its  may be c o n s i d e r e d  = 0 f o r (P1 (i) ,... , P K i ( i ) )  allows  regions that  positive of  t o the f o l l o w i n g :  the complete  ignorance  case.  volume  of  Ri.  This  case  e x c l u s i o n o f v a r i o u s combinations o f the P i ' s .  extention  different  within  ?B i ,  where C i s t h e i n v e r s e o f t h e  An  prior  f u n c t i o n f o r group i .  Thus,  3.  the  of  case  constant  the probability  P23 ( r e f e r r i n g  t o the  2 i s t o allow values  corresponding  space. 15-state  between .2 and .8, b u t a l s o t h a t  F o r example, model)  is  0<P<.2  =  1.5  .2<P<.8  =  .25 .8<P<1.0.  cn s e v e r a l  to  different  we might most  i t might.be o u t s i d e  Then a p o s s i b l e c h o i c e f o r F would be F = .25  F i t c take  think  probably  that  range.  118  4. of  another prior  special,  order  We  case  estimates  will  on  the  Case  1:  are  just  with  1 o b s e r v a t i o n s o f P and  two  Then P = E (P)  Now  So  I (l,n-l)  first  which  Cp** (1*1  =  f  l P  2 i s the  of  Q. I (1+1,  * * l * Q * * ( n - 1 ) dP  J \ > * * 1 * (1-P) * * ( n - 1 )  =  1 ! ( n - 1 ) !/(n+1) !  =  B (l,n-l)  dP  P = B ( 1 + 1, n^-l)/B ( l , n - l ) =  { (1+1) ! ( n - 1 ) ! / ( n + 2) } / { l ! ( n - 1 ) !/(n+1) !}  =  ( l + 1)/(n+2) .  the m u l t i n o m i a l  case  with  9  *** we  again  have P =  i=1 I (L1 .+ 1 ,L2,... ,1s) •— I ( L 1 , L 2 , ...,LS)  n-1)  1(1,n-1)  r . ***  For  i n which  c a l l them P and  )*Q** ( n - 1 ) dP  {Pi} = 1  admission  i . e . , P1>P2.  b i n o m i a l case  i n a row, n-1  case  probabilities,  c o n s i d e r the  probabilities  =  extends  Q  there (P+Q=1)  with  I ( L 1 , . . . ,Ls)  -X 1  1-P1 1-P1-.. .-Ps-1 f ... f d P s - 1 * P s - 1 * * L s - 1 * P s * * L s J0 J0  r  dP1*P1**L1  0  and Ps = 1 - P1 - ... - P s - 1 . Thus,  I(L1,...,LS)  =  L1 !L2!.. .Ls!/[L1+L2+.. . + LS+S-1 ] ! = B (L1 ,. . . , 1 s ) .  So,  P_l =  (L 1 + 1 ) / (n + S) ;  And,  Pi =  ( L i + 1 ) / (n + S) .  Case 2: A g a i n  we b e g i n  F(P)  Then  1(1,n-1)  =  with the b i n o m i a l  = 1/(b-a)  0<a<P<b<1  =0  otherwise.  case.  and  a*b  | P * * l * (1-P) ** (n-1) *dP a n-1 ***  =  B(l,n-1)  *P** ( l + i + 1 ) * (1-P) ** ( n - l - i )  *** i=0  B(l,n-1)  n-1 *** ®  **•  / n+1 j  j * | b * * ( l + i + 1 ) * (1-b) ** ( n - l - i )  i=0 -  a * * (l+i+1) * (1-a) *•*( n - l - i ) j  I (1 + 1, n-1) so  P  = 1 ( 1 , n-1)  For t h e m u l t i n o m i a l case  with a s i n g l e  restriction  a<P1<b,  120  find  the  same f o r m u l a  more r e s t r i c t i o n s integration, Case  Then  as above  n r e p l a c e d by n+S-2.  on t h e P i , we o b t a i n a more complex  b u t t h e same p r i n c i p l e s  3: C o n s i d e r t h e c a s e 0<PKa  = C2  a<Pl<t  = C3  b<P1<1.  n-L1+S-2 = j and d r o p p i n g  region  P1 =  {C3 + n+S  the s u b s c r i p t  from  +  {C3  j ***/n + S\ (C2-C3) 8 j |*b** (L + i + 2) * (1-b) ** ( j - i ) ***\j-i/ i=0  *** n+S  (C1-C2) ® *** i=0  3-i  (  i=0  +  Case  * a * * (L+i+2) * (1-a) ** ( j - i ) )/  1 n+S-1\ ***/n+S-1' + (C2-C3) 8 ( |*b** (L + i+1) * (1-b) ** ( j - i ) *#*  J  j ***/n+S-1\ (C1-C2) « | * a « * ( l + i + 1 ) * (1-a) * * . ( j - i ) }. ***\ j - i i=0  4: C o n s i d e r n-1  1 observations of P "  o f 1-P  of  restriction  find  L+1  With  hold.  with a s i n g l e  F = C1  calling  with  L 1 , we  121  Let  us  k  »  of Q  m-k  "  o f 1-Q.  further  reguire  that  P<Q.  Then t h e p o s t e r i o r  density  function i s H = C[ p * * l * (1-P) ** (n-1) ][ Q**k* (1-0) ** (m-k) ]  f o r 0<P<Q<1  = 0  J J  p -•  otherwise.  dP*P** ( 1 + 1 ) * (1-P) ** (n-1) 0  J  —  a p * p * * l * (1-P) * * (n-1)  1  P dQ*Q**k* (1-Q)** (m-k) P  f dQ*Q**k* (1-Q) * * (m-k)  0  m-k * * *  B (1+1,n-1)  -  «  i  *** p  \m-k-ij  i=0  .  .  m-k  B(l,n-1)  -s  *** »  ***  m+1 \  B ( l + k + i + 2 , n+m-l-k-i) * (  . m+1 \  B ( l + k + i + 1,n+m-l-k-i) * j  Im-k-i  i=0 In Each  o u r model  the f o l l o w i n g  relationship  given  prior  below  distribution  was  deemed  to  was hold  used. with  p r o b a b i l i t y .9. P23  > P20 > P5 > P7 > P9 > P11 > P13 > P15 > P17 > P2 | P3 - P 18 | < . 1  P26 > P21 > max(Pl8,P3) |P5 - P25| < . 1 . In  addition  relationships  we  know in  that  P2<1/64.  calculating  each  To Pi  apply was  the  above  computationally  122  untenable.  Consequently,  approximations.  recomputed  T h e s e new steps  estimates  differed  of  o f 80 80  fcr state  by  of  the  less  each  were u s e d of  the  (without  than  .001  and  Several  to estimate  f o u r models —  4-state  methods  state  (without  one  must  o f 20  *  the  Ni.*N.j  restriction into  account.  priors, (two  and  the  successive  were made from  b i r d s and  15-state E  flocking;  to test  16,000  probabilities  5-state  state E  the  system  In(with  removed).  v a l i d i t y of being  transition  10  runs  (with f l o c k i n g ) ,  removed),  to the  fcr.200  Thus, a t o t a l o f transition,  demonstrate t h a t the  random - i . e . t h a t i n g e n e r a l Nij  as  the  i n any P i ) .  are a v a i l a b l e  Markov model as an a p p r o x i m a t i o n  a  restrictions  probabilities  for flocks  following  a p p l y i n g none o f  the P i ' s converged  transition  flocking;  the  violated  p e r i o d s each f o r s o l i t a r y b i r d s .  flocking),  First  P i which  violated  until  p e r i o d s each  transitions  to  o f t h e P i ' s were used  repeated  Estimates runs  Then t h o s e  t a k i n g the  estimates  were  resorted  A l l t h e P i were e s t i m a t e d  above r e s t r i c t i o n s . were  we  the  modelled.  ccunts are  not  123  • r *** ® *** i=1  r *** where N i . =  N.j  {Nij},  ®  =  3=1  © ***  N =  ® ***  {Ni. }  i=1 This  i s similar  modification (Fienberg Table  presented  contingency  (see  Results  C  this for  method o f v a l i d a t i o n  second,  above.  and lemon  part  to  state  a two s t a g e  This  comparing squared  for  First  i . e . test  which  a  worked  Results probabilities  An  transition  demonstrating  probabilities  is  the u n d e r l y i n g s t r u c t u r e i n d i c a t e d that  we  to predict  show  the next  must  state.  tested  the  This i s  over  splitting  be  that  the hypothesis.that P i j i s constant by  the data  must  time  (Billingsley  these  example).  for  was. done  of  are impossible  involves  the data  t h e P i j f o r a l l q u a r t e r s of the time  test  suitable  (1972).  means  i s sufficient  process.  homogeneity, time.  account  with  a n a l y s i s are presented i n  o f the v a l i d a t i o n  Intuitively, this  current  r).  u s i n g an i n f o r m a t i o n measure i s  a Markov model w i t h c o n s t a n t  sufficient  ( i , j=1,2  table  transitions  of  Appendix  by C h a t f i e l d  The that  t o an r by r  1972).  alternative  {N . j }  j=1  t o e l i m i n a t e those  XXIV  3  r ***  r *** and  {Nij  i n q u a r t e r s and  utilizing  a. c h i -  1961).  tests  were c o n s t a n t  indicated  over  time  that in  the  transition  a l l cases  (p>.10)  12a  Clumping  flecking  X*  df=5  nonflocking X2 df = 1  0.07  0.74  1.a8  5. 2 a  9.a8  28.7 (<10~*)  443  932  573  2019  (<10-S)  (<10-s)  (<10-5)  .186 (>.1)  3.35 (<.070)  24.5  31.0  (<10-5)  (<10-5)  (<10-s)  16.7 (<10-*)  T a b l e XXIV,: Chi-sguared values and associated significance levels testing the h y p o t h e s i s t h a t the t r a n s i t i o n counts are quasi-independent; ie. P(A|B) = P ( A ) , u n l e s s c o n s t r a i n e d t o be 0. The tests have v e r y few d e g r e e s of f r e e d o m s i n c e so many o f t h e t r a n s i t i o n s a r e impossible.  125  except the  for  first  transitions  five  into  minutes  and  thereafter.  This  into  i s impossible  s t a t e E)  feet) in  has  appropriate  constant  third  yielded  test  methods  are  generated  test  set  with  chi-sguared  the  E with  f o r the  first  available.  both  a second  starting  state).  Tables.  XXV  .  p r e d i c t e d by  the  models  transition state arose  the  a  15  constancy  zero to for  minutes  the the  each)  T h i s model's  set of data  to  t h e Markov  will  these  to  counts,  model, and  presented  test  be  adequate  correlations  that  a  data second for  model  fcr  the  (in  obtaining given  below and several  i n accounting  T h i s apparent between  in  l e n g t h and  probabilities)  were u s e l e s s  (Table XXVI).  