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UBC Theses and Dissertations

Bird flocking as a foraging strategy Thompson, William Andrew 1974

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BIRD FLOCKING AS A FORAGING STRATEGY ' by WILLIAM ANDREW THOMPSON B.A., Pomona C o l l e g e , 1967 A DISSERTATION SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY I n t e r d i s c i p l i n a r y S t u d i e s i n M a t h e m a t i c a l E c o l o g y We a c c e p t t h i s t h e s i s as c o n f o r m i n g t o t h e r e q u i r e d s t a n d s r - d THE UNIVERSITY OF B R I T I S H COLUMBIA November, 1974 In presenting th i s thesis in p a r t i a l fu l f i lment of the requirements for an advanced degree at the Univers i ty of B r i t i s h Columbia, I agree that the L i b r a r y shal l make it f ree ly ava i lab le for reference and study. I further agree that permission for extensive copying of th is thesis for scho lar ly purposes may be granted by the Head of my Department or by his representat ives . It is understood that copying or pub l i ca t ion of th is thesis for f inanc ia l gain shal l not be allowed without my writ ten permission. Department of The Univers i ty of B r i t i s h Columbia Vancouver 8, Canada ° a t e / 3 Jh*. I9?V ABSTRACT To assess the s u r v i v a l value of b i r d f l o c k i n g , a s i m u l a t i o n model was c o n s t r u c t e d based upon known b i r d behaviours. The primary source of i n f o r m a t i o n f o r the model was p u b l i s h e d theory and data on s m a l l p a s s e r i n e s . By a v o i d i n g the confounding f a c t o r s of p r e d a t i o n (on the b i r d s ) and breeding, a p p l i c a b i l i t y i s r e s t r i c t e d to winter f l o c k s f o r a g i n g on low d e n s i t y prey p o p u l a t i o n s . S i m u l a t i o n experiments were performed to examine the e f f e c t s on f e e d i n g success of v a r i a t i o n i n the b e h a v i o u r a l parameters ( s e n s i t i v i t y a n a l y s i s ) , v a r i a t i o n i n prey d i s t r i b u t i o n , v a r i a t i o n i n f l o c k s i z e , d i f f e r e n c e s between b i r d s i n b e h a v i o u r a l parameters, v a r i a t i o n i n number of 'prey t y p e s ' (monomorphic vs. Polymorphic prey p o p u l a t i o n s ) , and v a r i a t i o n i n the r a t i o of abundance f o r two prey types. Success was measured two ways: 1) as mean capture r a t e and 2) as the p r o b a b i l i t y of a b i r d ' s making no captures i n twenty simulated minutes ('risk') . The l a t t e r measure may be more a p p r o p r i a t e under winter c o n d i t i o n s when a s i n g l e day's inadequate f e e d i n g may s e r i o u s l y d i m i n i s h a b i r d ' s l i k e l i h o o d of s u r v i v a l . To examine long-run model behaviour, methods were developed to approximate the s i m u l a t i o n with a Markov model. Sim u l a t i o n experiments v a r y i n g prey d i s t r i b u t i o n i n d i c a t e d that a two parameter c h a r a c t e r i z a t i o n of a prey p o p u l a t i o n (e.g. mean d e n s i t y and v a r i a n c e over mean d e n s i t y ) i s not i i i adequate to to e s t i m a t e the s e n s i t i v i t y a n a l y s i s r e v e a l e d the importance o f ' g i v i n g - u p time* (the time from l a s t prey c a p t u r e u n t i l the b i r d l e a v e s the immediate f o r a g i n g area) to f e e d i n g s u c c e s s . F l o c k i n g reduced ' r i s k 1 whenever prey were moderately t o h i g h l y c lumped. F l o c k i n g a l s o enhanced the mean c a p t u r e r a t e when prey were clumped and p o l y m o r p h i c (a po lymorphic prey p o p u l a t i o n means two or more major types of p r e y , not n e c e s s a r i l y of the same s p e c i e s ) . When the prey p o p u l a t i o n was monomorphic, f l o c k i n g n e i t h e r i n c r e a s e d nor decreased the mean c a p t u r e r a t e . TABLE OF CONTENTS CHAP. 1 INTRODUCTION 1 CHAP. 2 BIRD FLOCKING AND ITS EEHAVIGU RAI COMPONENTS 9 CHAP. 3 SIMULATION METHODOLOGY 21 CHAP. 4 THE SIMULATION MODEL 30 H a b i t a t 34 Movement 36 Feed ing 43 CHAP. 5 THE SIMULATION EXPERIMENTS 49 S e n s i t i v i t y a n a l y s i s 49 Mcnomorphic prey p o p u l a t i o n s 5.6 Popymorphic prey p o p u l a t i o n s 78 CHAP. 6 MARKOV MODELS 105 Methodology 105 I d e n t i f i c a t i o n of s t a t e s 108 E s t i m a t i o n of t r a n s i t i o n p r o b a b i l i t i e s 111 V a l i d a t i o n 122 V R e s u l t s 133 CHAP. 7 CONCLUSIONS 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 V 1 T A B L E S I B e h a v i o u r a l components of a search s t r a t e g y 33 I I Parameter v a l u e s used i n the s i m u l a t i o n 51 exper iments I I I S e n s i t i v i t y a n a l y s i s to t e s t whether parameter 52 v a r i a t i o n a l t e r s r e s u l t s IV S e n s i t i v i t y a n a l y s i s to t e s t whether g iven 53 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 i n r e s u l t s 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 r a t e per hour averaged over an 73 e n t i r e day ' s f e e d i n g VII Capture r a t e g i v e n i n d i v i d u a l v a r i a t i o n 76 VIII Risk g i v e n i n d i v i d u a l v a r i a t i o n 77 IX Des igns of po lymorphic prey exper iments 83 X Mean c a p t u r e r a t e s cn itononiorphic and 84 po lymorphic prey XI Reduct ion i n c a p t u r e r a t e by the p a r t i t i o n of 86 a prey p o p u l a t i o n i n t o two morphs XII Mean c a p t u r e r a t e s f o r v a r i o u s f l e c k s i 2 e s 87 f e e d i n g cn po lymorphic prey X I I I Ri sk f o r b i r d s f e e d i n g on moncmcphic and 89 po lymorphic prey via XIV Risk f o r v a r i o u s f l o c k s i z e s f e e d i n g on po lymorphic prey 90 XV F r a c t i o n o f the p r e d a t i o n s u f f e r e d by the l e s s a t t a c k e d of two morphs when the morphs are i n e q u a l numbers 92 XVI Apport ionment of prey c a p t r u e s between above-and be low-average b i r d s (b irds d i f f e r i n Frey d e t e c t i o n a b i l i t i e s ) 93 XVII Apport ionment of r i s k between above- and be low-average b i r d s ( b i r d s d i f f e r i n prey d e t e c t i o n a b i l i t i e s ) 94 XVIII Capture r a t e on po lymorphic prey i n p r o p o r t i o n s 1:1 and 9:1 expressed as percentage of c a p t u r e r a t e on s i m i l a r monomorphic prey 96 XIX Percentage , of a t t a c k s on prey type 1 i n a d i m o r p h i c prey p o p u l a t i o n f o r prey r a t i o s 1:1 and 9:1 97 XX Capture r a t e per i n d i v i d u a l prey on each mcrph i n a d imorph ic p o p u l a t i o n expressed as percentage o f c a p t u r e r a t e on s i m i l a r monomorphic p o p u l a t i o n 98 XXI Risk f o r d i f f e r e n t prey r a t i o s i n a d imorphic prey p o p u l a t i o n 99 XXII S t a t e space f o r the Markov models 110 XXIII T r a n s i t i o n s and t h e i r bounds f o r the Markov 112 models XXIV C h i - s q u a r e d t e s t s tha t t r a n s i t i o n counts are 124 "quas i - independent" f o r the Markov models v i i i XXV V a l i d a t i o n of Markov models by compar i son of t r a n s i t i o n p r o b a b i l i t i e s with a second s e t o f t r a n s i t i o n s 126 XXVI V a l i d a t i o n of Markov models by a comparison o f p r e d i c t e d s t a t e r e s i d e n c e t imes with a second se t of s t a t e r e s i d e n c e t imes 127 XXVII Asymptot i c c a p t u r e r a t e s as g i v e n by the Markov models 140 XXVIII Dominant e i g e n v a l u e s of the Markov models 141 XXIX V a r i a n c e over mean c a p t u r e r a t e as g iven by the Markov models 143 XXX Capture r a t e s and r i s k , monomorphic prey at v a r i o u s l e v e l s of c lumping (corresponds to F i g u r e s 4-6) 178 XXXI Mean c a p t u r e r a t e s f o r v a r i o u s f l o c k s i z e s , 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 each (corresponds to 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 v a r i o u s f l o c k s i z e s , prey i n twelve clumps of s e v e n t y - f i v e prey each (corresponds to F i g u r e 8) 180 XXXIII R i s k f o r v a r i o u s f l o c k s i z e s , prey i n t h i r t y -s i x clumps of t w e n t y - f i v e p i e y each (corresponds to 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 i n twelve clumps of s e v e n t y - f i v e prey each (corresponds to F i g u r e 10) 182 XXXV E s t i m a t e d t r a n s i t i o n p r o b a b i l i t i e s f or f l o c k i n g b i r d s f e e d i n g on monomorphic prey 191 ix XXXVI E s t i m a t e d t r a n s i t i o n p r o b a b i l i t i e s for 192 n o n f l o c k i n g b i r d s f e e d i n g cn monomorphic prey X FIGURES 1 A Search paths o f t h r e e b i r d s s e a r c h i n g f o r f i v e 3 8 minutes IB Area covered by a f l o c k of nine b i r d s 3 9 s e a r c h i n g f o r f i v e minutes 2 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 govern ing random 4 0 search movements 3 F l o c k d e t e r m i n a t i o n f o r d i f f e r e n t va lues of 4 4 PFLOK 4 Mean c a p t u r e r a t e as a f u n c t i o n cf prey 6 0 c lumping 5 V a r i a n c e over mean c a p t u r e r a t e as a f u n c t i o n 6 1 of prey c lumping 6 R i s k ( p r o b a b i l i t y of z ero c a p t u r e s i n twenty 6 2 minutes) as a f u n c t i o n of prey c lumping Mean c a p t r u e r a t e f o r v a r i o u s f l o c k s i z e s , 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 Mean c a p t u r e r a t e f o r v a r i o u s f l o c k s i z e s , prey i n twelve clumps of s e v e n t y - f i v e prey each 6 5 6 7 Risk f o r v a r i o u s f l o c k s i z e s , prey i n t h i r t y - 6 9 s i x clumps of t w e n t y - f i v e prey each 1 0 Bisk f o r v a r i o u s f l e c k s i z e s , prey i n twelve clumps of s e v e n t y - f i v e prey each 7 0 Mean capture r a t e per b i r d as a f u n c t i o n time f o r very h i g h l y clumped prey ( C = 9 . 4 8 ) Mean capture r a t e per b i r d as a f u n c t i o n time f o r h i g h l y clumped prey ( C = 5 . 2 4 ) Mean capture r a t e per b i r d as a f u n c t i o n time f o r randomly d i s t r i b u t e d prey (C=0.07) x i i DIAGRAMS (APPENDIX A) 1 S u b r o u t i n e TMSTP 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 S u b r o u t i n e IMTATE 6 Main program x i i i MATHEMATICAL NOTATION Owing to the l i m i t a t i o n s of the p r i n t e r cn which t h i s document has been produced , c e r t a i n s p e c i a l c o n v e n t i o n s have been adopted with regard to the mathemat ica l n o t a t i o n . . These are e x p l a i n e d below, a l o n g with some p o s s i b l y u n f a m i l i a r n o t a t i o n s . *** Summation i s i n d i c a t e d by « *** n * * * For example, ® {Ai} = A 1+A2 + . . . + A n . *** i=1 M u l t i p l i c a t i o n i s i n d i c a t e d by n ***** For example, * * {Ai} = * * i= 1 The symbols 8 and $ have been used to i n d i c a t e t h a t an element i s or i s not i n c l u d e d i n a s e t . , For example, XS[0,1] i m p l i e s t h a t 0<X<1, w h i l e , X£[0 ,1 ] i m p l i e s tha t e i t h e r X>1 or X<0. ***** * * * * A 1*A 2* . . . *A n, M a t r i c e s of numbers are surrounded by | • s . x i v .2 .81 T h u s , .1 . 6 r e p r e s e n t s a 3x2 m a t r i x . .5 . 1 | 1 CHAPTER QUE: INTRODUCTION The s u r v i v a l value of f l o c k i n g i n b i r d s has been d i s c u s s e d e x t e n s i v e l y i n the l i t e r a t u r e i n recent years (e.g. Crook S Goss-Custard 1972; P u l l i a m 1973; Krebs 1973a). V a r i o u s authors have proposed t h a t f l o c k i n g enables i n d i v i d u a l s to i n c r e a s e the e f f e c t i v e n e s s with which they e x p l o i t p a t c h i l y d i s t r i b u t e d food. To examine t h i s p r o p o s i t i o n , i n f o r m a t i o n and concepts from e x i s t i n g s t u d i e s were s y n t h e s i z e d to c o n s t r u c t a s i m u l a t i o n model of f l o c k i n g behaviour i n b i r d s . The study f o c u s e s on the s u r v i v a l value of f l o c k i n g i n p a s s e r i n e b i r d s which f l o c k while o v e r w i n t e r i n g i n c o l d c l i m a t e s . R e p r e s e n t a t i v e examples are the great t i t (Parus BS22£) which ove r w i n t e r s i n England (Hinde 1952) and the b l a c k -capped chickadee (Parus a t r i c a p i l l u s ) i n New England (Morse 1970). T y p i c a l l y , these b i r d s are t e r r i t o r i a l i n the breeding season and feed i n f l o c k s through the winter. During t h i s l a t t e r p e r i o d there.appears to be l i t t l e p r e d a t i o n pressure on the b i r d s (Morse 1970). In a d d i t i o n , i t should be noted t h a t i n severe winters f l o c k s i z e s tend t o be l a r g e r , while i n mild winters some b i r d s maintain nest s i t e s and t e r r i t o r i e s through the winter (Hinde 1952). T h i s appears to provide an e x c e l l e n t case f o r examining the hypothesis t h a t f l o c k i n g enhances feeding -. - • • • J success. The theory of f o r a g i n g or p r e d a t i o n s t r a t e g i e s of animals has been approached i n s e v e r a l ways. Here we w i l l be concerned 2 with two of these: one approach i n v o l v e s p r e d i c t i n g the d i e t s e l e c t i o n of a predator u s i n g some o p t i m a l i t y p r i n c i p l e s (e.g. Schoener 1971; HacArthur 1972; Charnov 1973; P u l l i a m 1974); the other i n v o l v e s breaking the p r e d a t i o n process down i n t o i t s component p a r t s and studying combinations of these ( H o l l i n g 1959a,b, 1965, 1966). The theory of d i e t s e l e c t i o n assumes that the predator i s faced with two d e c i s i o n s 1 ("predation d e c i s i o n s " ) while f o r a g i n g : 1) where to search and 2) whether to pursue (or eat) a given prey once i t has been l o c a t e d . D e c i s i o n r u l e s are d e r i v e d , u s u a l l y based upon the assumption that p r e d a t o r s attempt to maximize the r a t e of net c a l o r i c gain. However, other o b j e c t i v e f u n c t i o n s may be p l a u s i b l e ; f o r example, maximizing the r a t e of n u t r i e n t gain f o r l a r g e h e r b i v o r e s (Westoby 1974). Two common sense d e c i s i o n r u l e s are d e r i v e d from t h i s theory: 1) the predator s h o u l d search where the expected y i e l d per u n i t e f f o r t i s g r e a t e s t ; and 2) the predator should pursue (eat) any prey f o r which the expected y i e l d per u n i t f u r t h e r e f f o r t (e.g. e f f o r t = p u r s u i t time + handling time) i s g r e a t e r than the average y i e l d per u n i t e f f o r t f o r prey not yet l o c a t e d (e.g. e f f o r t .