for  A chi-  times  the  are. given  I t i s noteworthy  ( T a b l e XXV)  positive  be  is  predictions  ( s e t 2).,  p r e d i c t e d by  twc  model  compare. , r e s i d e n c e  transition  appeared  times  sufficient, Markov  f o r a g i v e n .sequence  times).  frequencies  residence> from  .times  (compares,  which  cases,  s e t 2 with.those  R e s u l t s of  (compares r e s i d e n c e  is  t o compare t r a n s i t i o n  i s performed  residence  of  constancy  simulated  (entry  (within  i n c r e a s e from  model  (set 1).  f o l l o w i n g s e c t i o n . a method  expected  assumption  (five  Markov  In  i s performed  s t a t e . i n data  (statistically)  since imitation  p e r i o d and  in  results.  whether a g i v e n  test  the  a linear  fourth guarters  those,  constant  a -bird*s.neighbor .  Replacing  compared. w i t h  squared 2,  until  from a s e t of d a t a  then  each  and  satisfactory  To  are  into.  remained  i s no.surprise  made a c a p t u r e .  transitions  second,  effect  s t a t e E which i n c r e a s e d r a p i d l y  in XXVI of for  predicting discrepancy transition  126  Clumping 0.07  0.74  1.48  5.24  9.48  flocking 15x15 df=15  14. 3 (.54)  9.0 (.88)  7.4 (.95)  27.0 (.03)  22.6 (.09)  nonflocking 14x14 df=10  2.67 (.99)  13.7 (.18)  17.1 (.07)  19.9 (.03)  15.2 (. 12)  flocking 5x5 df=11  20. 2 (.04)  5.5 (.90)  5.9 (.88)  22.4 (.02)  18.7 (.06)  nonflocking 4x4 df=6  2.47 (.87)  6.2  -12.0 (.06)  15.6 (.02)  4.40 (.62)  M 0 )  Table XXV.: Comparison of expected number o f t r a n s i t i o n s a s g i v e n by Markov model (from d a t a s e t 1) w i t h t h e t r a n s i t i o n counts data set 2. Numbers i n t h e t a b l e a r e c h i - s g u a r e d v a l u e s , associated significance levels below in parentheses, c l u m p e d n e s s i s measured w i t h L l o y d ' s i n d e x of mean c r o w d i n g .  the from with Prey  127  Clumping 0.07  0.74  1. 48  5. 24  9.48  flecking 15x15 df=14  1. 27 (>.99)  9. 43 (.80)  1. 96 (> .99)  14 .1 (. 45)  12.4 (.54)  nonflocking 14x14 df=13  1. 50 (>.99)  16.9 (.20)  14 .2 (. 36)  17 .9 (. 16)  3.37 (>.99)  flocking 5x5 df=4  70.8* (<.00001)  174.7* (<. 00001)  77 .6* (< .00001)  44 .8*. (< .00001)  11.5 (.021)  nonflocking 4x4 df=3  .554 (.90)  11.77* (.008)  11 .51* (.009)  14 .13* (. 003)  .194 (.98)  T a b l e XXVIj. Comparison of expected state r e s i d e n c e t i m e s a s g i v e n by t h e Harkov model (from d a t a s e t 1) w i t h the residence times from data set 2, giving chi-sguared values and a s s o c i a t e d s i g n i f i c a n c e levels, i n parentheses. Starred values indicate t h a t t h e a s s o c i a t e d model was r e j e c t e d f c r f u r t h e r a n a l y s i s .  128  frequencies  into  transitions Although starred causes 1.  into  these  a single j  are  a l l biased  i n T a b l e XXVI),  that  the p r e d i c t i o n s of  in., t h e  same  f o r further  i t i s instructive  direction.  a n a l y s i s (the t c examine t h e  for failure:  In t h e s m a l l models time  and  (5x5) w i t h  prey  substantial.overestimate In  handling  time  preference  Analysis  t h e . prey  resulted  in  ( s t a t e E)  o f prey  clumping,  the lack of  (4x4 and 5x5) r e s u l t e d  capture  a  by  a  bird  with  will  present  a  brief,  processes  and  in a  an prey  (state 0).  Of The Markov  In t h i s discussion  ranges  the e l i m i n a t i o n of  memory  of i m i t a t i o n  i n t h e s m a l l models of  flocking,  preference  the. intermediate  overestimate  section, of  Models we  Markov  which a r e u s e f u l . i n be  j , such  models.must be r e j e c t e d  entries  handling  2.  state  illustrated  a n a l y s i n g a Markov using  the  4x4  mathematical  some a n a l y t i c model.  latter  will  with  uniform  prey  Markov  process  with  model  The  procedures  distribution.  For our purposes, the f o l l o w i n g  we w i l l  consider a  properties:  1.  A discrete  time  p a r a m e t e r . (t=0,1,2,...) ;  2.  A discrete  s t a t e . s p a c e S=[S1,S2,...,Sr ];  3.  Stationary transition  probabilities,i . e .  P[ Xn=Sj ,X ( n - 1 ) = S i ] = P i j = c o n s t a n t  (n=1,2,..,).  129  r *** « *** j=1  Note t h a t  4.  {Pij)=  E r g o d i c s t a t e s -.- each s t a t e  state Such Let  i n a t most r  X (n) =  X(n)  c a n be  r e a c h e d from e a c h  other  transitions.  a Markov p r o c e s s . i s us w r i t e  1.  as a  commonly termed  a Markov  chain.  vector  (X1,X2,.. .,Xr) (n)  where X i = P [ X ( n ) . = S i ] and  Si i s  the  r-vector  with  1  in  column  i  and  O's  elsewhere. r *** ®  Thus,  { Xi} =  ***  1.  i=1 The of  vector  X(n)  e x p r e s s e s the occupancy  t h e s t a t e s a t t i m e t=n.  transition  probabilities, X(n)  =  letting we  probabilities  P = [Pij],  the  f o r each  matrix  of  have  X(n-1)P  = X (n-2)P**2 = Just  as  P  i s t h e m a t r i x o f one  P (n) i s t h e m a t r i x small matrix:  model  X(0)P**n.  of  without  n-step flocking  step  transition  transition has  the  probabilities,  probabilities. following  The  transition  130  .9855 .1029 .3270 .. 0  A =  where s t a t e s 1-4 The  matrix  correspond P can be  m a t r i c e s each m u l t i p l i e d • r *** ©  P•=  ***  0 .8330 0 .3150  .0145 0 .6670 0  to states  decomposed  A-D. intc  by an e i g e n v a l u e  {Li*¥i}  ~  and  Li  P <n) =  and  the  eigenvalues  Y i are associated  = Yi  cf P  ***  then  { L i * * n * Y i } . *°  ~  (solutions  cf the e q u a t i o n  for i=j.  are s o l u t i o n s  p * U i = P * L i and V i * P = C i - = Vi*Ui  I f zero  That i s ,  for i#j  Y i a r e g i v e n by  vector)  1 0  constant  matrices  Y i ..= C i * U i * V i  where U i and v i ( r e s p e c t i v e l y  and  o f P.  cf  Li*P),  with Yi*Y j = 0  The  sum  i=1  are  Vi*P =  a  r *** ©  i=1 where  0 .0641 .0060 .6850  1  a column  vector  and  a  row  of the equations Li*P  (the i n n e r product  c f O i and V i ) .  i s an e i g e n v a l u e o f m u l t i p l i c i t y (I1 = L2=.. ,=Ls=0;Ls+ 1, . .. ,Lr#0) , . r. . . . . . .  s, i . e .  P(n) =  0 { L i * * n Y i } + Zn, *** ~ i=s+1 where Zn's a r e c o n s t a n t m a t r i c e s f o r n=0,1,...s-1 n>s. F o r t h e c a s e where a n o n z e r o e i g e n v a l u e has g r e a t e r t h a n one s e e Appendix D.  and Zn=0 f o r multiplicity  131  (Feller, New  f o r P a Markov m a t r i x  1.  |Li|<1  2.  Li  =  Without  (i=1,2  ,...,  pp.  380-84)  have  r);  1 for precisely loss of  we  1957  one  generality  value  we  can  c f i , (i=1 say  11 =  t • • •r)i •  1.  r *** Then  P(n)  = Y1  © ***  +  {Lk**n*Yk}.  k=2 If  |Lk|<1  (k=2 ,... ,r) , t h e n  Thus, fcr from  each any  Y1  may  steady  state.  For  state  t o any  state  i  t h e rows o f Y1  also  finding  the  starting  probability), Y1  gives  P (n) — >  be  the system  Y1  as n  state  an  ~>infinity.  transition  ergodic  process  j i n r s t e p s or  are  interpreted  identical.  .as  less  (transition has  positive  Each element Y i j of  the long-term  i n s t a t e S j g i v e n t h a t the  Si.  probabilities  probability  system  began  of in  , The  terms  behavior latter  Lk**n*Yk  f o r |Lk|<1 and case,  rows  (k=2,...,r)  cyclic  of Y1  still  however, t h e r e i s n o . l o n g e r period  m,  The follows.  where m i s t h e decomposition First  we  behavior  represent for  state  s o l v e Vi*A  In  example  = Li*A.  the  times;  but r a t h e r c y c l e s  s m a l l e s t i n t e g e r such our  |Lk|=1.  r e p r e s e n t mean r e s i d e n c e  a steady  of  transitory  of  t h a t Lk**m=1.  matrix  proceeds  In g e n e r a l t h i s  as  eguation  132  has  r solutions  where A i s an r b y  r  matrix.  The  1 1  solutions  are: L1 = 1 .0000  V1 =  ("13.119, -  .033, -- .571, -.018)  L2 =  .6533  V2 •=  (" 1.086, -  . 150,  L3 =  .5984  V3 =  (- 2.620,  .808,  L4 =  .9188  V4 = (  1.474, -1.225,  0728 0728 *™ • 0728 ™" • 0728  -.0361 -.0081 .8282 .0809  .0048 .0032 .3310 -.5951 .1282 -.0149 -1.2040 -.8019  .0024 .0024 .0024 .0024  .0415 .0415 .0415 .0415  .0013 .0013 .0013 .0013  .0393 .0088 .8992 -.0878  .0054 .0012 -.1242 -.0121  -.0416 -.0094 .9526 .0931  -.0031 -.0007 .0709 .0069  .0013 -.0866 -.0335 .3149  -.0039 .2674 . 1036 -.9726  -.0003 .0183 .0071 -.0666  .0029 -. 1991 -.0771 . 7243  .0048 -.8770 -.0220 •1.1818  -.0040 .7290 .0183 .9823  .0003 -.0505 -.0013 -.0681  -.0011 . 1985 .0050 .2675  V-i  =  •  1.150,  .0 86)  .055,  -.602)  .085,  -.334)  -  Finally A(n)  .9547 .9547 .9547 .9547  =  +.6533**n  +.5984**n  +.9188**n  The expected of Let  above  decomposition  of  r e s i d e n c e times f o r each  P  may  state  be  used  i n a sequence o f  the  states  l e n g t h n. Tij(n)=E[number  transitions  into  Sj i n n steps  See footnote and A p p e n d i x D f o r t h e c a s e s l e s s than r independent s o l u t i o n s . 1 1  to f i n d  1  0  | X0=Si]  when t h e r e a r e  133  and  T (n) = [ T i j ] . n *** T(n) = ®  Then  {P**k}.  *** k=1  With  L1=1,  T (n) =  n *** 8  r  *** 8  ***  ***  k=1  i=1  r ***  n  *** 8  9  ***  {Li**kYi)  ~  {Li**kYi}  k=1  i=1  ,r  ***  = nYl +  8  { L i * Y i ( 1 - L i * * n ) / ( 1 - L i ) }.  ~  ***  ~  i=2 For  o u r example T(80)  Since state with  we f i n d  A, we  76.507 66.240 74.383 63.323  =  the  that .153 8.833 .319 9.823  3.250 2.768 5. 119 2.635  simulation generating  used t h e f i r s t  the r e s i d e n c e  times  line  found  .090 2.059 . 179 4.219  data  s e t 2 was  o f t h e above i n data  matrix  started in to  compare  s e t 2.  Results The  Markov  d y n a m i c s and l o n g  second  level  run  behaviour  model of  was  used t o examine t h e  feeding  success.  These  134  phenomena directly  could  not  from t h e  prey  feeding  or  e m p l o y i n g •. t h e  described  above,  additional  value  and  cost  Whereas . i n  rates... and  previous  variance  feeding . success  perspective run  permits e x p l i c i t  were  estimated  indicated  in  clumping  and  failure,  given  average  among b i r d s here  explicitly  a n a l y s i s of  f o r the  f o r a range of  will  so.  be  were  provides  the  through  path  time.  learning rates  f l o c k i n g and  five  with  probability D2  a  E) a)  of This  and  long  values.of  DO)  Examining  features  prey, preference  increases.  spatial  effect  of  flocking  with  nonflocking  capture  for  a  flocking  b i r d s , an  states  p r e y on :  decreases;, successive  These are  (14  preference  i n d i c a t i o n of i n t e r f e r e n c e .  matrices As  a prey  prey while  b)  attempts  consistent  birds reveals.that.the  without prey  models  and  prey c l u m p i n g . . Comparisons of  bird  not  apparent.  capturing  As  was  transition  are  case  prey clumping.  five  the  p r o b a b i l i t y of  two  nonflocking  results for the.larger  several  the  of c a p t u r i n g —•>  only  given.  appendix  increases:  searching  and  matrices  possible,  15.state)  (given  DO  chapter  a n a l y s i s presented  above, s i m p l i f i c a t i o n . t o f o u r or  generally  >  were  potentials.  Transition  and  hierarchical insights  in feeding  an a l t e r n a t i v e v a n t a g e by c o n s i d e r i n g and  answered  of f l o c k i n g f o r d i f f e r e n t the  t o compare s t r a t e g i e s , t h e  search  adequately  By  distributions.  daily used  i n t o the  economically  simulation.  m o d e l l i n g . method obtained  be  the (C2  with  matrices  probability  i s smaller  for  — the for of the  135  Repeating o f two t y p e s the  large  1 2  the s i m u l a t i o n experiments a similar  models  transition  set of r e s u l t s  were  matrices  However, i m p o r t a n t  were, t h e  differences  individual  to feed, we,find  a prey i n c r e a s e s over asymptote,  the  realizable  learning  processes  indicator  the asymptote learning at  curve  zero a t time  that f (t)  that  feeding  a fleck  (dominant)  the  atove. analysis  rate,  potential  limit  may  may  to a  The  bird  rate  therefore  curve.  limit.  This  be c o n s i d e r e d t o be  of. t h e  and t h e c u r v e  learning  of b i r d s f l i e s  when a l l  of. increase i n  be  graphing  The . g e n e r a l  considered  an  the approach  to  form  of  the  i s a monotonically increasing.function  beginning  z e r o , and bounded  we  above.  ..In o u r example ;  curve  was  =? a (1 - | L | * * t ) , . where a - i s t h e a s y m p t o t e state  given  in  asymptotic  an a c c u r a t e r e p r e s e n t a t i o n o f t h i s  steady,  only  the chance o f . a b i r d ' s c a p t u r i n g  ..are c o m p l e t e d .  learning, a  birds.or  run c a p t u r e  up t o t h i s  of  revealed  t i m e . u p t o some  long  the f u l l  capture rate  Again  and t h e p a t t e r n s i n the  same, a s , t h o s e  were  population  •  When e i t h e r area  was.found.  satisfactory,  which f o l l o w s .  new  with a prey  probabilities  eigenvalue  and  o f .the  L  is  e i g e n v a l u e i s always e q u a l t o 1 ) .  the. function  d e r i v e d from the  the . second,  transition, Examples  .largest  m a t r i x . (the cf  found  these  largest  learning  Two d i s t i n c t t y p e s . w e r e u s e d , nutritionally e g u i v a l e n t but d i f f e r i n g v i s u a l l y and i n m i c r o - h a b i t a t . Thus, a b i r d w i t h p r e y preference f o r prey type 1 w i l l not f i n d l o c a l l y a v a i l a b l e prey o f t y p e 2 and v i c e v e r s a . The o v e r a l l d e n s i t y and distribution was a s i n t h e p r e v i o u s e x p e r i m e n t s . 1  2  136  curves these  are  graphed  Figures  and  following 1..  in of  Figures  Tables  of  their  learning  higher  major p r e y  rate.  The r e s u l t s  of this  i s  most  effective  rate  than  there  is little  so l o n g a s o n l y  addition  i t should  noted  there  are  but  especially  in  polymorphic,  clumped  in  birds,  feeding  difference  capture  do rates  not  learning where  on  f o r low v a l u e s  food  of prey  are considered. of r a p i d  types,  we  flocking.enhances  strategy  being  clumping,  periods of time. prey,  o f prey  a  f o r s h c r t p e r i o d s of time i n  clumping. when  and a t low d e n s i t y .  results  in  convergence  t h a t the value  major  at  tut later  a d d i t i o n , i t i n c r e a s e s t h e maximum  effective  difference  initially  efficiences  Again  f o r high values  simulation  i n the  d e n s i t y makes i t p o s s i b l e  two  particularly  the  the b i r d s reduce  values o f prey  f o r longer  set of results.  reduces the  t r a d e o f f are v i s i b l e  The  even  search  be  i n t h e same a r e a  different rate,  the  their  Conversely,  gain  d e c l i n e s as i n c r e a s e d , prey  When  flocking  but  o f . t h e e n v i r o n m e n t and  concentrations i s important.  2.  reveals  of  increase  nonflocking  prominent a t high  sampling  clumping  type,  birds  by t h e n o n f l o c k i n g b i r d s .  feeding  XXVIII  feeding, rate,  where we s e e f l o c k i n g  flocking  and  a and L ; t h a t i s , by f l o c k i n g realizable  average  passed rate  both  maximum  Figures,  XXVII  Examination  points:  When t h e r e i s a s i n g l e  values  11, 12 and 13.  prey  learning  to  remain  , find  the  a very  learning  feeding  rate,  Thus f l e c k i n g  i sa  populations  are  I t i s noteworthy  themselves  In  reveal  between f l o c k i n g  that  that the  and n o n f l o c k i n g  13 7  4CH  T i m e (minutes) Figure 11: Capture rate per b i r d as a -function of time f o r very highly . clumped prey (09.48) . Labels £ and nf i d e n t i f y tine curves for f l o c k i n g and nonflocking birds" respectively. L a t e l s 1 and 2 i d e n t i f y the curves f o r 1 and 2 prey types.  138  Time (minutes) Figure 12: Capture rate per b i r d as a function of time f o r highly clumped prey (G=5.24). Labels as i n Figure 11.  139  Figure 13: Capture rate per b i r d as a function of tixne f o r randomly d i s t r i b u t e d prey (C=.07), Labels.as i n Figure 11.  n o  Clumping 0.07  0.74  1 .48  5.24  9.48  flocking nonflocking  1 prey type  . 057 .059 ,  . 125 . 137  182 198  . 156 . 222  .176 . 183  flocking nonflocking  2 prey types  .056 .052  . 142 .118  129 106  . 135 .088  .154 .077  Table XXVIII Asymptotic capture r a t e s e x p r e s s e d i n prey per b i r d ( i n t h e s i m u l a t i o n model t h e p r e y a r e c o n s i d e r e d to a p p r o x i m a t e l y 1/2 gram each) .  per be  minute large,  14 1  Clumping 0.07  0.74  1.48  5.24  9.48  flocking nonflocking  1 prey type  .717 .859  .915 .943  .933 .944  .918 .959  .957 . 971  flocking nonflocking  2 prey types  . 