=. search + p u r s u i t + handling , t i m e s ) . One major i m p l i c a t i o n of t h e s e ' d e c i s i o n r u l e s i s t h a t given the knowledge of the average s e a r c h , p u r s u i t and h a n d l i n g times f o r each type of prey and the average c a l o r i c i No i m p l i c a t i o n of c o n s c i o u s d e c i s i o n making i s intended here. A b i r d which i s "programmed" to respond to s p e c i f i c s t i m u l i i n s p e c i f i c ways may nonetheless be considered as a d e c i s i o n maker. c o n t e n t o f each type o f p r e y , the p r e d a t o r should d i v i d e the prey types i n t o two c a t e g o r i e s . In one c a t e g o r y (the ' o p t i m a l s e t ' ; Charnov 1973) are those prey which the p r e d a t o r s h o u l d always t r y to c a p t u r e ; i n the o ther are those prey which the p r e d a t o r shou ld always i g n o r e . Futhermore , t h i s breakdown i n t o p r e f e r r e d prey and i g n o r e d prey should not change with the p r e d a t o r ' s l e v e l o f hunger . However, t h i s l a t t e r p r e d i c t i o n i s c o n t r a r y to the known ev idence t h a t a s a t i a t e d p r e d a t o r becomes much more s e l e c t i v e i n i t s c h o i c e s (Schoener 1971). In d i s c u s s i n g t h i s f a i l u r e , P u l l i a m (1974) p o i n t s cut a major shor tcoming of t h i s form of p r e d a t i o n t h e o r y , the assumption tha t the i n d i v i d u a l a n i m a l has the necessary i n f o r m a t i o n to make o p t i m a l d e c i s i o n s . S i n c e i n d i v i d u a l an imal s must forage i n an u n c e r t a i n and o f t e n changing w o r l d , i t appears t h a t o p t i m a l f o r a g i n g t h e o r y at bes t p r o v i d e s an . i d e a l i z e d s tandard of performance a g a i n s t which to examine the a c t u a l f o r a g i n g b e h a v i o u r s o f a n i m a l s . While f o r a g i n g t h e o r y p r o v i d e s an i d e a l i z e d concept of what an o p t i m a l f o r a g e r shou ld do, i t does not c o n s i d e r the means by which an a n i m a l may a c h i e v e these ends . That i s , how i n o p e r a t i o n a l terms does the i n d i v i d u a l a n i m a l s e l e c t the area i n which to s earch and s e l e c t the prey to pursue and eat? A second approach to p r e d a t i o n t h e o r y , which does c o n s i d e r the b e h a v i o u r o f the p r e d a t o r , i s B o i l i n g ' s component a n a l y s i s ( H o l l i n g 1959a ,b , 1965, 1966). T h i s work has been concerned p r i m a r i l y with the f u n c t i o n a l response o f a p r e d a t o r to prey d e n s i t y , i e . a the changes i n capture r a t e accompanying changes i n p^rey d e n s i t y . The components which i n f l u e n c e the p r e d a t i o n d e c i s i o n s are (for each prey t y p e ) : 1. Rate of s u c c e s s f u l search 2. Time of exposure 3. Handling time 4. Hunger , 5. Learning (by the predator) 6 . I n h i b i t i o n by the prey 7. E x p l o i t a t i o n ( d e p l e t i o n of prey) 8. I n t e r f e r e n c e between p r e d a t o r s 9. S o c i a l f a c i l i t a t i o n 10., avoidance l e a r n i n g by the prey. , Of these components, l e a r n i n g , s o c i a l f a c i l i t a t i o n and i n t e r f e r e n c e w i l l be of primary i n t e r e s t i n c o n s i d e r i n g the value of f l o c k i n g as a f o r a g i n g s t r a t e g y . a l l t h r e e have a major i n f l u e n c e on the r a t e of s u c c e s s f u l s e a r c h . Time of exposure, handling time and hunger w i l l not be considered e x p l i c i t l y , hence they are h e l d f i x e d i n the b i r d f l o c k i n g model. I t i s a c c e p t a b l e to assume hunger i s constant only because the s i m u l a t i o n runs are f i x e d to cover a s h o r t p e r i o d of time (twenty simulated minutes), during which time the b i r d s are u n l i k e l y to become s a t i a t e d . E x p l o i t a t i o n i s an i n e v i t a b l e r e s u l t of the p r e d a t i o n process. In the s i m u l a t i o n experiments i t s e f f e c t s w i l l only be l o c a l , as the time h o r i z o n i s kept s h o r t enough to prevent s e r i o u s e x p l o i t a t i o n of the e n t i r e prey 5 p o p u l a t i o n . F i n a l l y , i n h i b i t i o n and l e a r n i n g by the prey w i l l not be c o n s i d e r e d . Of c e n t r a l importance i n making pr e d a t i o n d e c i s i o n s i s the dete r m i n a t i o n of the encounter r a t e s f o r . e a c h prey type. These must be estimated from the p r e d a t o r ' s past e x p e r i e n c e s . F a c t o r s which may a l t e r the estimated encounter r a t e s a r e : 1. Prey d e n s i t y 2. Prey d i s t r i b u t i o n ( i t i s assumed that the predator has a f i x e d p e r c e p t u a l f i e l d ) 3. Conspicuousness of prey 4. Predator l e a r n i n g to d e t e c t camouflaged prey - e.g. search image (de R u i t e r 1952; Tinbergen 1960; Dawkins 1971a,b) 5. Predator l e a r n i n g m i c r o h a b i t a t of prey - e.g. "prey niches" (Royama 1970) 6. Predator movement p a t t e r n - movement r a t e , t u r n i n g r a t e and giving-up time 7. I n t e r f e r e n c e with other p r e d a t o r s . In a d d i t i o n , past estimates of encounter r a t e s may be biased due to e x p l o i t a t i o n of the prey ( i g n o r i n g other sources of change i n the prey p o p u l a t i o n ) . Thus, the predator has a v a i l a b l e an a d d i t i o n a l s e t of d e c i s i o n v a r i a b l e s and de c i s i o n s ' to make (we w i l l c a l l these "secondary p r e d a t i o n d e c i s i o n s " ) . And the c h o i c e amongst these secondary predation d e c i s i o n s w i l l have a major i n f l u e n c e on the very i n f o r m a t i o n used to make the primary p r e d a t i o n d e c i s i o n s . Given the complexity c f t h i s g e n e r a l d e c i s i o n process, i t appears most p r o f i t a b l e t o examine known 6 behaviours of a predator to see how these d e c i s i o n s under u n c e r t a i n t y are made and how the s o c i a l i n t e r a c t i o n s which occur i n f l o c k s may l e a d to more e f f e c t i v e d e c i s i o n making. Krebs (1973b) presents a good review of the b e h a v i o u r a l aspects of t h i s problem. As i n d i c a t e d p r e v i o u s l y the b i r d s must make d e c i s i o n s r e g a r d i n g where to search and which prey to s e l e c t . A l l of these and t h e i r incumbent d e c i s i o n s may be i n f l u e n c e d by l e a r n i n g . When a b i r d i s i n a f l o c k , i t s . d e c i s i o n s (manifested i n i t s behaviours) w i l l be i n f l u e n c e d by behaviours of near neighbors as w e l l as by i t s own experiences. I t i s a t t h i s p o i n t t h a t f l o c k i n g may be v a l u a b l e i n terms of f o r a g i n g s u c c e s s . The pooled i n f o r m a t i o n of the f l o c k members i s l i k e l y t o provide 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 . The c o s t s to the i n d i v i d u a l of being In a f l o c k are l i k e l y to be r e a l i z e d from more r a p i d prey e x p l o i t a t i o n and g r e a t e r i n t e r f e r e n c e . Thus, under d i f f e r e n t environmental c o n d i t i o n s we can expect that d i f f e r e n t forms of s o c i a l , o r g a n i z a t i o n may be s e l e c t e d f o r . In t h i s t h e s i s I w i l l attempt to e l u c i d a t e those c o n d i t i o n s under which f l o c k i n g i s of net b e n e f i t to the i n d i v i d u a l f o r a g i n g b i r d . Chapter 2 w i l l . d i s c u s s the b e h a v i o u r a l and e c o l o g i c a l a spects of the problem i n a g e n e r a l context with p a r t i c u l a r r e f e r e n c e to the f o l l o w i n g t o p i c s : an examination of the p r e v a i l i n g i d e a s r e g a r d i n g the s u r v i v a l value of f l o c k i n g ; and a 7 d i s c u s s i o n of other i n v e s t i g a t i o n s concerning the p r e d a t i o n components d i r e c t l y r e l e v a n t to t h i s study. Chapter 3 w i l l review major problems of the s i m u l a t i o n methodology and w i l l d i s c u s s some a n a l y t i c methods which may be a p p l i e d to i n t e r p r e t a t i o n of s i m u l a t i o n models. Chapter 4 w i l l present a d e t a i l e d d e s c r i p t i o n of the model, i n c l u d i n g the sources of the v a r i o u s behaviours and parameters b u i l t i n to the model. Flow c h a r t s of major computer s u b r o u t i n e s w i l l be i n c l u d e d . Chapter 5 w i l l present the s i m u l a t i o n experiments, r e s u l t s and d i s c u s s i o n . Three groups of s i m u l a t i o n experiments w i l l be d i s c u s s e d : the f i r s t ones i n v e s t i g a t e parameter s e n s i t i v i t y ; the second examine the value of f l o c k i n g when prey p o p u l a t i o n s are monomorphic; and the t h i r d ones i n v e s t i g a t e the value of f l o c k i n g when prey are polymorphic. Throughout, the terms value and s u r v i v a l value are measured i n terms of the i n d i v i d u a l b i r d , net of the f l o c k . In examining these t o p i c s , the study w i l l be r e s t r i c t e d to low average prey d e n s i t i e s such as those l i k e l y to be a v a i l a b l e to s m a l l p a s s e r i n e s o v e r w i n t e r i n g i n c o l d c l i m a t e s (e.g. great t i t s w i n t e r i n g i n England). Chapter 6 w i l l d i s c u s s the a p p l i c a t i o n of Markov models f o r a n a l y s i s of s p e c i a l f e a t u r e s of s i m u l a t i o n models. The method w i l l be d i s c u s s e d and employed to examine ques t i o n s of long range and t r a n s i e n t f e e d i n g t r e n d s . T o p i c s w i l l i n c l u d e : 8 e s t i m a t i o n of t r a n s i t i o n p r o b a b i l i t i e s ; v a l i d a t i o n of the de r i v e d t r a n s i t i o n matrix; and a n a l y t i c manipulations of the Markov model to o b t a i n i n s i g h t i n t o long run and s h o r t term behaviour. F i n a l l y , chapter 7 w i l l summarize some of the more i n t e r e s t i n g r e s u l t s , make some suggestions f c r f u t u r e i n v e s t i g a t i o n , and provide a short commentary on the e f f e c t i v e n e s s of the methods employed i n the study., 9 CHAPTER TWOj. BIRD FLOCKING AND ITS BEHAVIOURAL COMPONENTS This chapter w i l l d i s c u s s the s u r v i v a l value o f b i r d f l o c k i n g . Emphasis w i l l be on the concepts and evidence which w i l l l a t e r be developed i n t o a model of a f o r a g i n g f l o c k . Since the g o a l of t h i s study i s the examination of hew f l o c k i n g may enhance f e e d i n g success, c o n s i d e r a t i o n s of p r e d a t i o n on the b i r d s have been excluded. T h i s e x c l u s i o n appears j u s t i f i e d on two grounds: 1) p r e d a t i o n on the b i r d s and f o r a g i n g successs are l a r g e l y independent (with the exc e p t i o n d i s c u s s e d belcw); and 2) there appears to be l i t t l e p r e d a t i o n on the b i r d s during the winter (Morse 1970). F o l l o w i n g the d i s c u s s i o n cf f l e c k i n g w i l l be a review of previous i n v e s t i g a t i o n s concerning the b e h a v i o u r a l components i d e n t i f i e d i n Chapter 1 as r e l e v a n t to a c o n s i d e r a t i o n of b i r d f l o c k i n g . T hese.include the e f f e c t s of prey d i s t r i b u t i o n on capture r a t e , the e f f e c t s of prey capture on movement, i n t e r f e r e n c e between p r e d a t o r s , giving-up time, l e a r n i n g and i m i t a t i o n ( s o c i a l l e a r n i n g ) . The adaptiveness of b i r d f l o c k i n g has i n t e r e s t e d e c o l o g i s t s f o r over h a l f a century (eg. M i l l e r 1922). Here the word • f l o c k 1 w i l l be r e s t r i c t e d t o mean a f o r a g i n g f l o c k , an aggregation of f o r a g i n g b i r d s with some s o c i a l a t t r a c t i o n between i n d i v i d u a l s . In o p e r a t i o n a l terms f l o c k s are g e n e r a l l y i d e n t i f i e d by the cohesion of the birds.' movements. As the s u b j e c t has been reviewed by s e v e r a l authors (eg. Rand 1954; Crook 1965), o n l y a s h o r t d i s c u s s i o n of major ideas w i l l be 10 given here. F l o c k i n g may have s u r v i v a l value f c r a b i r d ty 1) reducing the b i r d ' s s u s c e p t i b i l i t y to p r e d a t i o n , or 2) by i n c r e a s i n g the b i r d ' s success i n f o r a g i n g . While v a r i o u s i n v e s t i g a t o r s (Lack.1968; Goss-Custard 1970; Vine 1971; Hamilton 1971; B u s k i r k 1972; Lazarus 1972; Pu l l i a m 1973) have argued t h a t f l o c k i n g i s s e l e c t e d f o r by p r e d a t i o n pressures on the b i r d s , i t w i l l be d i s c u s s e d here only i n so f a r as i t d i r e c t l y a f f e c t s f e e d i n g success. S e v e r a l means have been i d e n t i f i e d by which f l o c k i n g may enhance f e e d i n g e f f i c i e n c y . 1. Reduce the time b i r d s watch f o r p r e d a t o r s . , Since time spent watching out f o r approaching predators i s time taken away from f o r a g i n g , i t i s d e s i r a b l e f o r the b i r d s to reduce the time spent i n t h i s necessary a c t i v i t y without s a c r i f i c i n g the a b i l i t y a l r e a d y present f o r d e t e c t i n g these predators. Murton, Isaacson & Westwood (1971) have noted t h a t s o l i t a r y wood-pigeons spend more time " l o o k i n g around" (head r a i s e d ) than those i n f l o c k s . Hence, the . b i r d s i n f l o c k s spent a g r e a t e r p r o p o r t i o n of t h e i r f o r a g i n g time a c t i v e l y engaged.in f o r a g i n g . Powell (1974) demonstrated a s i m i l a r phenomenon f o r s t a r l i n g s , n o t i n g a l s o t h a t the mean r e a c t i o n time t o . s i g h t i n g a model hawk was s m a l l e r f o r the b i r d s i n a f l o c k (of 10) than f o r s i n g l e b i r d s . As i n d i c a t e d e a r l i e r , p r e d a t i o n on the b i r d s i s not i n c l u d e d i n the model. , Hence, t h i s h y p o t h e s i s w i l l not be examined. 2. Copy the a c t i o n s of a s u c c e s s f u l b i r d . T h i s may i n v o l v e 11 the copying o f prey types (Burton 1971) cr the copying of prey ' s i t e s * ( i . e . s p e c i f i c m icrohabitats) (Krebs e t a l . 1972; Krebs 1973a). Copying behaviour i s the primary means i n c l u d e d i n the model by which f l o c k i n g may be b e n e f i c i a l . I t i s d i s c u s s e d more thoroughly l a t e r i n the chapter and again i n Chapter 4. Reduce the time b i r d s spend f o r a g i n g i n areas of low prey d e n s i t y ( M i l l e r 1922; Cody 1971). "The speed with which a f l o c k moves may be ad j u s t e d r a t h e r p r e c i s e l y to provide an optimum p e r i o d of feed i n g i n each area with a minimum of wasted e f f o r t : due to i n d i v i d u a l s moving i n t o areas p r e v i o u s l y f o r a g e d . " (Short 1961). While t h i s hypothesis i s not d i r e c t l y b u i l t i n t o the f l o c k i n g model, i t i s l i k e l y to r e s u l t from the e f f e c t s of s o c i a l i n t e r a c t i o n s and f o r a g i n g success on movement. Increase the time a v a i l a b l e f o r f o r a g i n g by reducing the time l o s t i n a g g r e s s i v e a c t s of s o l i t a r y b i r d s (Earash 1974). As p r e v i o u s s t u d i e s (eg. Hinde 1952) had i n d i c a t e d that a g g r e s s i o n was low i n winter f l o c k s , t h i s hypothesis was not c o n s i d e r e d . F l u s h out i n s e c t s (Swynnerton 1915; .Erosset 1969). T h i s hypothesis i s probably only a p p l i c a b l e to t r o p i c a l b i r d s , as temperate-zone b i r d s g e n e r a l l y f l o c k i n the winter when a d u l t i n s e c t a v a i l a b i l i t y i s low (Yapp 1970). The value of f l o c k i n g i s presumably c l o s e l y l i n k e d with the 12 d e n s i t y and d i s t r i b u t i o n (temporal and s p a t i a l ) of a v a i l a b l e prey (e.g. Brown 1964; Brown S Orians 1970). I t i s g e n e r a l l y proposed t h a t t e r r i t o r i a l i t y may be favoured when food i s evenly d i p e r s e d , and f l o c k i n g when food i s patchy. Under the l a t t e r c o n d i t i o n s the disadvantage of s h a r i n g food when i t i s found i s thought to be outweighed by the i n c r e a s e d a b i l i t y t c l o c a t e r i c h food sources (more eyes s e a r c h i n g ) , the p o s s i b i l i t y of i m i t a t i n g m i c r o h a b i t a t p r e f e r e n c e of s u c c e s s f u l b i r d s , and the r e d u c t i o n of time spent i n marginal f e e d i n g areas f o r a b i r d i n a f l o c k . For i n d i v i d u a l f o r a g i n g animals the g u e s t i c n of how host or prey d i s t r i b u t i o n a f f e c t s the attack r a t e of p a r a s i t e s or p r e d a t o r s has been examined by s e v e r a l i n v e s t i g a t o r s . On t h e o r e t i c a l grounds Rogers (1972) suggested t h a t when search i s random, host d i s t r i b u t i o n does not a f f e c t the capture r a t e . Rashevsky (1959) argues t h a t clumping of prey should reduce the a t t a c k r a t e , while Paloheimo (1967, 1971a,b) demonstrated that clumped prey s u f f e r a higher a t t a c k r a t e than randomly d i s t r i b u t e d prey when a l l prey i n a clump are captured once cne i s found. In a s i m u l a t i o n study of random search, Murdie (1971) found a s l i g h t d e c l i n e i n capture r a t e as prey became more clumped. However, i n Murdie's study the capture of one prey did not ensure the capture of the remaining prey i n the clump. A s e r i o u s d e f i c i e n c y i n these s t u d i e s i s that t h e i r c o n c l u s i o n s apply only to random search. Since the p a t t e r n of search g r e a t l y a f f e c t s the importance of prey d i s t r i b u t i o n , i t 13 i s necessary t o c o n s i d e r the evidence that p r e d a t o r s do not search randomly. Many predators ( p a r a s i t e s ) a l t e r t h e i r search p a t t e r n a f t e r the capture of a prey item.. Numerous s t u d i e s have shown t h a t i n s e c t s (Laing 1937; F l e s c h n e r 1950; Banks 1957; Dixon 1959; Bansch 1966; H a s s e l l 1968) and b i r d s (Croze 1970; Smith 1974) a l t e r t h e i r search p a t t e r n immediately a f t e r the capture of a prey. . By i n c r e a s i n g the r a t e of t u r n i n g a f t e r a prey capture, the predator c o n c e n t r a t e s i t s search i n the area of recent success ( 1 a r e a - r e s t r i c t e d search'; Crcze 1970). T h i s change i n behaviour l a s t s up to a few minutes, a f t e r which the predator resumes i t s , former search mode (often a p p a r e n t l y random). Thus, whenever prey a re aggregated, a r e a - r e s t r i c t e d search w i l l be adaptive, i n c r e a s i n g the mean capture r a t e . In s e v e r a l s t u d i e s i t has been demonstrated that the at t a c k r a t e i s higher on clumped prey than on randcm prey. These i n c l u d e s t u d i e s i n which the pr e d a t o r s were i n s e c t s ( G r i f f i t h & H o l l i n g 1969; H a s s e l l 1971), b i r d s (Tinbergen, Impekoven S Franck 1967; Smith 1974), and f i s h ( I v l e v 1961; Eeukema 1968). A r e l a t e d l i n e o f o b s e r v a t i o n s are those which compare the times which predators spend i n areas of d i f f e r e n t prey abundances. In f a c t many pred a t o r s spend a d i s p r o p o r t i o n a t e l y l a r g e p o r t i o n of t h e i r hunting time i n the areas of higher prey d e n s i t y ( i n s e c t s - H a s s e l l 1968, 1971 ; ..birds - Gcss-Custard 1970; Smith 6 Dawkins 1971). Such behaviour i s no s u r p r i s e given t h a t so many p o t e n t i a l prey p o p u l a t i o n s (both animal and 14 vegetable) have clumped d i s t r i b u t i o n s (eg. Parsons & U l l y e t t 1936; Smith 1939; Burnett 1958; Dybas S Davis 1962; B e r t h e t S Gerard 1965; T a y l o r 1971). Wh.en.ever pr e d a t o r s aggregate, they are l i k e l y to i n t e r f e r e with one another. I n t e r f e r e n c e between s e a r c h i n g predators has been i n v e s t i g a t e d by s e v e r a l r e s e a r c h e r s using i n s e c t p a r a s i t e s as the predators ( G r i f f i t h S H o l l i n g 1969; Rogers & H a s s e l l 1974; Cheke 1974)., I t has been demonstrated that i n t e r f e r e n c e dees occur a t n a t u r a l d e n s i t i e s (provided the predators are given a c h o i c e o f d i f f e r e n t prey d e n s i t i e s to a t t a c k ; Rogers 6 H a s s e l l 1974), t h a t aggregation i n areas of high prey d e n s i t y occurs d e s p i t e i n t e r f e r e n c e , but that p a r a s i t e s leave areas of high d e n s i t y sooner than areas of low p a r a s i t e d e n s i t y . Thus i n t e r f e r e n c e adds s t a b i l i t y to a predator-prey i n t e r a c t i o n . Cheke (1974) a l s o observed a r e l a t i o n s h i p . , between i n t e r f e r e n c e , prey d i s t r i b u t i o n and a t t a c k e f f i c i e n c y . When p a r a s i t e d e n s i t i e s were low, the a t t a c k e f f i c i e n c y was h i g h e r on randomly d i s t r i b u t e d hosts. But when p a r a s i t e d e n s i t i e s were hi g h , a t t a c k e f f i c i e n c y was higher on clumped hosts. As the prey i n one area are depleted, there comes a poin't at which the predator can expect to do b e t t e r t o search i n a new f o r a g i n g area than t o s t a y i n the depleted one. Thus, the g u e s t i o n of how long a predator should continue s e a r c h i n g i n an area before l e a v i n g f o r another a r i s e s . Croze (1970), studying the predation h a b i t s of c a r r i o n crows, proposed t h a t the crows 1 5 gave up and l e f t an area when the elapsed.time s i n c e the l a s t prey capture exceeded some- t h r e s h o l d . The model f i t Croze's data f a i r l y w e l l with a mean gi v i n g - u p time of 6.5 minutes. The n o t i o n t h a t p r e d a t o r s should have a giving-up time i s -one of the p r e d i c t i o n s of o p t i m a l f o r a g i n g theory. (Charnov 1973). Furthermore, t h i s theory p r e d i c t s t h a t the g i v i n g - u p time should vary with the o v e r a l l mean prey d e n s i t y . That i s , when prey are common, giying-up times should be r e l a t i v e l y s h o r t ; while when prey are s c a r c e , g i v i n g - u p times should be c o m p a r i t i v e l y long. Charnov (1973) has i n t e r p r e t e d Gibb's s t u d i e s of t i t p r e d a t i o n on Ernamonia (Gibb, 1958,1960,1966) i n t h i s l i g h t . T ests of t h i s hypothesis have been i n c o n c l u s i v e . Smith S Dawkins (1971) found no d i f f e r e n c e s , i n g i ving-up times between areas of d i f f e r e n t prey d e n s i t y , while.Krebs, Byan & Charnov (1974) did f i n d the p r e d i c t e d changes with changes i n prey d e n s i t y . The giving-up time may a l s o vary with predator d e n s i t y ( H a s s e l l 1971). While i t i s w e l l known t h a t the number of a p a r t i c u l a r prey type eaten by a predator i s not a simple f u n c t i o n of the prey abundance or d e n s i t y , n e i t h e r the above f a c t o r s nor d i f f e r e n c e s between prey i n d e n s i t y , s i z e , conspicuousness, p a l a t i b i l i t y and d i f f i c u l t y of capture can wholly account f o r the o b s e r v a t i o n s t h a t , predators tend.to concentrate a d i s p r o p o r t i o n a t e amount of a t t e n t i o n on the more abundant prey types ( H o l l i n g 1965). There seem to be s e v e r a l forms of l e a r n i n g u n d e r l y i n g t h i s tendency to concentrate a t t a c k s on the more abundant prey type (s) . These 16 i n c l u d e 1. 'search image f o r m a t i o n ' (Tinbergen 1960; Dawkins 1971a,b), and 2. 'niche hunting* (Royama 1970; Smith S Dawkins 1971). Both forms of l e a r n i n g lead to short-term prey p r e f e r e n c e s , i n c o n t r a s t to ' f a m i l i a r i t y * (e.g. A l l e n S C l a r k e 1968) and i t s c o u n t e r p a r t the rejection,: of u n f a m i l i a r prey (e.g. Rabinowitch 1968) which are long-term phenomena. Search image formation r e f e r s to the predator's l e a r n i n g to d e t e c t a prey type which i s c r y p t i c , and t h e r e f o r e . i n i t i a l l y d i f f i c u l t to d e t e c t . , De R u i t e r (1952) found t h a t when s t i c k c a t e r p i l l a r s (Geometridae) were mixed .with the a p p r o p r i a t e s t i c k s , j a ys (Garrulus g a r r u l u s ) and c h a f f i n c h e s (Irincjilla coelebs) had g r e a t d i f f i c u l t y i n f i n d i n g , t h e prey. Once a b i r d had chanced upon one c a t e r p i l l a r , however, i t r a p i d l y found o t h e r s . Thus, . i t appears t h a t the birds.., learned.,-,. how to d i s t i n g u i s h the c a t e r p i l l a r s from the s t i c k s . S e v e r a l other s t u d i e s provided i n d i r e c t evidence that b i r d s form sear c h i n g images. Croze (1970) performed a f i e l d study of crows pre y i n g on b a i t s hidden under pa i n t e d mussell s h e l l s . When l a i d out on a s h i n g l e beach a l l three c o l o u r s of s h e l l proved to be c r y p t i c . Croze found t h a t the b i r d s preyed s e l e c t i v e l y on the c o l o u r of s h e l l on which they .had s t a r t e d (the i n i t i a l f i n d was one made conspicuous by the .experimentor). Burton (1971) l a i d , out s e v e r a l types of c r y p t i c g r a i n on f i e l d s . . o f c l o v e r and s t u b b l e . He found that about .three g u a r t e r s of the, wood-pigeons. fee d i n g t h e r e had s p e c i a l i z e d on a s i n g l e type of f e e d f o r the 15-30 minutes of the experiments. Den Boer .