702 .933  .930 .942  .932 .947  .921 .961  .934 .961  T a b l e XXVIII:. Dominant e i g e n v a l u e s - t h e e i g e n v a l u e which d e t e r m i n e s t h e r a t e of convergence t o the steady s t a t e c o n d i t i o n f o r the two large models (15x15 and 14x14) a t f i v e v a l u e o f p r e y c l u m p i n g . Note t h a t s m a l l e r v a l u e s of t h e d o m i n a n t e i g e n v a l u e indicate faster learning rates.  142  birds 3.  i s in fact  a long-term  A multiplicity  depresses  as  w e l l as  of d i s t i n c t  the l e a r n i n g  curves  types  birds  flocking  are may  flocking be  than  favored  in  (providing  p r e y ) , but . t h i s r e d u c t i o n i n prey the  a short-term prey  population  some p r o t e c t i o n  susceptibility when  when  the.  phenomenon.  they  prey  f o r the  i s s m a l l e r when  are s o l i t a r y .  populations  Thus,  are  highly  polymorphic.  In  addition  to transient  Markov.model y i e l d s presents tc  the  For  these  first  the  cases  risk  when  prey  increases, birds.  is  about-  birds.  the average  rather  3  For  mathematical  r a n d o m l y ; , .but.  risk  is  as  somewhat  the  One  must  variance over  than  measuring  beyond  clumping  higher  With  two  risk  than  prey for  we the  there appears  this  to  phenomenon  mean measures s p r e a d risk  development see Appendix  both  which r i s k , d o e s not  interpret,  the  the  f o r the n o n f l o c k i n g  However, f o r f l o c k i n g . b i r d s clumping  to  In  prey  the.same p a t t e r n i n l o n g - t e r m  substantially.,  XXIX  nonflocking birds  ..high v a l u e s o f c l u m p i n g .  level.of.prey  since  study.  and  i n c r e a s e s much.more r a p i d l y  cautiously  1  equal f o r f l o c k i n g  risk  at  the  experiments.  i s similar  simulation  distributed  a critical  increase  risk  are  risk  Table  simulation  long-term  the  long-term.  rates,  mean c a p t u r e r a t e c o r r e s p o n d i n g  in,the  case the  reported f o r  substantially  nonflocking be  risk  prey  capture 1 3  measurement  In a d d i t i o n  short-term find  risk  long-term  variance estimates.  as v a r i a n c e o v e r  single  short-term  long-term  and  F.  cf  being  around below  143  Clumping 0 .07  0. 74  1. f8-  5.24  9.48  flocking nonflocking  1 prey type  0 .91 0 .96  3. 18 4. 37  4. 32 5. 55  4.73 9.66  10.76 15. 95  flocking nonflocking  2 prey types  0 .93 0 .95  4. 06 2. 28  4. 41 5. 48  5.35 12.01  5.54 12. 24  T a b l e XXIX:. Variance over mean c a p t u r e r a t e ( f o r t w e n t y f o r f i v e v a l u e s of prey clumping.  minutes) or  "risk"  144  average.  In  comes from  birds feeding  single for  this  case  prey c a s e . b i r d s  about  h a l f of the  the  major c o n t r i b u t i o n t o t h e  below t h e a v e r a g e feeding  variance.,,  rate  above the average  where rate  measure for  the  accounted  145  CHAPTER SEVEN! CONCLUSIONS This study its  began w i t h  behavioural  a discussion  components.  be c o n s i d e r e d a s a f o r a g i n g constructed  of  experiments  birds  with t h a t  o f f o r a g i n g and some  To examine whether f l e c k i n g stategy,  a  simulation  foraging i n flocks.  From  of  might  model  was  the s i m u l a t i o n  model and i t s s u b s e q u e n t . a n a l y s i s i n t e r m s  o f a Markov model s e v e r a l c o n c l u s i o n s c a n be drawn. 1.  Flock behaviour  c a n be m o d e l l e d  as the  sum  of  individual  behaviours. 2.  Prey  3.  Prey  distribution distribution  describe within  clump  Giving-up likely  Flocking reduces  time  Flocking rate  least  e.g.  t h r e e parameters  average  d e n s i t y , average  density,  number o f prey  to  average  items  (mass)  has a,major e f f e c t under t h e b i r d s '  upon s u c c e s s .  control,  As  i t i s  i t should vary  with  p r e y . d e n s i t i e s and d i s t r i b u t i o n s . . i s advantageous i n severe  the r i s k . t o  maintainence, 6.  upon f e e d i n g s u c c e s s .  .  to.be  different 5.  .requires , at  i t adequately:  per clump.. 4.  has a major e f f e c t  a  bird  particularly  ,neither  of  but  because  insufficient  when p r e y  flocking  when p r e y a r e p o l y m o r p h i c  However, i f t h e t w e n t y  flocking  feeding  type o r  t h e mean c a p t u r e  prey  are  randomly  does i n c r e a s e mean c a p t u r e  (several  minute  for  a r e clumped.  i n c r e a s e s nor d e c r e a s e s  when p r e y a r e o f a s i n g l e  distributed;  weather  types)  period  rate  and clumped.  cf  the  simulation  146  experiments remain large must 7.  is  and r i c h ) ,  a  (either  new  increasing greater  to  plateau  This  begins  . The c r o s s - o v e r  'learning*  minutes,  d e p e n d i n g upon  zero,  rate  alone.  capture  i s  However,  individual  rate.  flocks will  single  point  feeding  at  monomorphic,  long-term  i s long,  conclusions  b i r d s o n l y , when p r e y a r e  are  I f the  be f a v o r e d ;  individuals  will  be  seems t o be i n t h e r a n g e o f  prey  d i s t r i b u t i o n and d e n s i t y  least).  birds.  consider  how a p p l i c a b l e t h e s e  First,.these results  to birds whose.characteristics  birds  b) a r e s u b j e c t form  model . s h o u l d  conclusions  concerning match t h e s e  be  applicable  i n cold climates; to l i t t l e  short-term  prey  predation  should  cf  model.  the  (in winter) t o  . during  preferences;  are  flocking  which:  a) o v e r w i n t e r  c)  rate  i n a patch.is.shorty  t h e average s t a y  particular,the  small  above  or i n f l e c k s ) begin  capture  prey  stay  Now we must  In  When  average  real  the .  forflocking  have a h i g h e r  (at  of  plateau.  i s higher  birds  (e.g. clumps o f p r e y a r e  .  their  foragers  10-30  area  individually  some  the time that r e a l  f o r b i r d s i n f l o c k s than f o r those  favored.  apply  .  area,  polymorphic.  if  than  then the l a s t  be m o d i f i e d .  in  to  less  f o r a g i n g i n one s m a l l  When b i r d s  the  much  this  period;  147  d)  f e e d on m a r g i n a l  e)  copy e a c h  Examples o f s u c h England  and  copying)  birds  of  One s h o u l d  The  a r e clumped  important  prey  i s m i l d , t h e model p r e d i c t s . no  suggests one all  that  type  .in flocks.  (or s u f f i c i e n t l y . h i g h  r a t e , . . w i l l be s l i g h t l y  flocking  clarify  in  these  ( p r e y - p r e f e r e n c e and  for  these  or  both,  then  expect  The a n a l y s i s  to ether  the  o r the  flocking. selection  u s i n g Markov  models  i f p r e y . d e n s i t i e s a r e u n u s u a l l y h i g h and m a i n l y o f  t h e model has n o t p r o v i d e d any when  even  advantage  would  that  a bird  b u t one . t y p e ) ,.. then., f l o c k i n g ,  capture  that  i n many  I f p r e y . a r e . n e t clumped  c i r c u m s t a n c e s , one precedence.  winters  predictions  types,  winter  p r e s s u r e s t o take  which  in  a n d . i f the winter i s severe  will  .these  winters  under l a b o r a t o r y c o n d i t i o n s , b u t  conditions.  i f prey  forage  note  birds  Under  d e n s i t y i s lew;  t i t which  behaviours  demonstrated  or t h e r e a r e s e v e r a l  great,  chickadee  the c r i t i c a l  have>been  are, that  the  black-rcapped  not a s y e t under f i e l d birds  are  America.  cases, .several  prey  others foraging preferences.  the  p a r t s o f North  resources, i . e .  has  survival  the issue.of,when  can a f f o r d  by... r e d u c i n g  to ignore  the. average  disadvantageous.  Finally,  simple  statements  value  flocking  succinct for.birds,  may have  while of  i t has a i d e d t o  survival  value  in  t e r m s o f f o r a g i n g success., '* 1  i t i s p o s s i b l e , o f c o u r s e , t h a t f l o c k i n g nc l o n g e r has any s u r v i v a l value f o r b i r d s . T h i s s o c i a l b e h a v i o u r may be extant s i m p l y b e c a u s e a n i m a l b e h a v i o u r s a r e o f t e n slow t o e v o l v e ( G e i s t 1974 ) . 1 4  148  .An. a t t e m p t rates  with  artificial the  those baits.  t h r u s h e s had  faster., -  was  .064  . cases  per  metre  conditions  the  prey  the  were  the h i g h e r d e n s i t y t h e  of.  subsistance foraging  to the r e a l  lowprey  level  by  activities.  when t h e  prey  f o r the  . 299  prey  detection  densities,  The  chances i n d i c a t e  beween f l o c k i n g  ccnsiderabley  harder  and to  if.prey  preference,  it  prey  value  are a l s o  not  metre.  