(1971) found t h a t the 17 p r o p o r t i o n s of two c o l o u r morphs of the l a r v a e of Bujgalus £iniarus taken by c a p t i v e t i t s (Parus spp.) v a r i e d with the background. A f t e r b i r d s had been t r a i n e d to take the c r y p t i c mcrph (green l a r v a e on a green background), they took more green than yellow l a r v a e when presented with equal numbers on a green background. The best d i r e c t evidence t h a t b i r d s do form s e a r c h i n g images comes from Dawkins' (1971a,b) l a b o r a t o r y experiments with domestic c h i c k s . Two.prey types, green and orange r i c e , and two backgrounds, green and orange pebbly s u r f a c e s , were used to i n v e s t i g a t e whether c h i c k s form s e a r c h i n g images. When presented with the two types of g r a i n of a s i n g l e , background, conspicuous g r a i n s . were taken immediately, c r y p t i c g r a i n s only a f t e r a l a g of s e v e r a l minutes. Once the c h i c k s began f i n d i n g c r y p t i c g r a i n s , the a t t a c k r a t e on the c r y p t i c g r a i n s i n c r e a s e d r a p i d l y . T h i s s e a r c h i n g image was net maintained without reinforcement, being f o r g o t t e n w i t h i n 24.hours (and wi t h i n j u s t a few minutes under some c o n d i t i o n s ) . Search image formation was i n i t i a l l y proposed by Tinbergen (1960) to e x p l a i n the p r e d a t i o n p a t t e r n s cf t i t s on f o r e s t i n s e c t s ( a c t u a l l y Tinbergen determined what prey parent b i r d s gave to t h e i r young). . He found that at low prey d e n s i t i e s , a c c e p t a b l e prey tended to be ignored, making up a s m a l l e r p o r t i o n of the b i r d ' s d i e t than t h e i r r e l a t i v e p r o p o r t i o n i n the f o r e s t . As a prey type became more common, the b i r d s would a t 18 some po i n t switch to t a k i n g i t i n great abundance. Thus, the p a r t i c u l a r prey type would then be taken i n g r e a t e r p r o p o r t i o n than i t s p r o p o r t i o n amongst a l l prey p o p u l a t i o n s . Tinbergen's e x p l a n a t i o n was t h a t the b i r d s formed a ' s p e c i f i c s e a r c h i n g image* f o r the prey type, but o n l y a f t e r the co n t a c t frequency was s u f f i c i e n t l y high. Other r e s e a r c h e r s (Mcok, Meek S Heikens 1960) have gathered a d d i t i o n a l evidence which supports t h i s view. An a l t e r n a t i v e i n t e r p r e t a t i o n of Tinbergen's and s i m i l a r data i s taken by Royama (1970). While h i s data are b a s i c a l l y s i m i l a r to Tinbergen's, he r e j e c t s , the s e a r c h i n g image hyp o t h e s i s , arguing i n i t s place t h a t the b i r d s l e a r n i n which 'n i c h e s ' (microhabitats) to search f o r prey. The b i r d l e a r n s the p r o f i t a b l e 'niches' and spends most of i t s hunting time t h e r e , but co n t i n u e s to sample other 'niches' to see i f t h e i r p r o f i t a b i l i t y changes.. Thus, the b i r d w i l l , have the opt i o n to change i t s f e e d i n g p a t t e r n s whenever a p r e v i o u s l y u n p r o f i t a b l e niche becomes p r o f i t a b l e . For the purposes of t h i s study i t w i l l not be necessary to d i s t i n g u i s h the two mechanisms - 'searching image f o r m a t i o n ' and 'niche h u n t i n g ' , s i n c e both produce much.the same net e f f e c t , a s h o r t term p r e f e r e n c e f o r a p a r t i c u l a r . prey type (a 'prey p r e f e r e n c e ' ) . E x a c t l y what experiences are necessary f o r the formation o f t h i s preference are not c l e a r . Models c f a t t e n t i o n t h r e s h o l d s such as those of R o l l i n g (1965) and Dawkins(1969a,b) 19 appear q u i t e l i k e l y . The evidence i s that the preferences can be formed with as few as a s i n g l e c o n t a c t (De R u i t e r 1952; l e v e l o f hunger may a f f e c t the r a t e of prey preference formation -H o l l i n g 1965), and they can be e x t i n g u i s h e d w i t h i n a few minutes i f not r e i n f o r c e d (Dawkins 1971a,b). An a d d i t i o n a l form of l e a r n i n g i s p o s s i b l e when b i r d s f o r a g e i n a f l o c k , s o c i a l l e a r n i n g . S o c i a l l e a r n i n g as used here covers two i n t e r r e l a t e d phenomena. A b i r d may copy, another by 1. s e a r c h i n g i n a s i m i l a r p l a c e (eg. same branch) or 2. s e a r c h i n g f o r a s i m i l a r prey item. For our purposes i t w i l l not be necessary to d i s t i n g u i s h these cases , (whereas i n a study i n v o l v i n g mimicry i t might w e l l be necessary to make the d i s t i n c t i o n ) . In l a b o r a t o r y c o n d i t i o n s copying has been demonstrated i n t r a s p e c i f i c a l l y ( f o r .great t i t s - Krebs, MacRoberts 6 C u l l e n 1972) and i n t e r s p e c i f i c a l l y ( f o r , c h i c k a d e e s , Parus spp. - Krebs 1973a). Burton's f i e l d o b s e r v a t i o n s suggest that s i m i l a r phenomena occur,under n a t u r a l c o n d i t i o n s as w e l l . T h i s copying i s manifested f o r no more than a minute or two un l e s s r e i n f o r c e d . T h i s i s d i s t i n c t from the s o r t of i n f o r m a t i o n s h a r i n g proposed t o e x p l a i n the adaptiveness of c o l o n i a l n e s t i n g (Horn 1968; Ward 1965). A d e t a i l e d d e s c r i p t i o n of how the f a c t o r s . d i s c u s s e d here were s y n t h e s i z e d to c r e a t e a model of f o r a g i n g b i r d f l o c k s w i l l be g i v e n i n Chapter <4. I t i s preceded by. a d i s c u s s i o n of the s i m u l a t i o n methodology and some a n a l y t i c techniques f o r a n a l y s i s of s i m u l a t i o n r e s u l t s . 21 CHAPTER THREE:. SIMULATION METHODOLOGY An a b s t r a c t model i s a s e t of concepts and the expressed r e l a t i o n s h i p s among them. In a mathematical model these concepts and r e l a t i o n s h i p s are mathematical ones. To use a mathematical model as a model f o r r e a l , phenomena, one must supply a s e t of r u l e s f o r t r a n s l a t i n g the p r o p o s i t i o n s of the mathematical theory i n t o p r o p o s i t i o n s about the r e a l phenomena (Parzen 1960). Commonly, the f o r m u l a t i o n of a mathematical model begins with the t r a n s l a t i o n of p r o p o s i t i o n s regarding the r e a l phenomena i n question i n t o mathematical p r o p o s i t i o n s . The mathematical model may then be manipulated to y i e l d a d d i t i o n a l p r o p o s i t i o n s , which are i n f a c t i m p l i c i t i n the mathematical theory (Kowal 1971). T r a n s l a t i n g these back i n t c p r e p o s i t i o n s concerning the r e a l phenomena y i e l d s a set of new c o n c l u s i o n s . To the extent that these p r e d i c t i o n s of the model are v e r i f i e d , the model may be judged a p p r o p r i a t e . Simulation i s the dynamic execution or manipulation of a model (Barton 1970). Computer simulation.models are a s p e c i a l case of mathematical models. I f the the b a s i c p r o p o s i t i o n s of the mathematical model (the model's s t r u c t u r e ) are too complex t o be manipulated by standard mathematical techniques, two o p t i o n s are a v a i l a b l e f o r the a n a l y s i s of the model. One i s to make v a r i o u s s i m p l i f y i n g assumptions r e g a r d i n g the model, u n t i l the model i s reduced to a workable l e v e l cf complexity. Deductions made from t h i s s i m p l e r model then 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 model. T h i s approach has the advantage that one can o f t e n make simple, c o n c i s e and o f t e n broad statements about the model. However, the high degree of a b s t r a c t i o n of such, models o f t e n l i m i t s t h e i r a p p l i c a b i l i t y . A l t e r n a t i v e l y , one may work out the i m p l i c a t i o n s c f va r i o u s p a r t i c u l a r cases ( s p e c i f i c parameter values) of the o r i g i n a l model. Computer s i m u l a t i o n can make the manipulation of a complex model f e a s i b l e . While t h i s method f r e q u e n t l y f a i l s to permit the broad statements which a r i s e upon s i m p l i f i c a t i o n of the model, i t does provide c o n c l u s i o n s which are more l i k e l y to be r e p r e s e n t a t i v e of the r e a l phenomena.. In a d d i t i o n , model p r e d i c t i o n s which are not v e r i f i e d may provide a b a s i s f o r a l t e r i n g the model, whereas with the s i m p l i f i e d model the new o b s e r v a t i o n s may have no d i r e c t c o u n t e r p a r t i n the model. S e v e r a l s o r t s of mathematical models are commonly used i n ecology and animal behaviour. Walters (1971) pro v i d e s a good d i s c u s s i o n of the use of mathematical models i n ecology, One common type of e c o l o g i c a l models has been p o p u l a t i o n dynamics models,,, These may be models of a s i n g l e p o p u l a t i o n c r of s e v e r a l i n t e r a c t i n g p o p u l a t i o n s . Many o f . t h e s e models are based upon g e n e r a l i z a t i o n s of the L o t k a - V o l t e r r a equations (Lotka 1925; V o l t e r r a 1931) to account f o r such phenomena as random capture of prey by pr e d a t o r s ( B a r t l e t t 1957) and the age s t r u c t u r e of the p o p u l a t i o n and i t s e f f e c t s upon f e c u n d i t y and m o r t a l i t y (e.g. L e s l i e 1945; Frank 1960; Pennycuick 1969). Such models have been extended to account f o r s t o c h a s t i c 23 elements (Kerner 1957) and have been analyzed e x t e n s i v e l y i n terms of t h e i r s t a b i l i t y (e.g. F r e d r i c k s o n et a l . 1973; May 1973). A second common type of e c o l o g i c a l model i s the "compartment" model. ,• The system i s d i v i d e d i n t o a number of compartments through which energy and . m a t e r i a l s flow. For example, an ecosystem can be s p l i t by t r o p h i c l e v e l . By d e s c r i b i n g the flows of energy between _ t r o p h i c l e v e l s (compartments), the o v e r a l l f low of energy through and w i t h i n the system may be s t u d i e d . Patten (1971).and Van Dyne (1969) c o n t a i n good examples of t h i s approach to model b u i l d i n g . A t h i r d type of e c o l o g i c a l model i s the model cf a process. Examples are models of p r e d a t i o n and of co m p e t i t i o n (e*g. Lotka 1925; V o l t e r r a 1931; N i c h o l s o n S B a i l e y 1935; Eeverton S Holt 1957; Watt 1959; Rcyama 1971). . The most r e f i n e d v e r s i o n of t h i s approach i s H o l l i n g ' s "experimental components a n a l y s i s " ( H o l l i n g 1959a,b 1965,. 1966)., T h i s i n v o l v e s breaking down an e c o l o g i c a l process i n t o simpler subprocesses or "experimental components", examining these components e x p e r i m e n t a l l y , modelling the components i n d i v i d u a l l y , and f i n a l l y i n t e g r a t i n g the component models to produce a model of the o r i g i n a l process. The b i r d f l o c k i n g model i s of t h i s type.with the q u a l i f i c a t i o n that the p u b l i s h e d l i t e r a t u r e on the ecology and behaviour of p a s s e r i n e s has s u b s t i t u t e d f o r the performance of experiments to a s c e r t a i n parameter val u e s f o r the components. By c o n t r a s t , mathematical models i n animal behaviour have seen l i t t l e use.. The major exc e p t i o n has been the use of Markov 24 models to d e s c r i b e sequences of behaviour, (e.g. Hiepkema 1961; Nelson 1964 ; Altmann 1968; C h a t f i e l d . S Lemon 1970 ; F e n t r e s s 1971; S l a t e r 5 O l l a s o n 1972; Straw 1972; S l a t e r 1973; Eaker 1973). A v a r i a n t on t h i s approach w i l l be d i s c u s s e d l a t e r i n t h i s chapter. The d e c i s i o n t o use a s i m u l a t i o n model to a i d the i n v e s t i g a t i o n of some r e a l phenomena i n v o l v e s two d e c i s i o n s . F i r s t one must p r o v i d e reasons f o r c o n s t r u c t i n g any mathematical model, and second one must demonstrate that an a n a l y t i c model i s not f e a s i b l e . There are s e v e r a l reasons one might wish to c o n s t r u c t a mathematical model of some system ( G a r f i n k e l 1965). These i n c l u d e : . 1. To organize masses of i n f o r m a t i o n i n t o a workable theory; 2. To p r o v i d e i n s i g h t i n t o how the given system operates; 3. To help i d e n t i f y experiments which may d i s t i n g u i s h among s e v e r a l t h e o r i e s . In a l l these r e s p e c t s a mathematical model has the p o t e n t i a l advantage over a v e r b a l model of g i v i n g q u a n t i t a t i v e r e s u l t s . Thus, , i t s p r e d i c t i o n s may be more d i r e c t l y t e s t a b l e . However, mathematical models are not always simple t c t e s t as the necessary r e a l world measurements may be d i f f i c u l t to o b t a i n . Kowal (1971) pre s e n t s a p r a c t i c a l s e t of g u i d e l i n e s f o r the c o n s t r u c t i o n of a mathematical model: »1) n a t u r a l n e s s with which the. mathematical theory r e p r e s e n t s the r e a l - w o r l d phenomenon, e.g., continuous versus 25 d i s c r e t e f u n c t i o n , one versus s e v e r a l v a r i a b l e s , 2) a b i l i t y to generate p r e d i c t i o n s , 3) comprehensiveness and e s t h e t i c s , 4) t r a c t a b i l i t y of the mathematics, 5) c o n s i s t e n c y with other e x i s t i n g models." As w i l l be seen l a t e r , the f i r s t , second and l a s t of these c r i t e r i a have r e c e i v e d the major emphasis i n g u i d i n g the c o n s t r u c t i o n of the b i r d f l o c k i n g model. Let us examine the proposed use of a . b i r d f l o c k i n g model i n l i g h t of the above c r i t e r i a . The questions t c be asked using the model are t h r e e f o l d . F i r s t , can a. s y s n t h e s i s of known i n d i v i d u a l b i r d behaviours generate f l o c k i n g behaviours? I f not, then there must be some key behaviours, not yet i d e n t i f i e d , which are i n t e g r a l t o f l o c k behaviour. These might be behaviours of i n d i v i d u a l s whose s i g n i f i c a n c e has been underestimated or behaviours which are only e l i c i t e d when the b i r d s are i n groups. .. On the other hand,.a s u c c e s s f u l s y n t h e s i s of i n d i v i d u a l behaviours to generate f l o c k behaviours would i n d i c a t e t h a t no key elements were ign o r e d . Second, what are the r e l a t i v e impacts of d i f f e r e n t b e h a v i o u r a l components on f o r a g i n g success? I d e n t i f i c a t i o n of key behaviours over which the b i r d s have some c o n t r o l may lead t o e x p e r i m e n t a l l y t e s t a b l e p r e d i c t i o n s , r e g a r d i n g how*these behaviours might change with changing circumstances. T h i r d , what are the c o n d i t i o n s ( i f any) under which b i r d s i n f l o c k s forage more s u c c e s s f u l l y t h a t s o l i t a r y b i r d s ? P r e d i c t i o n s about such c o n d i t i o n s may then be 26 t e s t e d e x p e r i m e n t a l l y . For a l l three q u e s t i o n s a s i m u l a t i o n methodology i s a p p r o p r i a t e . The f i r s t question demands.that we c o n s t r u c t some s o r t of model. Owing t o the complexity of the r e l e v a n t p h y s i c a l and b i o l o g i c a l environment and of the b i r d s ' behaviour, i t seems necessary that the model be a s i m u l a t i o n model. The second qu e s t i o n a l s o can be best approached using a model. While i t might be t h e o r e t i c a l l y p o s s i b l e t o s e l e c t i v e l y manipulate each of the r e l e v a n t behaviours of i n d i v i d u a l b i r d s i n an experimental s e t t i n g , the p r a c t i c a l . d i f f i c u l t i e s are overwhelming. The t h i r d q u e s t i o n i s the one most amenable to f i e l d and l a b o r a t o r y study. In t h i s case.a major d i f f i c u l t y has been i n a s s e s s i n g and manipulating the prey p o p u l a t i o n . I t i s here t h a t the model may be v a l u a b l e i n i d e n t i f y i n g experiments which are l i k e l y t o be c r i t i c a l f o r e v a l u a t i n g the r o l e of f l o c k i n g as a f o r a g i n g s t r a t e g y . In c o n s t r u c t i n g the model, emphasis was placed upon the d i r e c t n e s s of t r a n s l a t i o n between r e a l - w o r l d phenomena and the model elements. T h i s l e d to what may be termed a "mechanistic" model. Each b i r d and each prey item i s represented e x p l i c i t l y i n terms of i t s l o c a t i o n and r e l e v a n t c h a r a c t e r i s t i c s . For the prey these i n c l u d e . conspicuousness and s i z e (in terms of handling t i m e ) . For the b i r d s these i n c l u d e r a t e of movement, a b i l i t y to d e t e c t prey, a b i l i t y to l e a r n , r e p e r t o i r e of responses to other b i r d s and memory of past events. The model 27 i s c o n s i s t e n t with other e x i s t i n g models i n that each component was modelled on the b a s i s of p u b l i s h e d data and theory. Since chance elements e n t e r i n t o determining, many of the b i r d s ' behaviours, the model i s a s t o c h a s t i c one. As such, the mathematics are not t r a c t a b l e . Hence, the model must be manipulated p r i m a r i l y as a computer s i m u l a t i o n mcdel. The comprehensiveness of the model has alr e a d y been d i s c u s s e d i n Chapter 2. As c e r t a i n p o t e n t i a l l y important elements have been excluded from the model (pre d a t i o n o f the b i r d s ) , i t may be t h a t the model i s not s u f f i c i e n t l y comprehensive to e x p l a i n the value of f l o c k i n g . The t e s t of comprehensiveness w i l l be whether the model generates e x p e r i m e n t a l l y t e s t a b l e p r e d i c t i o n s regarding the adaptive value of f l o c k i n g . f T h i s p o i n t w i l l be r a i s e d again i n Chapter 7 when the u s e f u l n e s s of the,model i s examined. Of the various, behaviours i n the model, only one , movement, has r e c e i v e d much a t t e n t i o n from s i m u l a t i o n .modellers. S t o c h a s t i c models of i n d i v i d u a l animal movements g e n e r a l l y c o n s i d e r step s i z e and t u r n i n g r a t e as the key v a r i a b l e s . I f these are independent of both p o s i t i o n . a n d time, then animals w i l l be d i s t r i b u t e d as a negative e x p o n e n t i a l d i s t r i b u t i o n around the common s t a r t i n g p o i n t ( K i t c h i n g 1971). Ey l e t t i n g step s i z e and t u r n i n g r a t e vary with p o s i t i o n , i t i s p o s s i b l e . t o s i m u l a t e more complex movement p a t t e r n s . Holgate (1971) and S i n i f f B Jensen (1969) have s u c c e s s f u l l y s i m ulated the movements of an animal over i t s home range, and Rohlf S Davenport (1969) were able to simulate k l i n o k i n e t i c , o r t h o k i n e t i c and t r o p o t a x i c 28 behaviours. A n a l y t i c A n a l y s i s Of A Si m u l a t i o n Model The d e r i v a t i o n of a n a l y t i c models of complex b i o l o g i c a l systems i s o f t e n u n f e a s i b l e . The necessary data i s u n a v a i l a b l e , and f r e q u e n t l y the necessary measurements.cannot be made e i t h e r f o r economic or p r a c t i c a l reasons. In such cases a v a i l a b l e data i s o f t e n amenable to the development of a s i m u l a t i o n model. In chapter 6 a methodology to f a c i l i t a t e the i n t e r p r e t a t i o n of r e s u l t s from complex b i o l o g i c a l s i m u l a t i o n s i s developed and demonstrated. The method i s most u s e f u l i n those cases where the modelling approach r e q u i r e s such a degree of d e t a i l as to c o n s t r a i n the f e a s i b i l i t y o f long s i m u l a t i o n runs. The method i s then a p p l i e d to reexamine the b e n e f i t s of f l o c k i n g i n the monomorphic and polymorphic prey s i t u a t i o n s . . S i m u l a t i o n s of animal s o c i a l behaviour and dynamics of po p u l a t i o n must o f t e n t r a c e , f o r each animal, a long a r r a y of environmental f a c t o r values o c c u r r i n g at a given moment of the s i m u l a t i o n as w e l l as t r a c e the impacts of lagged v a r i a b l e s . In a d d i t i o n , as s p a t i a l c h a r a c t e r i s t i c s o f t e n assume an important r o l e i n molding the behaviour of the i n d i v i d u a l organism as well as c o n s t r a i n i n g the s o c i a l p a t t e r n s which are f e a s i b l e , models must account f o r the l o c a t i o n of each i n d i v i d u a l a t each moment i n time as we l l as the a t t r i b u t e s of each l o c a t i o n . These requirements make s u f f i c i e n t l y d e t a i l e d s i m u l a t i o n s expensive to 29 run and d i f f i c u l t , t o analyse thoroughly. I t i s p o s s i b l e i n such cases to develop second l e v e l models based upon the s i m u l a t i o n model t o f u r t h e r s i m p l i f y the system and magnify s p e c i a l phenomena of i n t e r e s t . One may c o n s i d e r t h i s s t r a t e g y of c o n s t r u c t i o n a h i e r a r c h y o f models as analogous to c o n s t r u c t i n g a microscope with a s e r i e s cf l e n s e s which f i l t e r the o b j e c t image of the r e a l world, p r o v i d i n g f o r p e r s p e c t i v e s which i l l u m i n a t e c e r t a i n f e a t u r e s of the o b j e c t . Each model pr o v i d e s a d i f f e r e n t " c u t " and m a g n i f i c a t i o n , hence p r o v i d e s a more e f f e c t i v e t o o l i n answering p a r t i c u l a r q u e s t i o n s . The p a r t i c u l a r a n a l y t i c " l e n s " which w i l l be developed (Chapter 6) i s a Markov process model through which input/output r e l a t i o n s h i p s o b t a i n e d from the s i m u l a t i o n are viewed. The Markov p r e s e n t a t i o n of the model makes . i t amenable to some powerful mathematical techniques which permit a n a l y s i s of system long run dynamics and s t a b i l i t y . 30 CHAPTER FOUR:. THE SIMULATION MODEL T h i s chapter presents a d e t a i l e d d e s c r i p t i o n of the s i m u l a t i o n model. The model i n c o r p o r a t e s data and ideas from the l i t e r a t u r e on f l o c k i n g and fe e d i n g behaviour of b i r d s . The model i s s t r u c t u r e d p r i m a r i l y on the. b a s i s o f i n d i v i d u a l behaviours and the i n t e r a c t i o n s between p a i r s cf b i r d s . Only 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 b i r d responds to a group of b i r d s . Thus, the extent to which the model •behaves* l i k e a r e a l f l o c k should t e l l us how w e l l cur understanding of p a r t i c u l a r b e h a v i o u r a l mechanisms can e x p l a i n the phenomenon of f l o c k i n g . Three main b e h a v i o u r a l components are b u i l t i n t o the model: movements, preference f o r prey type r e s u l t i n g from experience, and s o c i a l l e a r n i n g . Two types of movements are c o n s i d e r e d . 1. Short movements of an i n d i v i d u a l b i r d l o o k i n g f o r food. These movements are i n f l u e n c e d not only by the i n d i v i d u a l s search f o r food, but a l s o by i t s a t t r a c t i o n and r e p u l s i o n from neighboring b i r d s i n the f l o c k . 2. Integrated movements of the whole f l o c k which r e s u l t i n the b i r d s moving from one area to another (Hinde 1952)., The. second b e h a v i o u r a l component of the mcdel, short-term preference f o r a p a r t i c u l a r prey type, i s i n c l u d e d to take i n t o account the e f f e c t s of v a r i o u s mechanisms such as search image 31 formation (Tinbergen 1960; Dawkins 1971a,b; Croze 1970), area r e s t r i c t e d search (Croze 1970) and 'niche* hunting (Rcyama 1970). , Although t h e r e i s some argument as to the r e l a t i v e importance of these d i f f e r e n t mechanisms (Krebs 1973b), they a l l produce the e f f e c t of short-term preference f o r one type of prey. The t h i r d type of behaviour i n c l u d e d in- the model i s s o c i a l l e a r n i n g . Murton (1971), and Krebs et a l . (1972) have presented evidence which supports the idea t h a t b i r d s i n a f l e c k tend to copy one another i n terms of f e e d i n g behaviour. . In Murton's study the copying was of food types; i n t h a t of Krebs e t a l . copying was of l o c a t i o n or nature of food h i d i n g p l a c e s . T h i s s o c i a l l e a r n i n g supposedly p r o v i d e s the b a s i s f o r the s u r v i v a l value of f l o c k i n g , s i n c e each i n d i v i d u a l may b e n e f i t from the experience of o t h e r s . The value of f l o c k i n g i s presumably c l o s e l y l i n k e d with the d e n s i t y and d i s t r i b u t i o n (temporal and s p a t i a l ) c f a v a i l a b l e prey. I t i s g e n e r a l l y proposed t h a t t e r r i t o r i a l i t y may be favoured when food i s evenly d i s p e r s e d , and f l o c k i n g when food i s patchy (e.g. Brown S Orians 1970). Under the l a t t e r c o n d i t i o n s the disadvantage of s h a r i n g food when i t i s found i s thought to be outweighed by the i n c r e a s e d . a b i l i t y t o l o c a t e r i c h food sources (more eyes s e a r c h i n g ) , the p o s s i b i l i t y of i m i t a t i n g m i c r o h a b i t a t p r e ference of s u c c e s s f u l b i r d s , and the r e d u c t i o n of time spent i n marginal f e e d i n g areas f o r a b i r d i n a f l o c k . 32 Of the ways f l o c k i n g may enhance f e e d i n g success (Rand 1954; Crook 1965), t h i s study w i l l focus on enhancement of s e a r c h i n g e f f i c i e n c y . The s e t of behaviours a s s o c i a t e d with the search f o r prey may be s a i d to d e f i n e a search s t r a t e g y by t h e i r aggregate a c t i o n . From a v a r i e t y of s t u d i e s 2 the b e h a v i o u r a l elements of a s e a r c h s t a t e g y were s e l e c t e d (Table I ) . These b e h a v i o u r a l components.were broken down , i n t o two c a t e g o r i e s : those a f f e c t i n g movement and t h o s e , a f f e c t i n g prey preference. In a d d i t i o n . Table I i n d i c a t e s the e f f e c t of an i n c r e a s e i n each of these b e h a v i o u r a l components (as manifested i n terms of observable behaviours whenever p o s s i b l e ) . In a s i m i l a r way we may d e f i n e the prey environment on the b a s i s of those prey c h a r a c t e r i s t i c s which i n f l u e n c e the l i k e l i h o o d of a given prey being, found and captured. P r e v i o u s s t u d i e s ( c i t e d above) i n d i c a t e t hat r e l e v a n t prey c h a r a c t e r i s t i c s are s i z e , conspicuousness, p a l a t a b i l i t y , d i f f i c u l t y of c a p t u r e , and s p a t i a l and temporal d i s t r i b u t i o n . By using a s i m u l a t i o n methodology, we are a b l e to handle the complexity of the prey environment and the s t o c h a s t i c nature of the search process. The c h i e f drawback of such an approach 2 On great t i t s - B e t t s (1955), Gibb (1954, 1960), K l u i j v e r (1951), Krebs et a l . , (1972), l a c k (1966), Owen (1954), Rcyama (1970), Smith & Dawkins (1971), L.Tinbergen (1960): North American 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): European thrushes - Smith (1971): wood-pigeons -Murton (1971), Murton, Isaacson S Westwcod (1971). 33 (a) 1a Parameter Hopping r a t e 1b Length of hop 1c Turning r a t e Strength of i n t e r - b i r d a t t r a c t i o n s I n d i v i d u a l d i s t a n c e Movement Besu l t of Increase Increase area searched, search l e s s through Increase area searched, search l e s s thorough Decrease area searched, search more thorough Increase crowding wi t h i n f l o c k s Space b i r d s f a r t h e r a p a r t w i t h i n f l o c k s Giving-up time P r o p e n s i t y to j o i n a f l i g h t I n t e g r a t e d t h r e s h o l d f l i g h t Increase f l i g h t s time be tween Increase f l o c k cohesion Beduce f l o c k cohesion (b) Prey preference Prey c o n t a c t t h r e s h o l d Beduce development of prey prefence (reduce l e a r n i n g r a t e Minimum rei n f o r c e m e n t l e v e l f o r prey p r e f e r e n c e Copying a b i l i t y Maximum d i s t a n c e f o r copying Reduce memory span of prey preference Increase l e a r n i n g r a t e f o r b i r d s i n a f l o c k Increase l e a r n i n g r a t e f o r b i r d s i n a f i c c k Table I_j_ B e h a v i o u r a l components of a search s t r a t e g y and t h e i r f i r s t -order e f f e c t s when i n c r e a s e d . 