per  1.79  The -  .45  minute; prey  per  minute.  of  flocking  i s as  and  Experiments  to find  certain  both  i f the  to  do  to  cases to  increasing  the  If  with  comparative prey and  results  are  without  prey.density  these  to  difficulty  only for birds  that  per  near  waking... h o u r s restricted  the s i z e  the  are r e s t r i c t e d  of  for birds  are harder is  .59  solitary . foraging. find  in  desity  substantial-change for  p r e f e r e n c e , ..then, t h e e f f e c t But,  used  when b i r d s a r e f e e d i n g  model., no  prey  results  densities  randomly.  prey  90-100%  results  i n . the  the  moved  sg.  model.predicted  The  devoting  per  on  since  p r e y and  that  distributed  concerning  feeding  prey  same as  b i r d s ' 2.30  are of approximately  detect . postulated  prey  (nearly  of feeding  be . a d j u s t e d  were o b t a i n e d a t two  have u n i v e r s a l a p p l i c a b i l i t y .  reduced.  to  a s m a l l e r h a n d l i n g time  However, t h e r e s u l t s  prey  had  m i n u t e compared t o t h e . r e a l b i r d s !  for  results  f o r thrushes  were r e a s o n l b l y good a t t h e l o w e r  m i n u t e a s compared  not  (1974)  s i m u l a t i o n e x p e r i m e n t s ) , and  predictions  while  by S m i t h  S e v e r a l parameters  prey per s g .  both  prey  found  Simulation results  previous In  made t o compare .model p r e d i c t i o n s  were  without apply.  149  Likely  such  flocking  birds.,  season. high  a situation The  model  to give  enough t o s u p p o r t  model.  which A  comparing  comparison  feeding  foraging  over  areas one  occur  a territorial.system  example, when.other  with  the success  means o t h e r - t h a n predators  the  are reproducing  feeding. flocks  take  changes  the  considerable copying.takes  model  may  the.birds' foraging.  For  a significant highly  portion.of  mobile  explain  f l o c k s of the temperate  Those flocking postulated  conclusions, are  for this  movements  reasonable  likely  which  to  have  model.  of : the  description  species f l o c k s i n terms  of  multi-species  i n the  and  that  laboratory).  some moderate s i z e d  mixed-  zone.  do - n o t wider  concern  t h e . value  applicability  -than  On t h e b a s i s . o f t h e . g e n e r a l  flock, of  Finally,  niche . overlap  (at l e a s t  the  c r when t h e  differentiation . in  feeding  help  each  prey  may be i n a p p l i c a b l e .  place  15  in  K r e b s ( 1 9 7 3 a ) . s u g g e s t s t h a t a c t u a l l y many have  species  of  the r e s u l t s  i s significant.niche  interspecific Thus,  i s p o s s i b l e by  o f 15 s i n g l e b i r d s  . m o d e l . i s n o t . d i r e c t l y a p p l i c a b l e t o mixed  which t h e r e  i n the  I n , a d d i t i o n , t h e model i s  p r e y , . .when t h e p r e y . p o p u l a t i o n ( s ) . a r e prey  are  s a y .15 b i r d s f o r a g i n g i n f l o c k s o v e r  o f t h e 15 s i t e s .  by  the  nc s e l e c t i o n  a r e embedded  applicable t o , s i t u a t i o n s where.substantial  abundance  to  densities  system, s i n c e  territoriality  with  the success-of  separate  when, p r e y  a territorial  might-favor  an a d v a n t a g e  i s not a p p l i c a b l e t o the b r e e d i n g  I t may.also,be i n a p p l i c a b l e  pressures  not  would t e n d  .the  flock  model , a p p e a r s behaviour,  at  tc least  of that  pattern give at  a a  150  superficial  level.  behavioural .the  This,  It  should,  individua1 bird  concerns  be  hence,-  c i r c u l a r - reasoning  process.  of  success,  upon  the  time.  conclusion (average Chapter  2  my  possibilities  One.is  t o expand  the area.and  expand  the  consideration mortality and  made  varieswith, constant) It  by a  supporting  the  feeding  sensitive to  elsewhere,  based  1973).  prey  The  distribution  i s n o t a_new r e s u l t (see  has  n o t . , been  generally  d e s c r i p t i o n o f a prey  to characterize  i t for  comparing  stategies.  to increase to  held  be i n s u f f i c i e n t  Several study.  density  rate  provides  prediction that  however, ..that ..a t w o - p a r a m e t e r  population foraging  attack  from  a r e net c r e a t e d  p r i n c i p l e s . (Charnov  f o r references).  recognized,  .its  no  t c t h e model  o f whether b i r d s . f l o c k , i s v e r y  different  that prey  that  missing  input  A d d i t i o n a l evidence  T h i s . p r e d i c t i o n , has been  entirely  are  that  while t h e o u t p u t  the.conclusions  model, i s  regardless  giving-up  hopefully,  r e i t e r a t e d that  behaviours,  behaviours;  validity  indicates,.  parameters of c r u c i a l - i m p o r t a n c e  model.  flock,  .  of  t h e model c a p a b i l i t i e s , s p a t i a l l y ; number o f p r e y c o n s i d e r e d . ,,temporally..  hunger,  energy  .This  requirements  the  weight  over  a period  determine weight  distributions o f about  initial  and  f o r each d a y ) .  this i.e.  A second i s  .might  require  ( t e m p e r a t u r e ) and  ( e . g . b i r d s below a c e r t a i n w e i g h t . d i e ;  temperature  different  .model  are .available, f o r continuing  daily  feeding  Comparisons  of  chance o f s u r v i v a l o f t h e b i r d s  10 d a y s f o r d i f f e r e n t weather  prey d e n s i t i e s a n d - d i s t r i b u t i o n s  conditions,  might  lead t c  151  more p r e c i s e h y p o t h e s e s . are  conditions  individuals  (e.g.. h i g h e r  have, higher  Markov  models  patches  might  be s u c h is  time  as  areas  "best"  giving-up  with  whether f l o c k s this  time.  created  by  different time  tend  predictions  flock  of  the  might, e x p l o r e ,  the  Birds.adjust their  .density  how  about by  giving-up  this times  measure  could  parameters  differ  value? a model,  i t should  model has  been  The . f o l l o w i n g  model .might be the  the  measure  birds.  on  create  determine  u s e f u l the  real  To  Then,  this "best" such  prey  flocking  one  foragers. time  and  that  times.  d e n s i t i e s . and  just  large of  behaviour,  solitary,  the  set  ,prey  giving-up  h y p o t h e s e s which w e r e . g e n e r a t e d  basis  tested.. of  recent  success.  ,in  flocks  average than  capture  do,solitary  type. Birds  and  second  i s the_possibility  toward  in  3.  time;  prey  for  experimentally  Birds  A  solitary  using  densities  First..one  of  be d e t e r m i n e d  2.  prey  b e f o r e , c o n t i n u i n g t o use  making  Analysis  conditions.  giving-up  "best" value  feeding  rates.  o f b i r d s whose g i v i n g - u p  However,  1.  d e n s i t y ) under  high  More i n t e r e s t i n g  several  from  which  that  a i d i n the " a d j u s t m e n t "  giving-up  prey  there  to..make a m o r e . t h o r o u g h i n v e s t i g a t i o n  between  distribution.  whether  a s e t of  i m p o r t a n c e o f giving--up relationship  m i g h t a l s o examine  capture  suggested  possibilities  may  One  ... . in  polymorphic  flocks prey  neither  more n o r . f e w e r  b i r d s when t h e r e .  capture  is a  prey  on  s i n g l e , prey  .. • more . p r e y  p o p u l a t i o n than  do  when  attacking  a  t h e same b i r d s f o r a g i n g  152  alone. 4.  5.  Birds i n flocks for.a  fixed  Birds  in  than  do  are  not•an  f o r a g i n g p e r i o d than flocks  solitary  entering Models . and  have a s m a l l e r v a r i a t i o n  a new  increase their birds  o r g a n i z e our  techniques  in,themselves.  ideas  birds  capture  f o r a g i n g alone.: ,  c a p t u r e r a t e more  first  few  rate  minutes  rapidly after  a  foraging area.  mathematical end  f o r the  do  in  and  select  such  Bather  as  those  they.are  presented  tools  which q u e s t i o n s t c  ask.  to help  here us  153  REFERENCES Allen,J.A. & Clarke,B. by w i l d p a s s e r i n e s ,  CITED  (1968) E v i d e n c e f o r a p c s t a t i c s e l e c t i o n N a t u r e 220:501-02.  Allen,J.A. (1972) Evidence for stabilizing and apostatic s e l e c t i o n by w i l d b l a c k b i r d s . N a t u r e 237 :348-49. . Altmann,S.A. (1968) Sociobiology c f Rhesus monkeys I I I : the b a s i c c o m m u n i c a t i o n n e t w o r k . 