34 i s the p o s s i b i l i t y of i g n o r i n g v i t a l aspects cf the behaviour and ecology of the modelled animals. In the c u r r e n t study t h i s danger does not appear to be severe, owing to the number and e x t e n s i v e nature of the a v a i l a b l e s t u d i e s . In compensation we are able to study a h i g h l y complex prey environment and t c c o n t r o l p r e c i s e l y experimental c o n d i t i o n s , e n a b l i n g the many r e p e t i t i o n s of experiments necessary to examine a process i n which chance p l a y s a major r o l e . The, model simulates the movements and feeding a c t i v i t i e s of .a. small p a s s e r i n e (e.g., great t i t ) on a 15 second b a s i s , experiments l a s t i n g 20 simulated, minutes. A b i r d ' s behaviour i s determined by i t s search s t r a t e g y , whose e x p r e s s i o n i s modified by the b i r d ' s recent f e e d i n g h i s t o r y and by the a c t i v i t i e s of members of i t s f l o c k . A d e t a i l e d d e s c r i p t i o n of the model f o l l o w s . H a b i t a t The model i s designed to simulate f l o c k s of s m a l l wcodland b i r d s o u t s i d e the breeding season, which e l i m i n a t e s problems a s s o c i a t e d with n e s t i n g . The modelled b i r d s feed i n a two-dimensional world, e i t h e r i n the canopy of t r e e s or on the ground beneath t r e e s . S i n c e the f o l i a g e cf t r e e s o v e r l a p s c o n s i d e r a b l y i n t y p i c a l woodlands, i t was deemed s u f f i c i e n t to i n c o r p o r a t e continuous stands of t r e e s i n which the b i r d s f o r a g e . The prey are d i v i d e d i n t o d i s t i n c t 'prey types' (or morphs) on the b a s i s of conspicuousness, s i z e and p a l a t a b i l i t y . P r i o r to each run of the model, the d i s t r i b u t i o n and abundance 35 of each prey type i s s e l e c t e d , w i t h i n these l i m i t s the computer program maintains a r e c o r d of number of i n d i v i d u a l prey o f each type i n each c e l l of a g r i d c o v e r i n g the t e s t a r e a . A s c a l e commensurate with the s c a l e of b i r d s ' s e a r c h i n g a c t i v i t i e s , hence r e q u i r i n g as f i n e a g r i d as p o s s i b l e , was r e c o n c i l e d with computer storage c o n s t r a i n t s . A g r i d of 3 by 3 f e e t was found to be most s a t i s f a c t o r y . The f o r e g o i n g was implemented i n the computer model by the i n i t i a t i o n s u b r o u t i n e s TINIT and FINIT. Subroutine TINIT e s t a b l i s h e s , t h e l o c a t i o n , of the stands of t r e e s from input data. For s i m p l i c i t y the stands are assumed to be sguare. The prey are i n t r o d u c e d i n t o the model i n two steps. F i r s t input data are read to e s t a b l i s h the c h a r a c t e r i s t i c s of each c f the prey t y p e s . Then subroutine FINIT e s t a b l i s h e s . t h e d i s t r i b u t i o n and abundance of each prey type from a d d i t i o n a l i n p u t d a t a . Program o p t i o n s i n c l u d e p r e c i s e assignment of prey to s p e c i f i c l o c a t i o n s as w e l l as s e v e r a l random d i s t r i b u t i o n s (uniform, normal, e x p o n e n t i a l ) . These can be a p p l i e d throughout the t r e e s , i n a p a r t i c u l a r stand, or w i t h i n a r e s t r i c t e d area i n a stand, thus producing a wide range of prey d i s t r i b u t i o n s ranging from spaced-out to h i g h l y aggregated. Foll o w i n g the d i s t r i b u t i o n of the prey, s u b r o u t i n e s FMAP and FANAL p r o v i d e i n f o r m a t i o n about the prey d i s t r i b u t i o n . FMAP p r i n t s a map f o r each stand of t r e e s showing the l o c a t i o n s of the prey._ FANAL performs a s t a t i s t i c a l a n a l y s i s c f the prey 36 d i s t r i b u t i o n i n each stand of t r e e s f o r each prey type, i n c l u d i n g an index of clumping (*mean crowding*, Lloyd 1967; a l s o r e f e r r e d to as C throughout the remainder of the t e x t ) . Movements Movements of b i r d s are c l a s s i f i e d i n t o two broad c a t e g o r i e s : i n d i v i d u a l r e l o c a t i o n movements and i n t e g r a t e d movements. The f i r s t . c l a s s of movements i s e x p l a i n e d by var i o u s f a c t o r s a f f e c t i n g a s i n g l e b i r d . These i n c l u d e a) the search f o r food, b) a t t r a c t i o n to other b i r d s , and c) r e p u l s i o n from other b i r d s . B i r d s which are a c t i v e l y handling prey do not move. , For the remaining b i r d s movement i s s t i m u l a t e d by environmental and s o c i a l s t i m u l i . As the b i r d s move, these s t i m u l i change i n d i r e c t i o n and i n t e n s i t y . Hence, the end r e s u l t of movement fo r . e a c h 15-second p e r i o d must be c a l c u l a t e d as an approximation to the i n t e g r a t i o n of the response v e c t o r s . The f a c t o r s which enter t h i s c a l c u l a t i o n are as f o l l o w : 1. D i r e c t i o n of p r e v i o u s movement - b i r d s shew preference f o r c o n t i n u i n g i n the same d i r e c t i o n between f e e d i n g episodes (e.g. Cody 1971). 2. L o c a t i o n of s u c c e s s f u l b i r d s - a b i r d i s a t t r a c t e d to another b i r d i f the l a t t e r has j u s t captured a prey (Krebs e t a l . 1972) . 3. L o c a t i o n of a l l other b i r d s - a b i r d i s r e p e l l e d from another b i r d when t h e i r s e p a r a t i o n i s l e s s than a t h r e s h o l d d i s t a n c e ( i n d i v i d u a l d i s t a n c e ) . T h i s d i s t a n c e i s reduced when a 37 b i r d ' s a t t e n t i o n i s c l o s e l y focused upon some a c t i v i t y such as fe e d i n g (Crook 1961). These f a c t o r s combine i n the c a l c u l a t i o n as f o l l o w s . The movement response of one b i r d t o another ( ' s o c i a l movement') i s c a l c u l a t e d by the formula: where MR = movement response, D = d i s t a n c e between the b i r d s , a = s c a l i n g parameter used to f i t the ..curve to data. The movement i s along the l i n e between the two b i r d s . Thus, with MR p o s i t i v e (D<T) the b i r d s move a p a r t , and with MB negative (D>T) the b i r d s move c l o s e r t o g e t h e r . The. maximal response to a t t r a c t i o n takes p l a c e when D-T = 1/a. The t o t a l movement response of a s i n g l e . b i r d to the f l o c k was found by the v e c t o r -a d d i t i o n of responses to each of the other b i r d s . / S e a r c h i n g ('non-social') movement was considered to be s t o c h a s t i c . V a r i a t i o n i n d i r e c t i o n and v e l o c i t y during a 15-second p e r i o d was modelled on a p r o b a b i l i s t i c b a s i s i n c o r p o r a t i n g preference f o r maintaining d i r e c t i o n and s t a t i s t i c a l f e a t u r e s of v a r y i n g f o l i a g e d e n s i t y . On t h i s b a s i s a p r o b a b i l i t y d i s t r i b u t i o n was c a l c u l a t e d and used t o determine a movement v e c t o r i n response t o the environmental s t i m u l i . T h e . p r o b a b i l i t y d i s t r i b u t i o n used i s g i v e n i n F i g u r e 2 and examples of the r e s u l t i n g movement are given i n Fi g u r e s 1 ( a , b ) . 3 3 T h i s f u n c t i o n y i e l d s the Maxwell v e l o c i t y d i s t r i b u t i o n f o r random movement i n two dimensions. MR i f D>T and both b i r d s s e a r c h i n g f o r prey, otherwise T = t h r e s h o l d d i s t a n c e s e p a r a t i n g a t t r a c t i o n from r e p u l s i o n , and A t / / K-/ / . * : *'v-. * \ i . *o" . . . N 1 S F i g u r e l a : A t y p i c a l s a m p l e of th ree 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 as the b i r d s s e a r c h f o r p r e y . The a r e a d e p i c t e d is 60 ft. x 6 0 ft. ^ The t h r e e b i r d s m o v e 55 , 64, and 47 feet ( r e a d i n g lef t .to r igh 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 a r e a p p r o x i m a t e l y 42 , 42 , and 30 s q u a r e feet r e s p e c t i v e l y . ( © — — y . — _ A3; —17^7. z Figure 1 b This figure illustrates the area covered by a flock of 9 birds search-ing for 5 minutes. . The area depicted is 60 ft. x 90 ft. The shaded area indicates the region searched by three birds whose search areas ove rlapped conside rably. In this flock the birds search 2/3 of the area they would have searched separately, thus allowing intensive search of some areas. As success increases flock density increases, result-ing in greater overlap of search areas. . 30 ,20 A rt £: O 0, 10 . oo 0 f(r) = 2Sre O = 1/2 V 0 P (,|r -.ul- < 2a) = .963 16 18 Figure 2: 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 density function used to simulate search-ing movements, given f o r several values of the parameter (3 . This function gives the p r o b a b i l i t y that a b i r d w i l l move r feet i n 15 seconds. Thus the p r o b a b i l i t y that a b i r d w i l l move no more than, x feet i s the i n t e g r a l of f ( r ) from 0 to x. To the r i g h t of the figu r e are given u., • the average distance moved; a, the standard deviation of movement distances; and the p r o b a b i l i t y that a b i r d move more than two standard deviations more than the average. The value 6 = .06 was used i n the simulation exper-iments (also i n Figures l a and b). c 41 These movements (searching and s o c i a l ) were c a r r i e d out i n s u b r o u t i n e TMSTP (Diagram 1, Appendix A). F i r s t the s e a r c h i n g movements were si m u l a t e d by drawing two numbers a and b from a normal d i s t r i b u t i o n ( 0 , s 2 ) . Using these two numbers as the x and y components o f the b i r d ' s movement, the b i r d moves a t a r- ^ v e l o c i t y \| ( a 2 +b 2) /T where T i s the number of i t e r a t i o n s per minute (4 i t e r a t i o n s per minute i n a l l the s i m u l a t i o n experiments). To match t h i s with the p r o b a b i l i t y f u n c t i o n given above, we s e t s =. 1/TX2B1. A f t e r the movement response of each b i r d i s c a l c u l a t e d , the b i r d s are a l l moved to t h e i r new l o c a t i o n s . So f a r we have j u s t c o nsidered the s m a l l 'hopping' movements made by the b i r d s i n t h e i r search f o r prey. Now we must c o n s i d e r f l i g h t , which may be t r i g g e r e d by n c n s o c i a l or s o c i a l s t i m u l i . In the f i r s t case we need c o n s i d e r only a s i n g l e b i r d . When a b i r d ' s s e a r c h i n g and f e e d i n g a c t i v i t i e s b r i n g i t to the edge of a t r e e (or stand of t r e e s ) , i t i s l i k e l y to take f l i g h t t o another t r e e i n the v i c i n i t y . For example, Croze (1970) observed t h a t a crow w i l l continue s e a r c h i n g i n a r e s t r i c t e d area u n t i l the time s i n c e the l a s t prey capture exceeds some l i m i t {'giving-up t i m e ' ) , a f t e r which p o i n t the b i r d f l i e s to another area. In f l o c k s , once one b i r d takes f l i g h t , t h a t f l i g h t may serve as a stimulus to induce f l i g h t i n other b i r d s i n the f l o c k . T y p i c a l l y , the b i r d i n i t i a t i n g f l i g h t makes c e r t a i n c h a r a c t e r i s t i c c a l l s (Hinde 1952)., Other b i r d s 42 f e e d i n g s u c c e s s f u l l y are l e s s l i k e l y to j o i n than the r e c e n t l y u n s u c c e s s f u l ones. I f a s u f f i c i e n t number of b i r d s j o i n the f l i g h t , the l e v e l of c a l l i n g exceeds some t h r e s h o l d and a l l the b i r d s i n the f l o c k j o i n the f l i g h t . T h i s whole course of events may take up to a minute's time. The l o g i c which governs f l i g h t i s i n c o r p o r a t e d i n s u b r o u t i n e FLY. T h i s subroutine performs the f o l l o w i n g s t e p s : 1 . B i r d s whose s e a r c h i n g movements c a r r y them beyond the edge of the t r e e s w i l l f l y i n the c u r r e n t p e r i o d . 2. B i r d s which are h a n d l i n g food or which flew i n the c u r r e n t or p r e v i o u s p e r i o d do not f l y t h i s p e r i o d . 3. Other b i r d s f l y with p r o b a b i l i t y (N-G)/N i f N (periods s i n c e the l a s t prey capture) i s g r e a t e r than G (the giving-up t i m e ) . I f N<G, the b i r d does.not f l y . . 4. Once a f l i g h t i s i n i t i a t e d , b i r d s may j o i n i t i n t h i s p e r i o d or any of the t h r e e f o l l o w i n g p e r i o d s ( t o t a l of one minute). The p r o b a b i l i t y of f o l l o w i n g depends upon var i o u s t h r e s h o l d s , f e e d i n g success, and d i s t a n c e from the l e a d e r . 5. I f a s u f f i c i e n t number of b i r d s f l y , a l l b i r d s i n the same f l o c k f o l l o w , a t t h e i r f i r s t o p p o r t u n i t y ( i n t e g r a t e d f l i g h t ) . T h i s l o g i c a l sequence i s given i n d e t a i l i n the flow c h a r t of Diagram 2, appendix A. The v a r i o u s t h r e s h o l d s are s e t as parameters, and t h e i r values w i l l be d i s c u s s e d with the s i m u l a t i o n experiments (see Table I I ) . -To complete the d e s c r i p t i o n of f l i g h t , 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 of f l o c k as used i n the model. A f l o c k i s d e f i n e d as f o l l o w s : two b i r d s are i n the same f l o c k i f they are i n the same stand of t r e e s , and they are not more than PFLOK f e e t apart. The value of PFLOK (40 f e e t ) was estimated from o b s e r v a t i o n . F i g u r e s 3(a,b,c) present p i c t o r i a l l y the determination c f f l o c k s f o r s e v e r a l values of PFLOK. Subroutine FLOGG (see Diagram 3, Appendix A) performs the s o r t i n g of b i r d s i n t o f l o c k s . Feeding In modelling the feeding behaviour c f the b i r d s , i t i s necessary to take i n t o account o b s e r v a t i o n s from v a r i o u s s t u d i e s which.show t h a t b i r d s tend to concentrate on common prey types and tend t o take prey i n 'runs* of one type (e.g. L. Tinbergen, 1960; Royama, 1970). Tinbergen (1960) e x p l a i n s these phenomena on the b a s i s of • s p e c i f i c search images', formed by c o n d i t i o n i n g and maintained by success i n p r e d a t i o n . That i s , given some s u f f i c i e n t frequency of s u c c e s s f u l f e e d i n g on one prey type, a b i r d w i l l d i r e c t i t s f u t u r e searches on the b a s i s of v a r i o u s c h a r a c t e r i s t i c s of t h a t prey type. Once formed, the • s p e c i f i c search image* w i l l be r e t a i n e d so long as the b i r d experiences s u f f i c i e n t success i n i t s f e e d i n g . Rcyama (1970) presents a somewhat d i f f e r e n t t h e s i s , suggesting t h a t the b i r d s ' search i s d i r e c t e d towards l o c a t i n g 'prey-niches', s i t e s where prey are l i k e l y to be found. Thus, he emphasizes search by s i t e a t t r i b u t e s r a t h e r than search f o r prey, by t h e i r a t t r i b u t e s . 