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(1970) Regulation o f numbers ( A v e s : P a s s e r i f o r m e s ) , J.. Z o o l . loncL  i n the G r e a t t i t 162:317-33.  Krebs,J.R. (1971) T e r r i t o r y and b r e e d i n g d e n s i t j t i t , P a r u s major L., E c o l o g y 52: 2-22.  in  Newton,I. (1967) The a d a p t i v e r a d i a t i o n and f e e d i n g some B r i t i s h f i n c h e s , I b i s 109:33-98.  the  Great  e c o l o g y of  Perrins,C.M. (1970) P o p u l a t i o n s t u d i e s o f t h e G r e a t t i t , Parus major, i n P r o c.. Ady^ Inst.. Dynamics Numbers P c j u l ^ (edT P.J. den Boer S G.R. G r a d w e l l ) ""centre ~ f c r ~ Agricultural P u b l i s h i n g and D o c u m e n t a t i o n , Wageningen, pp.524-31.  165  P u l l i a m , H . R. S E n d e r s , F . (1971) The sympatric f i n c h s p e c i e s . Ecology Taylor,R.J. Monogr^  (1974) R o l e o f l e a r n i n g 44:89-104.  feeding ecology 52:557-66.  of  five  in insect parasitism,  Ecol,  APPENDIX A:  Diagram 1.  FLOW CHARTS OF MAJOR MODEL COMPOI^NTS  :Subroutine TMSTP  Diagram 2  Subroutine' FLY  Diagram 3  Subroutine FLOGG  Diagram 4  Subroutine EAT  Diagram 5 .  Subroutine LMTATE  Diagram 6  Main Program  Write out tin* time p e r i o d  CM,!, 'KAT'  _y C A I , h 'n.CX'.G' once e v e r y fonr p e r i o d s , and v l i c i i r v o r a b i r d Hew in tho. previous, period  W r i t e c u r r e n t stntus of each b i r d  C A M , 'Kl-Y  j  A cti vo bi rds move  M o v e m e n t t tati r, t ic r. computed  P r i n t m a p of b i r d locations if a b i r d flew or if the last map w i t t ' p r i n l c d N M A i * pc j iod s a j;o  y Inish (  168  SUHKOUTINT T I . Y '  (A) Yr » Have A l l l . i n l s l u ' m c h e r t . . . ! (or  No  i r.ts  c a r lie d  Oid hi i d f l y in t i n s o r p i cvions  period?  11a;;  b i r d ' s I  No  IIKIVC:;  (lif.M:  (  M'.\ i i h i u  lu-y o n d  vdy  ;  (  e  o f  1  l i m ' '  © Yr j  Is bi rd hand I inr. food? ,  I l i r d f l i r s v/ilh prob.vhilily P(fly) -•• ~N  if N > G  If U&G N = nc-iiods t . i n r c l> j rd a I r l.t st p r r y C - mi'iiriuiin r i v i n g up I line  - 0 where:  No  No  ©  Jtss.  Doc K bi rrl fly?  ©  Have . i l l turds been c ln.-c keel for fi'.l Inwi up  Yes  No Is .bird in i a m c flock as leader''?  No Dor s b i r d fly in tl.i  Set  'JhtB  Diagram 2  h l r d may  c d ?  de lay <• d follow . s w i t c h  fwlluw  A n y t i m e  i n t h e  V  on  n e x t  m i n u t e  -©  169  Diil Ui it) fly l:.M  i iod? II ni:iy f„l|,, in Mic iw\t 's - |n-i iixl.-. w  I «t bi r'll II.IIHU i  ]<il<l follows  p icy '.'  probability  Willi  J'(follow) -  y  N-  I  N where: K^ - a threshold  Docs l,i|:<] follow?  Jlircl follow... Icicle r  ll;iv»:  ciu.uj.li  birilt  lldwn  .Ml. i n t e r , I . I t e i l  u.  t n i ^ ' e r  flil;ht?  (ulcerated [|i-|;lit takes place  AW l> i rd i in s a me flor k .15 Ic.nlc r join tl,r; intrj ial*-fl Div;hl  I U vo J ] I bi r i l h e r n le :.ied for .lel.iye.l folluv.ii,,.?  If ili.l.iycl follnv. j. w i ; , I, „„ „ . 3 (•(: rio.lt, «il off v i  I" II-: I'i  M l ' .  . l e h . y , .1  I"ir\ish  r  r,,l|„-„ „wil, I. t,n?  170  No Is l,i III ha nil 1 i n|- prey '  l)i<] l i i n l  No  Dili  (ly tlii i. or j n e v i o u s p e r i o d ?  I b i s bird's f lir.ht in the  Hird  | , . i v r an integrated l,i K .< pi' ri.ul s ?  fk.ek  follows  with  probability'  N-K  ; •' N whr rc: ' K = a I l l I r 5 liolil  ...  ]>(follow) =  No  Y e s  D o r s b i i il follow?  ®  11 i rrl fol lows leade r  Delayed follow switch of(  171 M H . K O U T I U K 'I'l.OCC.  i-i;11r • I  A l l l.lnl:i  lo flock 7t ro  ••• 0  •  Test  NO  NO  11  i l d s j, j =1, . . . . N I l l l l l l S  lave a l l birds  /  vi: s  been Irs rd?  • •i--,f i i  Is b i r d j a l r e a d y a s s i g n e d to a flock?  K K = i ; i ; 1 1. A s s i f o i b i r d j to flock K K  Tect b i r d s i , t =1, . .. , .NIMH MS  Have a l l b i r d s  N O  ~\^  Yl..  r  been tested?  Has b i r d i ;il ready bo en ossil'ncd to  NO  the same floe k a s b i i d j?  Y ICS  NO  Is b i n ! i in the same, t i r e as hi r cl. j?  Ait  bird* i  NO  Ami  j  lui.i  Is bi i d i  than PFI.OK feet .'[.ait?  A  1 re in! y  a i; s if*ne cl to a f 1 nek'  s i[;n bi nl' i to the . e.uiu Hock a n b i n l j.  A H  Oonibi ne Ike b i r d * I n bi i d 1' n Noci-._w.itl, the bl ids ,,, bi nl I'II flock r i < ir,;; I be , i , , , , ! , , , , , d,l lo. k the  lover I of ll.e. tv/., I I oc 1: mil 11 be | | i .  Y  l  '  S  Finish  •172  SUWIOUTINF:  'I:AT' Y i:  NO  ll.ive all birds k e n checked?  NO  Y i:s  Is \> i i d li.'md liiij', prey Ibis p e r i o d ?  K i n d which t r r c the b i r d i s in  T h i s bird will fly tli i s pe r i od  Is the bi rd in a t r e e ?  T r a n s l a t e bird's coordinate into g rid loi.a t ion  location  NO Docs b i r d have .1 s e a r c h i n g i m a g e ?  KO  Docs b i r d have a t e m p o r a r y s e a r c h i m a g e ?  Hfvd  te a r c h e s only for prey  YKS  corresponding  to t h U image (i) P r o b a b i l i t y of s u c c e s s on p i e y type. I Is G i l pe r p i e. y in t lie g r i d c e l l  NO  Have a l l the m e m b e r s of p i r y type i  YKS  (n the g rid ( e l l bee :. te :.U.d?  . I) r A v/ i*a ndoui numbc r  NO  ])i;i|;) ;in i A ,  ,  <  Doe  0  bi rd c a | ii r e tills p 1 e y ?  Z>—©  173  ® 'Ul  rt!  t.r. J n:\tr t- for  ;i  11  |i r r  y ly pr a  findinf, p i r y type j v.-illi p I'ldi.i hi I ity )'  NO  y( r pre y of 1 y pr j in thr  j* i  id c c 1  Have all.prey types, been tested?  Have a 11 nir mbe r r. of prey type j  Y  KS  in the f; r id t e l l be r n te :de d V  llraw random number  NO Does b i r d capture this p r e y ?  T i l l s bird.fed  successfully,  R e m u v c f r o m thr £rid cr 11 the c a p t u r e d prey.  Assign  handling time to the hire] c o r re '.pond\n[[ type (NSTA'l  tu the prey )  NO "Dues b i r d have s e a r c h imaf-.e?  Set r . f i i r c h image c q o n l to I c m p o i .11 y D e . i r c h i r.'.v;e ( i f .my) a n d U i.-t 0 1 ;ir•/ ch JiM.-itji: t o t . h e p i i . y t.ype- c a p l u i i'-d t h i s l i e r i c d .  ®  YI-TS  174  ® The  b i r d failed to fr c <1 this p r r i o d .  5 ct tf iiijiu t J ry s e a r c h im.i[;c lo z e r o . ;  NO  1 ii rd f. iled (o feed for M I I U N p e r i o d s ?  ll.l s the  Y  l\S  -  —  Set s e a r c h imaee to 7.e ro  NO Have p.l 1 b i r d s been tested?  Is the b i r d ha:idlih|; p r e y ?  K  .  n  l  .  h  YI:S  NO Doer, b i r d have a s e a r c h ima|;e?  "~1 The  b i r d attempts to Imitate, anothe r  b i r d by invoking j u b r o u l i n e IM I AT K  i  Kit 11 it tun: INK IMTA TV.' 1  i l r i l I attempts In imitate nearest hnc  ( r:'. M i l l  iM'i -.' Id i a u r  NO  Yl'S ll.i vc .ill l)inls been .-h.-c U'il 7  c—.-  NO I>id b i r d j capture prey this p e r i o d ?  NO  Y-K: Arc  b i r d s i and j in thr s a m e tree  Is d i s t a n c e between b i r d s i and j l e s s than the m a x i m u m imitation d i s t a n c e ?  NO  /  Is b i r d j n i J i r r ti. b u d I than any b i r d s p r e v i o u s l y tested''  IM11IHI)  NO  Did b i r d i find a b i r d to t m i U t c ?  IUrd i. obtains t e m p o r a r y s e a r c h imajfc for th*: p r e y typo he o h u - r v r d IMI'dlUJ c;.pturiny',  )l i i I" <1  i  o b t a i n s  no  £ 4..T T C h  ( z e r o ) i l t U |U:  t e m p o r a r y  MU§  Finish  M A 1 N I'UOC'.ll AM  176-  H e . 1 d i n j u r . u i i i ' 11-1' a  Hi-,nl  i n l.i r i l a . a n i l I m M l i n i i S Miltnuilii.f  via  'ftlMT'  Initialize i'lii.ht an.I f a d i n g v a i i a b l e s  Head in itainlr- of t i e r s via ?.ub roiil jnr .' 1 1 .VI "|"'  Read in . i n (<,» i rna t i on t .i d i s t r i b u t e v.ia s ul) i m i l in r ' }' 1 \' IT  piey  1  Initialise and  M I K K K H immbci  the x lock J"l  p r ne i a.t'o r s  !.•; -- 0  1', oducr food maps via subroutine  S"il  'FMAP'  b i r d s into flocks v a sub rout inc 'PI.OGG'  P r o d u c e ma |> of b i r d locations via nit) rout i or ' I I M A I "  P r o d u c e statistics 1 a n.i l y s i s of prey d i s t l i hut i on s v i .i uh s rout im*' ' I'A N A 1,  1  J T I M K = J T I M P." f 1  C A D . TM S T P  NO  I , : i , s lied !  Is t be ! m i nl.i I inn  (J'l l\< I-.'V/.NI IM 1.)  O > • 11 • < • ( f i n a l '  I I M  A |',  I  M  Ai-,  lejulls I  A : ;  vi. i  K nb I l u l l i ne s  ,\ i. i. i : , i  A  I  177  APPENDIX B:  TABULAR INSULTS FOR SINGLE PREY TYPE EXPERIMENTS  Table XXX  Capture rate and r i s k f o r b i r d s foraging on one prey type  Table XXXI  Capture r a t e f o r d i f f e r e n t f l o c k s i z e s on 36 clumps of 25 prey each  Table XXXII  Capture r a t e f o r d i f f e r e n t f l o c k s i z e s on 12 clumps of 7.5 prey each  Table XXXIII  Risk f o r d i f f e r e n t f l o c k s i z e s on 36 clumps of .25 prey each  Table XXXIV  F i s k f o r d i f f e r e n t f l o c k s i z e s on 12 clumps of 75 prey each  178  Clumping 0. 07 0.13 0.25 0.33 0. 46 0. 62 1. 01 1.23 1.48 1.86 2.44 3.64 5. 24 9.48  Mean F 1. 30 1.55 1.27 2.12 2.04 1.40 2.23 3.23 2. 20 3.02 4.28 3.34 3.44 2.35  Caps. Nf 1, 08 1.60 1.76 1.46 1. 85 2.20 2. 16 2.93 2.63 2.72 2.78 3.36 3. 58 2.31  Var/Mean Caps. F Nf 0.98 0.88 1.52 1.96 2.19 2.89 3.58 2.42 3.70 4.99 2.98 4.04 3.57 5.80 4.20 7.09 4.45 9.16 7.04 6.27 6.14 8.95 5.69 8.71 5.83 7.83 6.62 10.72  Prob. Zero F Nf 0.29 0.32 0.25 0.30 0.34 0.37 0.36 0.39 0.38 0.44 0.46 0.40 0.38 0.49 0.37 0.53 0.47 0.53 0.49 0.52 0.38 0.63 0.46 0.62 0.45 0.59 0.56 0.73  Caps.  T a b l e XXX_: C a p t u r e r a t e s and r i s k f o r b i r d s f o r a g i n g on one p r e y t y p e f o r a range o f v a l u e s o f prey clumping. Each value r e p r e s e n t s an average o v e r 100 s i m u l a t e d f o r a g i n g e p i s o d e s , e a c h r e p r e s e n t i n g t w e n t y m i n u t e s . ' These v a l u e c o r r e s p o n d to F i g u r e s 4-6 where C = (16*Clumping) and C l u m p i n g i s Lloyd's index of mean crowding, a n . i n d i c a t o r of within-clump d e n s i t y . Capture rates are not s i g n i f i c a n t l y different (p>.10) between f l e c k e r s and nonflockers, however both measures of risk do differ s i g n i f i c a n t l y (p<.005) .  179  C  = I n (16*clumping)  Flock s i ze  0. 16  1. 39  1. 65  1. 99  2. 29  2.78  3. 38  4. 06  1 2 4 6 8 12 16  1.08 1.05 1.90 1.22 1.50 1.29 1. 48  1 .55 1. 40 1. 95 1 .67 1. 53 2.10 1. 30  1.65 1.00 1.75 1.50 2. 50 2.63 1. 50  1.85 2.40 3.35 2.00 1. 30 2.18 2.03  1. 50 2. 55 2. 25 1. 47 1. 53 1. 62 2. 91  2. 10 2.40 6.30 2. 13 2.03 2.82 2.76  3. 05 7. 50 3. 85 4. 27 4. 13 2. 67 2. 53  2. 78 . 2.25 7. 20 1.60 3. 00 1 .50 5. 03 2.02 3. 83 2.15 1. 84 3. 85 3. 30 1 .69  5.00  T a b l e XXXI:. Mean captures i n 20 m i n u t e s when p r e y a r e i n t h i r t y - s i x clumps of t w e n t y - f i v e prey each. T h e s e v a l u e s c o r r e s p o n d t c F i g u r e 7.  180  C» =  (16*clumping)  Flock size  0. 16  1.38  1 .89  2.53  3. 14  3.72  4. 44  5. 29  6.08  1 2 3 6 10 16  1. 08 1. 05 1. 44 1. 22 1. 40 1. 48  1. 30 1.75 2.10 1. 47 1.84 1. 44  2. 85 2.50 2.93 2. 42 2.41 1. 67  2.05 4. 10 1.67 2. 68 2.75 1.68  2. 80 7. 85 4. 43 3. 27 2. 64 2. 07  3.75 3.40 3.03 2.03 5.25 4.07  3. 45 4. 25 4. 70 2. 95 3. 72 4. 15  3. 00 2. 80 3. 40 2. 25 3. 69 4. 15  2. 60 1 .80 1. 56 2.08 1. 34 2.11  T a b l e XXXII:. Mean c a p t u r e s i n t w e n t y m i n u t e s f o r a g i n g on twelve clumps of s e v e n t y - f i v e prey each. T h e s e v a l u e s c o r r e s p o n d t c F i g u r e 8.  181  C» Flock size 1 2 4 6 8 12 16  =  (16*clumping)  0.16  1.39  1.65  1.99  2.29  2.78  3.38  4.06  5.00  32 25 20 18 28 24 19  30 30 20 20 30 27 28  40 40 30 43 10 7 20  45 20 25 27 40 23 20  50 50 15 37 28 28 11  45 50 10 27 35 10 11  45 30 45 3 28 18 14  70 30 35 13 45 8 5  75 85 75 58 55 52 31  T a b l e XXXIII.: Percentage zero c a p t u r e s i n t w e n t y m i n u t e s f o r a g i n g on t h i r t y s i x clumps of t w e n t y - f i v e prey each. These v a l u e s c o r r e s p o n d t o F i g u r e 9.  182  C Flock size 1 2 3 6. 10 16  1  =  {16*Clumping)  0. 16  1. 38  1.89  2. 53  3. 14  3.?2  4.44  5.29  6.08  32 25 23 18 26 19  30 35 47 33 26 24  30 40 27 20 17 27  55 50 37 30 15 29  65 30 33 35 31 29  55 50 50 63 14 12  70 55 43 57 31 18  75 75 67 67 38 17  85 75 87 68 68 37  T a b l e XXXIVj. P e r c e n t a g e z e r o c a p t u r e s i n twenty m i n u t e s foraging on twelve clumps of s e v e n t y - f i v e prey each. These v a l u e s c o r r e s p o n d to F i g u r e 10.,  183  APPENDIX Ci  INCOMPLETE CONTINGENCY TABLES  When some c e l l s combinations,  the  column e f f e c t s  i s no  to  analyse,  This  of a c o n t i n g e n c y usual longer  Let  Nij  possible.  slightly  f o r the i m p o s s i b l e  be  the  from  It i s possible,  the  by  i n rcw  =  r *** ® *** i=1  N. j =  {Nij},  j=1  and  for  r *** © *** i=1  N =  i=1,...,r  Furthermore,  we  9  ***  and  j=1,...,s.  let Xij = 0  f i t the  {N.j}  j=1  = Then  {Nij]  r *** {Ni. }  s e t of  1  1972).  accounting  i , column  let  Ni.  however,  (fienfcerg  u s u a l one  and  cells.  number o f o b s e r v a t i o n s  r *** © ***  impossible  f o r i n d e p e n d e n c e o f row  the t a b l e f o r "quasi-independence"  analysis differs  explicitly  analysis  table represent  i f cell(i,j)  must  be  empty  otherwise.  simultaneous  linear  equations  j ; and  184  Ni.  ***  =  ®  ***  {Ai*Bj*Xij}  j=1  N.j  r *** 6  =  ***  {Ai*Bj*Xij},  i=1 Then E i j = E[ # c o u n t s =  in cell(i,j)]  Ai*Bj*Xij.  *** The  with  sum  ©  ***  {[Nij-Eij ] /Eij} 2  ( r - 1 ) * (s-1)-K  empty c e l l s  degrees  i s asymptotically chi-squared,  o f freedom,  where K = the  number  ( t h o s e f o r which X i j = 0) .  Example Consider  the f o l l o w i n g ( a r t i f i c i a l )  contingency  table.  of  185  Hi.  Then  This  40  60  2  10  10  10  30  3  30  10  10  50  M.j  60  60  20  13=14 0  the following s e t of eguaticns.  A1*(B1+B2) = 60  B1* (A1+A2+A3) = 60  fl2* (B1+B2 + B3)  = 30  B2* (A1+A2 + A3)  A3* (B1+B2 + B3)  = 50  B3*(A2+A3) = 20.  system  is  Ai,Bj  Ai*B j *  underdetermined,  but  for  A1=1  = 60  any  two  sets  of  and A i ' , B j ' we have  = Ai»*Bj'.  we may s e t any o f t h e v a r i a b l e s t o  constant  Then  20  we s o l v e  solutions  Thus,  1  and s o l v e A2=3/8  an  f o r the remaining ones. A3=5/8  f o r the t a b l e o f expected  B1=30  values  nonzero  One s o l u t i o n i s  B2=30  we  arbitrary  find  E3=20.  Sc c h i  2  = 20.14  30  30  11.25  11.25  7.5  18.75  18.75  12.5  with  3 d e g r e e s of  freedc  187  APPENDIX Dj. REPEATED EIGENVALUES  Example j k R e p e a t e d When it  an  Eigenvalues  NxN m a t r i x  has fewer than  may be i m p o s s i b l e t o p r o c e e d  given  i n Chapter  6.  .6 .1 . 1  .2 .1 .6  p = has  .2 .8 .3  eigenvalues  AND ZERO EIGJNV ALUJS  with  N distinct  the matrix  eigenvalues,  decomposition  as  F o r example.  1 , .5 and .5.  When we s o l v e t h e e q u a t i o n s  P*Oi = ,5*P V i * P = .5*P, we f i n d  t h a t t h e dot product  represented powers  Ui*Vi  a s a sum o f c o n s t a n t  of their  = 0.  matrices  respective eigenvalues.  Hence, P**N c a n n o t multiplied  be  by t h e N ' t h  To s o l v e t h i s  problem,  we may use t h e method o f g e n e r a t i n g f u n c t i o n s (or  •z-transforis  Let  inf *** ®  P (z) =  1  ***  ; Howard  1960).  p**}j* ** z  (inf=inf i n i t y ) .  n  n=0 Then,  P ( z ) = I + P (z) *z*P  Hence, P (z) = [ I Now [ I - z * P ] of  constant  _ 1  (P**0 = I , t h e i d e n t i t y  z*P]-i.  c a n be decomposed  matrices  by p a r t i a l f r a c t i o n  Ri.  into  a sum  H i t i m e s e x p r e s s i o n s o f t h e form  ( 1 - L i * z ) * * ( - R i ) , where t h e L a r e t h e e i g e n v a l u e s multiplicity  matrix).  Taking  inverse transforms  c f P,  o f these  each  of  t e r m s , we  188  find  *** P**n =  e  F i (n)*Li** *Mi.  ***  n  ~  i where F i ( n ) . i s a p o l y n o m i a l For  t h e example  matrix  we  find  =  .20 .20 .20  .56 .56 .56  + .5**n  .8 -. 2 -.2  -.56-. 16n .44 + . 04n -.56+. 04n  Example 2\ Z e r o  Eigenvalues,  .24 .24 .24  as i n the text,  0**n = 0 =  Thus,  1  with  are  of  multiplicity  the understanding  f o r n=1,2,... f c r n=0.  -. 24+. 16n -.24-. 04n .76-. 04n  Multiplicity  When a l l t h e e i g e n v a l u e s proceed  i n n of degree Ri-1 c r l e s s .  that  1,  we  may  189  ,6 .6  ,5  .2 .2 .5  .2 .2 .0  7/12 7/12 7/12  **n  +  -1/12 -1/12 5/12  (-.2) **n  1/2 -1/2 -1  (0) **0  Example In  3_1 Z e r o this  Eigenvalues,  case  uniguely  defined.  N-S  nonzero e i g e n v a l u e s . by  decompostion  based  the  2/12 2/12 2/12  3/12 3/12 -15/12 -1/2 1/2 1  -2/12 -2/12 10/12  0 0 0  Multiplicity  the eigenvectors  not  determined  3/12 3/12 3/12  So, P may Then  S>2  f o r the z e r c e i g e n v a l u e s  be decomposed the  Zn  discrepancies  upon t h e n o n z e r o  i n terms of  (n=0,... ,S-1) between  eigenvalues.  P**n  these  may and  are  be the  APPENDIX E: .TRANSITION PROBABILITIES FOR THE MARKOV MODELS  Table XXXV  Transition probabilities f o r flocking birds feeding on one prey type .  