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, these . two hypotheses need net be mutually e x c l u s i v e (e.g. Turner, 1965). In the model f o r m u l a t i o n prey and h a b i t a t c h a r a c t e r i s t i c s form j o i n t l y the b a s i s f o r prey p r e f e r e n c e s . Search and capture are c a r r i e d out i n subroutine EAT (Diagram 4, Appendix A). Each b i r d ' s l o c a t i o n a s s o c i a t e s the b i r d with a p a r t i c u l a r g r i d c e l l whose i n i t i a l prey c o n t e n t s were e s t a b l i s h e d at., the beginning of the s i m u l a t i o n run. The g r i d s i z e of 3 by 3 f e e t was chosen as appproximating the area which one b i r d c ould search i n 15-seconds given a ' d i r e c t -d e t e c t i o n d i s t a n c e * (Croze 1970) of one f o o t . T h i s s e l e c t i o n r e p r e s e n t s an averaging of v a r i a t i o n i n d i r e c t - d e t e c t i o n d i s t a n c e with background as w e l l as v a r i a t i o n i n speed of movement and search w i t h i n each 15-second p e r i o d depending upon d e n s i t y of f o l i a g e , hunger, and other such e x t e r n a l and i n t e r n a l f a c t o r s . A b i r d has an o p p o r t u n i t y of d e t e c t i n g and c a p t u r i n g any one of the prey i n the g r i d c e l l . The b i r d *enccunters' the prey i n random o r d e r , d e t e c t i n g and c a p t u r i n g a prey with p r o b a b i l i t y P i where i r e p r e s e n t s the prey type. These p r o b a b i l i t i e s r e f l e c t such prey c h a r a c t e r i s t i c s , as s i z e , conspicuousness, p a l a t a b i l i t y and d i f f i c u l t y cf c a p t u r e . I f a prey i s captured,* .the b i r d ceases to s e a r c h , handling and consuming the prey f o r a given number of p e r i o d s . The captured prey i s removed from the system. Handling time may vary f o r the * 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 predator may not r e j e c t a prey item, and the prey cannot escape. 46 d i f f e r e n t prey t y p e s , depending p r i m a r i l y on s i z e . l u r i n g th^e handli n g p e r i o d s , the b i r d does not engage i n any ether a c t i v i t i e s . In a d d i t i o n , when a b i r d renews s e a r c h i n g , i t has a sh o r t term (1 period) p r e f e r e n c e f o r prey of the type j u s t c a ptured. I f a b i r d makes two s u c c e s s i v e captures cf the same prey type, a preference f o r that prey type i s formed (as suggested by H o l l i n g 1965), A b i r d with a prey preference goes about d e t e c t i n g and c a p t u r i n g prey i n much the same way as before, but the p r o b a b i l i t i e s f o r success are changed. Ey d i r e c t i n g the search a c c o r d i n g t o c h a r a c t e r i s t i c s of prey and h a b i t a t , the b i r d with prey preference f o r prey type i i n c r e a s e s i t s chances of d e t e c t i n g and c a p t u r i n g such prey from P i to Q i i . A d d i t i o n a l l y , the p r o b a b i l i t i e s f o r success on any of the remaining prey types ( Q i j , i * j ) are s u b s t a n t i a l l y reduced. In the model we assumed t h a t a prey preference e f f e c t i v e l y excludes the capture of a l t e r n a t i v e prey. Thus, we have: a) 0<Pi<Qii<1 (i=1,...,n) b) .Qij=0_ i # j ( i , j=1,... ,n) where n = number of prey types; P i = p r o b a b i l i t y of capture of a prey of type i by a b i r d without prey p r e f e r e n c e ; Q i j = p r o b a b i l i t y of capture of a prey c f type i by a b i r d with prey preference f o r type j . A b i r d with prey p r e f e r e n c e r e t a i n s the preference so long as prey are captured with s u f f i c i e n t frequency, i . e . the preference i s r e i n f o r c e d . I f a b i r d goes more than M p e r i o d s without success, then the prey preference i s 47 f o r g o t t e n . The c h o i c e of the parameter M can•have a major impact on the f e e d i n g behaviour e x h i b i t e d . For example, i f M i s very l a r g e , we have a b i r d which s p e c i a l i z e s on a s i n g l e prey type. As M decreases, the time to r e p l a c e an u n s u c c e s s f u l f e e d i n g stategy with a more s u c c e s s f u l one decreases, and the b i r d becomes more o p p o r t u n i s t i c . In the s e n s i t i v i t y a n a l y s i s the i n f l u e n c e of changing t h i s parameter i s examined. The f e e d i n g behaviour of the b i r d s i n the model i s a l s o i n f l u e n c e d by s o c i a l l e a r n i n g . F i n d i n g s such as those of Murton (1971) and Krebs et a l . (1972) i n d i c a t e that b i r d s i n a f l o c k i m i t a t e each ot h e r * s f e e d i n g behaviour, copying prey types or prey s i t e s . . I m i t a t i o n , i s i n c o r p o r a t e d i n t o the model by a l l o w i n g an u n s u c c e s s f u l b i r d with no prey p r e f e r e n c e to copy i t s nearest s u c c e s s f u l neighbor, provided that the neighbor i s s u f f i c i e n t l y c l o s e . The copying b i r d i s s a i d to acquire a 'temporary prey p r e f e r e n c e . ' I f the 'temporary prey p r e f e r e n c e ' i s employed s u c c e s s f u l l y i n the next p e r i o d , the b i r d a c q u i r e s the corresponding prey p r e f e r e n c e . T h i s l c g i c i s executed by s u b r o u t i n e IMTATE (Diagram 5, Appendix A) which s e l e c t s from a l l the b i r d s w i t h i n d i s t a n c e DIM of the u n s u c c e s s f u l b i r d the nearest neighbor which captured a p r e y . i n the c u r r e n t p e r i o d . The value of DIM r e p r e s e n t s the maximum d i s t a n c e at which a b i r d can adequately observe another b i r d ' s f e e d i n g behaviour to copy i t . : - • The remaining p o r t i o n s of the computer program handle the 48 inp u t of parameters (MAIN, Diagram 6 , Appendix A ) , i n i t i a l p o s i t i o n s o f the b i r d s (BINIT) and output (EMAF, FSTAT, OUTR, OUTS) . Subroutine BMAP p r i n t s a map of the b i r d ' s l o c a t i o n s . FSTAT prov i d e s capture s t a t i s t i c s and search s t a t i s t i c s . OUTR and OUTS provide summary s t a t i s t i c s f o r each b i r d i n c l u d i n g d i s t a n c e moved, number and length of f l i g h t s , f e e d i n g success on each prey type, prey preference and success of b i r d s f o r each of the f l o c k s which were formed dur i n g the s i m u l a t i o n . 49 CHAPTER FIVEj_ THE SIMULATION EXPERIMENTS Three groups of s i m u l a t i o n experiments w i l l be d i s c u s s e d here. The f i r s t examines the s e n s i t i v i t y o f the model to changes i n the b e h a v i o u r a l parameters. The aim here i s to d i s c o v e r which of the behaviours are c r i t i c a l i n determining f o r a g i n g success. The second examines the value cf f l o c k i n g to b i r d s feeding on a s i n g l e prey type (monomorphic prey p o p u l a t i o n ) . These w i l l examine the e f f e c t s cf varying the prey d i s t r i b u t i o n , of f i x i n g f l o c k s i z e and of i n t r o d u c i n g i n d i v i d u a l v a r i a t i o n ( d i f f e r e n c e s i n behaviour f o r d i f f e r e n t b i r d s ) . The t h i r d group examines, the value of f l o c k i n g when the prey p o p u l a t i o n has s e v e r a l prey types (polymorphic p r e y ) . These experiments examine the same q u e s t i o n s as the second group, but with two or more prey types a v a i l a b l e . S e n s i t i v i t y . A n a l y s i s In order to i d e n t i f y key parameters i n the model a s e n s i t i v i t y a n a l y s i s was conducted. The procedure i n v o l v e s making a s m a l l v a r i a t i o n ( u s u a l l y about 10?) i n one input parameter at a time and observing how much v a r i a t i o n i s c r e a t e d i n the output.. The s e t of r e s u l t s obtained with the parameter values used i n l a t e r experiments w i l l be r e f e r r e d to as the standard. Thus, f o r each parameter we have two s e t s of r e s u l t s c o rresponding to two l e v e l s of the parameter. We then t e s t the two hypotheses: (A) the r e s u l t s do not d i f f e r ; and (B) the 50 percentage change i n r e s u l t s i s not g r e a t e r than the percentage change i n the parameter. The aim was t w o - f o l d : 1) to i d e n t i f y parameters whose v a r i a t i o n has a major impact on the simulated r e s u l t s (A and B r e j e c t e d ) - determination of these parameters may r e g u i r e improved measurements i n l a b o r a t o r y cr f i e l d - and 2). t o i d e n t i f y those parameters whose v a r i a t i o n has l i t t l e impact on simulated r e s u l t s (neither A nor B r e j e c t e d ) - a s i m p l i f i c a t i o n of the model may then be p o s s i b l e . The a n a l y s i s was run by v a r y i n g the f o l l o w i n g : hopping r a t e , s t r e n g t h of i n t e r - b i r d a t t r a c t i o n , i n d i v i d u a l d i s t a n c e , p r o p e n s i t y to j o i n . a f l i g h t , i n t e g r a t e d f l i g h t t h r e s h o l d , memory span f o r prey p r e f e r e n c e , maximum d i s t a n c e f o r copying, maximum i n t e r - b i r d d i s t a n c e w i t h i n a. f l o c k , and prey d e t e c t i o n l i k e l i h o o d . Two measures of s e n s i t i v i t y were used: the v a r i a t i o n i n mean capture r a t e and the v a r i a t i o n i n p r o b a b i l i t y of .zero captures i n 20 minutes. Tables I I I and IV present the r e s u l t s cf t h i s . a n a l y s i s f o r a moderately clumped prey environment.and Table I I gives the parameter val u e s used to e s t a b l i s h the standard values.. Since these r e s u l t s were obtained by v a r y i n g a s i n g l e parameter at a time, they do not r e v e a l second and higher order i n t e r a c t i o n s . However, f o r s m a l l v a r i a t i o n s i n parameters these i n t e r a c t i o n s are g e n e r a l l y s m a l l . Mean capture r a t e (=success) was h i g h l y s e n s i t i v e (A and B r e j e c t e d ) to prey d e t e c t i o n l i k e l i h o o d without prey preference and to giving-up time. The p r o b a b i l i t y of zero captures i n 20 minutes (=risk) was s e n s i t i v e to giving-up time. In a d d i t i o n . 51 £§havioural component Rate of hopping and t u r n i n g Strength of i n t e r - b i r d a t t r a c t i o n I n d i v i d u a l d i s t a n c e Giving-up time P r o p e n s i t y to j o i n f l i g h t I n t e g r a t e d , f l i g h t t h r e s h o l d Prey c o n t a c t t h r e s h o l d to form preference Reinforcement r a t e to maintain p r e f e r e n c e Copying e f f e c t on prey preference Flockincj SCI=3.0 PEXP=15.0 f t . E5U^ 25£JSi£2 Comments Same P1TRB=1.0 f t . P2TRB=3.0 f t . PFL¥=2.0 min. PFOL=1.0 min. INTF=4 b i r d s PCT=2 items MHUN=2.0 min. Reduces PCT to 1 a f t e r copying -1. 0 Same Same Same 99.0 99 Same Same None If PEXP<0, nc a t t r a c t i o n s take place Nc f l i g h t s j o i n e d by no n f l o c k i n g b i r d s No i n t e g r a t e d f l i g h t f o r n o n f l o c k i n g b i r d s No copying f cr n o n f l o c k i n g b i r d s Maximum copying d i s t a n c e DIM=15.0 f t . -1.0 Table I I I Beh a v i o u r a l parameters used i n the simu l a t i o n . e x p e r i m e n t s . 52 V fl S 2 Z A1sig 510 c h i 2 A2sig Stand. — 1.58 5. 13 ~ — 58 — Prey d e t e c t i o n with p r e f . +6.1% 1.46 6.54 -.35 NS 64 0.75 NS Prey d e t e c t i o n without p r e f . +20* 2. 74 9. 95 2.97 ..003 45 3. 37 ..07 Giving-up time +10* 3.72 17.64 4.46 <..001 37 8.80 .003 Pr o p e n s i t y to j o i n f l i g h t +10* 1.74 6.71 0.46 NS 54 0.32 MS Pr o p e n s i t y to ignore f l i g h t i f f e e d i n g -11* 1.58 5.13 0.00 NS 58 0.00 NS Integ r a t e d f l i g h t t h r e s h o l d +25* 2. 39 10.55 2.04 ..021. 50 1.28 NS Memory span +12* 1.41 6.54 -.50 NS 67 1.72 NS Maximum copying d i s t . +10* 1.52 5.21 0.19 NS 61 0.19 NS Str e n g t h i n t e r - b i r d a t t r a c t i o n +10* 1.94 6.46 1.05 NS 51 0.98 NS I n d i v i d u a l d i s t . , (searching) +10* 1.40 5.?2 0.54 NS 55 0.18 NS I n d i v i d u a l d i s t . (feeding) +10* 1.64 7.38 0.17 NS 64 0.75 NS Spread w i t h i n f l o c k +10* 2.06 8.27 1.31 ..097 57 0.02 NS Table III: e n s i t i v i y a n a l y s i s to t e s t the hypotheses t h a t v a r i a t i o n i n the parameter produced no v a r i a t i o n i n mean captures (A1) and i n percentage zero captures (A2) . R e s u l t s are f o r twenty simulated minutes. N o t a t i o n : M = mean captures, S 2 = estimated variance i n ca p t u r e s , Z=[ N/(S 2 (p)+S 2 (s)) ]* 2 * [ M (p)-M (s) ] where (s) and (p) denote the standard and v a r i e d parameter r e s u l t s r e s p e c t i v e l y , 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 such a high (low) va l u e of Z under hypothesis A1 (standard normal v a r i a t e , one-sided t e s t ) , *0 •= percentage cf b i r d s making no c a p t u r e s , chi 2=[2N-1 ]*[%0 (p)-*0 (s) ] 2 / [ (*0 (p) +*0 (s)) * (200-*0 (p)-*0 (s) ], A2sig = p r o b a b i l i t y of such a high value of c h i 2 (1 degree of freedom) under hypothesis A2. 