Table XXXVI  T r a n s i t i o n p r o b a b i l i t i e s f o r nonflocking b i r d s feeding on one prey type  191  Prob. .  0.07  0.74  Clumping 1.48  5.24  9.48  1 2 3  .9653 .0164, .0183  .9652 .0116 .0232  .9695 .0107 .0198  .9809 .0054 .0137  . 9836 .0043 .0121  4 5  .9850 .0150  .7600 .2400  .7800 .2200  .7436 . 2564  .7917 . 2083  6 7  .9850 .0150  .8677 .1323  .8179 .1821  .8127 . 1873  .7917 . 2083  8 9  .9850 .0150  .8714 .1286  . 8482 .1518  .9161 .0839  .8327 . 1673  10 1 1  .9850 .0150  .8786 .1214  . 8486 .1514  .9552 .0448  .9507 .0493  12 13  .9850 .0150  .90 37 .0963  . 8854 .1146  .9653 .0347  .9553 .0447  14 15  .9850 .0150  .9483 .0517  . 8893 .1107  .9691 .0309  .9713 .0287  16 17 18  .9670 .0150 .0180  .8597 .0442 .0961  .8125 .0676 .1199  .9247 .0168 .0585  .8563 .0219 . 1218  19 20  1 1  1 1  1 1  1 1  1 1  21 22 23  .9004 .0649 .0348  .6643 . 2571 .0786  .4532 .4260 .1208  24 25  1 1  1 1  1 1  1 1  1 1  26 27  .9850 .0150  .7826 .2174  .6751 .3249  .4937 .5063  .2604 .7396  28 29 30  .9336 .0302 .0362  .5433 .1544 .3023  .4070 .1901 .4029  .3311 . 2838 . 3851  .5727 . 1453 .2820  .2216 .7210 -.0574  . 1484 .7558 .0958  T a b l e XXXVi Estimated values f o r the t r a n s i t i o n probabilities f o r the flocking birds feeding on one p r e y t y p e . These r e s u l t s were d e r i v e d from 8000 t r a n s i t i o n f o r e a c h v a l u e o f p r e y clumping.  192  Prob._  0.07  0.74  Clumping 1. 48  5.24  9.48  1 2 3  .9850 .0150 NA  .9837 .0163 NA  .9848 .0152 -NA •  .9906 .0094 NA  .9945 .0055 N.A.  4 5  .9440 .0560  .7142 .2858  .7373 .2627  .7160 .2840  .7513 . 2487  6 7  .9440 .0560  .8045 .1955  .7945 .2055  .7512 .2488  .7838 .2162  8 9  .9440 .0560  .8331 .1669  . 8333 .1667  .8373 . 1627  .7908 . 2092  10 1 1  .9440 .0560  .8518 .1482  .8659 .1341  .9028 .0972  .9358 .0642  12 13  .9440 .0560  .8555 .1445  .8732 .1268  .9194 .0806  .9465 .0535  14 15  .9440 .0560  .9051 .0949  .8912 .1088  .9379 .0612  . 9614 .0386  16 17 18  .9440 .0560 NA  .9506 .0494 NA  .9333 .0667 NA  .9436 .0564 NA  .9614 .0386 NA  19 20 21 22 23  1 1 .9811 .0189 NA  1 1 .6910 .3090 NA  1 1 .4503 . 5497 NA  1 1 , .2685 .7315 NA  1 1 . 2940 .7060 NA  24 25 26 27  1 1 .9440 .0560  1 1 .6897 .3103  1 1 .5535 .4465  1 1 .3321 .6679  1 1 .2045 .7955  28 29 30  NA NA NA .  NA NA NA  NA NA NA  NA HA NA  NA NA NA  T a b l e XXXVIi Estimated t r a n s i t i o n p r o b a b i l i t i e s f o r the n o n f l o c k i n g birds f e e d i n g on one p r e y t y p e . S i n c e t h e r e i s no i m i t a t i o n s t a t e f o r nonflocking birds, certain of t h e t r a n s i t i o n s l i s t e d a r e not applicable (NA), These results were derived from 8000 t r a n s i t i o n s f o r each v a l u e o f prey clumping.  193  APPENDIX To  F i DERIVATION derive  OF VARIANCE  Var(j;n), after  the v a r i a n c e i n  for  state  Vj ,  the v a r i a n c e i n r e c u r r e n c e time  2  j  TO MEAN  Markov c h a i n . let  n periods,  Following Feller  & be t h e r e c u r r e n t Un=P[S  define  we b e g i n  f  time  with the d e r i v a t i o n of  f o r state  j  in  a  finite  trial], time a t the n'th t r i a l ] .  00=1, inf  inf  ***  F(S)  residence  event,  occurs at the n t h  F0=0,  state  (1957),  Fn=P[ 6 o c c u r s f o r t h e f i r s t Also  RATIO  ® ***  {Fn*S**n}  =  n=0  *** 6 *** n=1  { Fn*S**n}  inf ***  © ***  0(S)  {Un*S**n}.  n=0 for  n>1 P[ & f i r s t  t i m e a t k<n, and a g a i n a t n ] = Fk*Un-k  P[8  time  first  a t n ] = Fn = Fn*U0.  inf Thus,  Un =  *** e  { Fk*Un-k}.  *** n=1 Multiplying U(S) Hence,  by  S**n  and summing  - 1 = F (S).U(.S) . .  U (S) =  1/(1-F(S))  from  n=1  to i n f i n i t y ,  we  find  Now inf *** ®  with  f o r 8 persistent  {Fi}= 1 ;  M =  Define  inf *** ® ***  Qn =  and n o n p e r i o d i c  thus  F i s a probability  F'O)  {i*Fi] =  inf *** ® ***  {Fn+j}  Q(S)  =  distribution,  t h e mean r e c u r r e n c e  Rn  =  j=1  inf *** ® *** n=0  we have  inf *** 6  ***  {Qn+j}  j=1  {Qn*S**n}  R(S)  =  inf *** ® *** n=0  inf *** AS S — > 1 , Q (S) -->  ® ***  {Qn}  inf  inf  9 4** n=0  8 *«*  n=0  ***  /  **« j=1  {Fn+j}  {Rn*S**n}.  time.  inf *** 9 *** k=1  {k*Fk}  = M;  inf and  B (S)  —>  4**  {Rn]  ©  n=0 inf  inf  ***  ***  ©  ®  ***  ***  n=0 inf  ***  D=1 . inf  ***  ®  9  ***  ***  n=0  inf *** ® *** k=2 =  [V  2  { Qn+  + u  j=1  inf  *** 9  ***  {Fn +j+i}  i=1  { k (k-1) *Fk/2 }  2  - u]/2.  T h u s , Q(S) and R(S) c o n v e r g e f o r |S|<1.  196  Now  Q(S) =  =  inf *** © *** n=0  {S**n  1/(1-S)  = 1/(1-S)  and  n *** 9 *** k=0  [1 -  {Fl}]}  - {SF1 + S 2 ( F 1 + F 2 ) + S 3 (F1+F2+F3) . .. ] - F ( S ) - S{SF1  + S2(F1+F2) +  =  1/(1-S) - F (S){ 1+S+S2 + . . . }  =  (1-F (s) ) / ( 1 - s ) ;  R (S) =  inf *** 9 *** n=0 inf **.* 9  ***  { S**n [  inf *** 6 *** j=1  { S**n [M -  n=0  = M/(1-S) -  {Qn + j } ] }  n *#• €  ***  {Ok  }]}  [  n *** 6  k=0 inf *** 6  ***  {S**n  n=0  = M/(1-S) -  .,.}  inf *** « *** k=0  ***  {Ck}]}  k=0  {Qk  [  inf *** € *** n=k  {S**n}]}  197  = M/(1-S) -  inf *** 8  {Q*S**k  ***  inf *** [ »  ***  k=0  {S**n]]3  n=0 inf *** 9  M/(1-S) - 1/{1-S)  {Qk*S**k}  *** k=0  =  (M-Q (S) ) / (1-S) .  R (S) Thus  M-Q  (S)  = MQ(S)  / (1-S)  M (1 - F (S) ) — (1-S)  H - Q(S)  M (1 - F (S) )  Now  inf *** 9  ***  { (Un - 1/R)S**n}  = U (s)  - 1/M(1-S)  n=0  =  1/(1 - F ( S ) )  = [ H - MS  -  1/M(1-S)  - 1 • F (S) ]/[M (1-S) (1 - F (S) ) ]  M, - [ 1 - F (S) ] / (1-S) --~ M[ 1 - F(S) ]  198  =  [ H  - Q (S) ]/[M (1 - S (S)) ]  = 8(S)/MQ <S).  Thus f o r S = 1  inf ***  { Un - 1/M} = Q 2 + M2 - H]/2H2.  « ***  n=0  For  a finite  state  Markov c h a i n  with  r *** «  *** k=2  1/M  Thus,  matrix  j  Un = P n [ j j ] = Y 1 [ j j ] +  and  transition  = 1/Mj  =  Y1[jj].  [ Ik**n*Yk[ j j ] }  ~  P, f o r a  given  inf  inf  **« €  ***  ***  {On - 1/M}  r  ***  ©  n=0  ®  ***  ***  n=0  k=2  {Lk**n*Yk[jj]}  ~  *** ® *** k=2 = [V  So  V  2  = M - M  2  {Yk[ j j ] / | 1 - L k ) }  - M + M ]/2M . 2  r• »** + 2M * ©  2  2  2  { Yk£ j j ] / ( 1 - I k ) }  ~  *** k=2 [  in Feller's  Alternatively,  V  2  notation  Sk/(Sk-1)  replaces  we may sum o v e r n = 1 , i n f ,  = -M + M  2  r *** + 2M * ® 2  1/ (1-Lk) ]  getting  { Yk[ j j ] L k / (1-Lk) }  k=2 [ Finally  i n Feller's  notation  1/(Sk-1)  r e p l a c e s L k / (1-Lk) ] .  we have  Var(j;n)  = n*V j/M j. 2  3  (Feller,  1957, p. 310)  Publications Thompson,  W . A . a n d V e r t i n s k y , T. (1975) polymorphic prey populations,  B i r d flocking revisited: J . A n i m . E c o l . (in p r e s s )  Thompson,  W . A . V e r t i n s k y , I . a n d K r e b s , J . R . (1974) The s u r v i v a l v a l u e of f l o c k i n g i n b i r d s : a s i m u l a t i o n m o d e l , J . A n i m . E c o l . 43: 78 5-8 2 0 .  Vertinsky,  T . , T h o m p s o n , W . A . a n d U y e n o , D . (1974) M e a s u r i n g ' consumer desire for p a r t i c i p a t i o n in c l i n i c a l d e c i s i o n m a k i n g , H e a l t h S e r v . R e s . . .Summer: 121-34. T h o m p s o n , W . A . , V e r t i n s k y , T. a n d K a n e , J . (1974) K S M - policy simulation: user's manual, TTCC Review. (in press) . T  Thompson,  Kane,  W . A . , V e r t i n s k y , I . a n d K a n e , J . (1973) Canadian i n d u s t r i a l p o l i c y - s i m u l a t i o n and a n a l y s i s , L o n g Range Planning December: 66-73.  J . , V e r t i n s k y , I . , a n d T h o m p s o n , W . (1973) K S I M : a .' methodology for interactive r e s o u r c e p o l i c y simulation, W a t e r R e s o u r c e s Res.9(1): 6 5 - 7 9 .  i Kane,  J . , T h o m p s o n , W . A . , a n d V e r t i n s k y , I . (1972) E n v i r o n m e n t a l s i m u l a t i o n and p o l i c y f o r m u l a t i o n - m e t h o d o l o g y and example (water p o l i c y for B r i t i s h C o l u m b i a ) , International S y m p o s i u m on M a t h e m a t i c a l M o d e l l i n g T e c h n i q u e s (ed A . K . B i s w a s ) E n v i r o n m e n t C a n a d a , O t t a w a , 1: 3 9 - 5 5 .  Kane,  J . , T h o m p s o n , W . , a n d V e r t i n s k y , T. (1972) Healthcare delivery: a policy simulator, Socio - Econ. P l a n . S c i . 6: 283-93:.  Thompson,  W . A . a n d V e r t i n s k y , I . ( u n d e r r e v i e w ) A p p l i c a t i o n of M a r k o v c h a i n s to a n a l y s i s of a s i m u l a t i o n of b i r d s ' f o r a g i n g .  

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