53 Stand. Prey d e t e c t i o n without p r e f . + 20% A l t . Stand. +20* Giving-up time +10% A l t . , Stand. +1051 Spread w i t h i n f l e c k + 10% A l t . Stand. , +10% In t e g r a t e d f l i g h t t h r e s h o l d , ; . . +25% A l t . Stand. +25% H 1.58 2.74 1. 90 3.72 1.74 2.06 1.74 2. 39 1.98 S2 5.13 9.95 7.39 17.64 6.21 8.27 6.21 10.55 8.02 2.02 4.04 B1sig i022 .0001 0.84 NS %0 58 45 48 37 53 c h i 2 E 2 s i g 0.22 4.96 NS .026 0.96 NS Table 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 t h a t v a r i a t i o n i n the parameter produced no more than the corresponding percentage change i n mean captures (B1) and i n percentage zerc c a p t u r e s (B2). R e s u l t s are f o r twenty simulated minutes; the ' a l t . Stand.' r e s u l t s are the expected r e s u l t s under hypotheses B1 and B 2. No t a t i o n : as i n Table I I I except t h a t the ' a l t e r e d standard' (as) r e p l a c e s the standard f o r each t e s t where M (as)= (1+a)M(s), S z (as)= (1+a) 2S2 ( S ) , %0 (as)= (1+a)-*%0 (s), a=V/100, and v = percentage change i n the parameter. 54 s i g n i f i c a n t changes i n success no g r e a t e r than the changes i n the parameters (A r e j e c t e d , B not r e j e c t e d ) were found f o r i n t e g r a t e d f l i g h t t h r e s h o l d and spread w i t h i n f l o c k , while such changes i n r i s k were found only f o r i n t e g r a t e d f l i g h t t h r e s h o l d . In a l l s i x of these cases, an i n c r e a s e i n the parameter i n c r e a s e d success and decreased r i s k . Before proceeding i t i s necessary to c o n s i d e r how these r e s u l t s a f f e c t the use of the model. The high s e n s i t i v i t y to prey d e t e c t i o n p r o b a b i l i t i e s , i m p l i e s that c o n c l u s i o n s regarding a b s o l u t e f e e d i n g r a t e w i l l only apply where prey are very n e a r l y as d i f f i c u l t to detect as those considered i n the model. A range of, prey d e t e c t i o n l i k e l i h o o d s must be examined before g e n e r a l i z i n g statements r e g a r d i n g absolute f e e d i n g r a t e s of f l o c k i n g and n o n f l o c k i n g b i r d s . In c o n t r a s t to prey d e t e c t i o n p r o b a b i l i t i e s , which i n the model are p r i m a r i l y c h a r a c t e r i s t i c of the prey, giving-up time i s s p e c i f i c t o the b i r d s and may well be a behaviour which they can modify. The high s e n s i t i v i t y of both r i s k and success to g i v i n g - u p time suggests t h a t b i r d s should t r y to optimize i t . Using an e n t i r e l y d i f f e r e n t approach, Charnov (1973) reached the same.conclusion, with the f u r t h e r note t h a t a b i r d should be able to a l t e r i t s g i v i n g - u p time i n a s h o r t p e r i o d of time (e.g.; under one hour). Krebs, Ryan .& Charnov (1974) found as p r e d i c t e d that black-capped chickadees (Parus a t r i c a j j i l l u s L.). t r a i n e d at one food d e n s i t y w i l l reduce t h e i r giving-up times when t h e i r food d e n s i t y i s g r e a t l y i n c r e a s e d . In the s i m u l a t i o n experiments which f o l l o w a 55 s m a l l giving-up time was used (2 minutes) d e s p i t e i t s suboptimal value (as demonstrated by the s e n s i t i v i t y a n a l y s i s ) because as gi v i n g - u p time i n c r e a s e s , i t s e f f e c t s on n o n f l c c k i n g b i r d s are l e s s pronounced than on f l o c k i n g b i r d s . Thus, any advantage found f o r f l o c k i n g b i r d s should become even g r e a t e r with a longer giving-up time (at l e a s t f o r giving-up times no greater than the o p t i m a l v a l u e ) . While no such c l a i m can be.made f o r cases where the giving-up time i s l e s s than the one we use, the s e l e c t i o n appears reasonable i n l i g h t , of measured g i v i n g - u p times: 6.5 minutes f o r c a r r i o n crows (Corrus corone L.) (Croze 1970); 11.5 minutes f o r gre a t blue herons (Ardea herodias L .) (Krebs 1974). A f i n a l note on v a l i d a t i o n concerns f l e c k movement. As i n d i c a t e d i n the model d e s c r i p t i o n , f l o c k behaviour i s generated (with the e x c e p t i o n of i n t e g r a t e d f l i g h t ) by the c o l l e c t i o n of i n d i v i d u a l b i r d s responding to environmental and i n t e r n a l cues and to the other i n d i v i d u a l b i r d s . That i s , no behaviour besides the i n t e g r a t e d f l i g h t i s generated by a b i r d ' s r e a c t i o n to (awareness of) the f l o c k . Thus, the observed c o n s i s t e n c y of the f l o c k movements generated i n the s i m u l a t i o n experiments with f l o c k movements observed i n the f i e l d p r o v i d e s a p o s i t i v e and independent s u b s t a n t i a t i o n of the model's v a l i d i t y . 56 Monomorphic Prey P o p u l a t i o n s In the experiments which f o l l o w , we w i l l concern o u r s e l v e s with the advantages i n terms of feed i n g success which accrue to i n d i v i d u a l b i r d s through f l o c k i n g when the prey are of a s i n g l e type. Four s e t s o f s i m u l a t i o n experiments were performed. The f i r s t compares f l o c k i n g and n o n f l o c k i n g b i r d s on s e v e r a l measures of success over a wide range of prey d i s t r i b u t i o n s . In the second the success of v a r i o u s f l o c k s i z e s are examined. The t h i r d and f o u r t h s e t s examine the impact cf i n t r o d u c i n g d i f f e r e n c e s i n behaviour between d i f f e r e n t b i r d s . The b i r d s are considered to vary i n t h e i r a b i l i t i e s to d e t e c t prey, and the f i r s t two s e t s of experiments are repeated under these c o n d i t i o n s . The e f f e c t of fl o c k i n g ; on feeding, with a monomorrihic p_rey. In the f i r s t s et of s i m u l a t i o n experiments we examine how f o r a g i n g success changes when we vary the behaviours i n v o l v i n g s o c i a l a t t r a c t i o n . . In one group of experiments the b i r d s show normal f l o c k i n g ( s o c i a l a t t r a c t i o n , c o p y i n g ) , while i n the ether group there i s no f l o c k i n g behaviour ( t h i s c o n s t i t u t e s the c o n t r o l group). For the parameter values used i n these and subsequent experiments r e f e r back to Table I I . The b i r d s forage i n an area of 120 by 120 yards. I n i t i a l s i t e s e l e c t i o n i s random f o r each b i r d and independent of the other b i r d s . Thus, 57 the b i r d s have no i n i t i a l p r e f e r ence w i t h i n the a r e a . Within the f o r a g i n g area are s c a t t e r e d 900 ' p r e y 1 , a l l of a s i n g l e prey type (prey c h a r a c t e r i s t i c s are s e t cut i n Table V). The v a r i a b l e i n the prey environment i s the p a t t e r n of s p a t i a l d i s t r i b u t i o n . In each case the 900 prey are d i v i d e d i n t o t h i r t y - s i x equal clumps (twenty-five prey each) whose centres are randomly l o c a t e d i n the f o r a g i n g area. These l o c a t i o n s are kept f i x e d throughout both s e r i e s of experiments. The area covered by a clump, and consequently the average d e n s i t y within a clump, i s v a r i e d between experiments. D e n s i t i e s t e s t e d ranged from 0.07 to 9.48 prey per square yard within clumps (wi t h i n -clump d e n s i t y was estimated with Lloyd's index of mean crowding, Lloyd 1967), corresponding to areas from 400 to 2.5 square.yards per clump. I t should be noted t h a t the number of prey.items per clump was not v a r i e d i n t h i s s e t of experiments. To determine the importance of f l o c k i n g to b i r d s during mid-winter, i t i s necessary to e s t a b l i s h c r i t e r i a f c r e v a l u a t i o n of f e e d i n g s u c c e s s . During t h i s p e r i o d , f e e d i n g a c t i v i t i e s are p r i m a r i l y d i r e c t e d toward, maintenance r a t h e r than growth or r e p r o d u c t i o n (Owen 1954). As the mean capture r a t e may not be a s u f f i c i e n t measure of f e e d i n g success - a s i n g l e day's i n s u f f i c i e n t f e e d i n g may s i g n i f i c a n t l y i n c r e a s e a b i r d ' s chance of m o r t a l i t y s i n c e weight l o s s e s of 10% i n one night are not uncommon ( K l u i j v e r 1952) - we c o n s i d e r the r i s k of i n s u f f i c i e n t f e e d i n g as a second measure of f e e d i n g success. Twc measures of r i s k were i n i t i a l l y used, one the r a t i o of v a r i a n c e to mean i n ££§2 c h a r a c t e r i s t i c s Value i n model Comments S i z e Handling time P r o b a b i l i t y of d e t e c t i o n and capture by b i r d : a) without prey p r e f . b) with prey p r e f . 0.5 grams approx.. 20 gram/day f o r maintenance 30 seconds 0.25 0.75 Accounts f o r conspicucus-ness, 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 Table V_: Prey c h a r a c t e r i s t i c s . 59 f e e d i n g r a t e , the other the p r o b a b i l i t y o f o b t a i n i n g no food i n a f i x e d p e r i o d of time. R e s u l t s R e s u l t s of s i m u l a t i o n experiments ( f i v e runs each) r e p r e s e n t i n g twenty minutes of a c t i v i t y are s e t out i n Figures 4-6 (see Appendix B f o r these r e s u l t s i n t a b u l a r form). In F i g u r e 4 we see t h a t the mean.capture r a t e i n c r e a s e s with the degree of clumping of the prey (as might be expected, e.g. G r i f f i t h s & H o l l i n g 1969), appearing to l e v e l o f f or even drop at extreme clumping (although the high c o r r e l a t i o n c o e f f i c i e n t s 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 l i n e a r f o r the range of C v a l u e s t e s t e d ) . Both f l o c k i n g and n o n f l o c k i n g b i r d s show the same p a t t e r n of i n c r e a s e (the two r e g r e s s i o n l i n e s are not s i g n i f i c a n t l y d i f f e r e n t , p>.10). F i g u r e s 5 and 6 show t h a t both measures of ' r i s k * i n c r e a s e l i n e a r l y i n C* , the log a r i t h m of food clumping. In both cases the r e g r e s s i o n l i n e s f c r f l o c k i n g and n o n f l o c k i n g b i r d s d i f f e r s i g n i f i c a n t l y (p<.001) with the s l o p e f o r n o n f l o c k i n g b i r d s 1.?5 times t h a t f o r f l o c k i n g b i r d s . s Thus,in the model, f l o c k i n g b i r d s do not have a higher mean capture r a t e than n o n f l o c k i n g b i r d s , but they do s u f f e r a lower r i s k when f e e d i n g cn clumped food. And as 5 S i n c e the two measures of r i s k g i v e such s i m i l a r r e s u l t s , only one i s r e p o r t e d - percentage of b i r d s making zero captures i n 20 minutes - i n other experiments. < 5.5. 5-aL o-o MEAN CAPTURES IN 20 MIN Flocking: r = .75 Nonflocking: r = .84 0-5 1-0 5*5 6-0 C ~ L N ( l G * C ) Figure 4: The graph plots mean capture rate against C' = In (16C) where C is Lloyd's index of mean crowding, an indicator of within clump density. Points marked "x" are results with flocking, those marked " t " are without flocking. The regression lines plotted through these points are not significantly different. The.goodness of f i t i s indicated by the correlation coefficients r = .75 (flocking) and r = .84 (nonflocking). .cr> o F i g u r e 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 p e r i o d a g a i n s t C (as 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 r e p r e s e n t n o n f l o c k i n g 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 - s l PROBABILITY- OF ZERO CAPTURES IN 20 MIN o-al 0-71 •0-6. 0-51 o-aL 0< Nonfloc k. i n s z r = .96 y = .26 + .037.x 0-1-O - Q l 0-0 0-5 1-0 1-5 2-0 2-5 3 -0 3-S- 4-0 4-5 5-0 C = LNC16*€) ' F i g u r e 6: 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 making no c a p t u r e s i n twent y m i n u t e s a g a i n s t C' (as 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 r e p r e s e n t n o n f l o c k i n g 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). 5-5 S-O cn 63 clumping of food i n c r e a s e s , t h i s d i f f e r e n c e i n r i s k a l s c i n c r e a s e s . The i m p l i c a t i o n s of these r e s u l t s w i l l be d i s c u s s e d along with the set examining the success of d i f f e r e n t f l o c k s i z e s . In the extreme case of clumping f o r these experiments, when each clump oc c u p i e s a s i n g l e g r i d square, the r e g r e s s i o n formula p r e d i c t s the p r o b a b i l i t y of making zero captures i n 20 minutes to be 0.57 f o r f l o c k i n g b i r d s and 0.78 f o r n c n f l o c k i n g b i r d s . \ In t h i s case, however, we can model, the f e e d i n g of the n o n f l o c k i n g b i r d s as a B e r n o u l l i sampling scheme i n which a b i r d samples (searches) u n t i l e i t h e r e i g h t y draws are completed without success (20 simulated minutes of searching) or the b i r d succeeds (makes a c a p t u r e ) . The p r o b a b i l i t y of f a i l u r e i n a given p e r i o d i s the sum of the p r o b a b i l i t y of p i c k i n g an empty g r i d square (14364/14400) p l u s the p r o b a b i l i t y cf p i c k i n g a g r i d square with twenty-five prey but not d e t e c t i n g any. of them (0.75 z 5*36/14400). Assuming the samples are independent, we have the p r o b a b i l i t y of f a i l u r e = [(14364 + 0 . 7 5 2 5 * 3 6 ) / 1 4 4 0 0 ] 8 0 = 0.82. which i s i n e x c e l l e n t agreement with the value p r e d i c t e d by the r e g r e s s i o n equation. 64 Flock s i z e , p_rey. d i s t r i b u t i o n , and f e e d i n g success The second s e t of experiments examines the r e l a t i o n s h i p between prey d i s t r i b u t i o n and f e e d i n g success f o r d i f f e r e n t f l o c k s i z e s . Whereas i n the f i r s t experiments f l o c k i n g b i r d s formed t h e i r f l o c k s ' n a t u r a l l y * i n accordance with t h e i r modelled behaviours, i n t h i s s e t of experiments the f l o c k s i z e f o r each experiment i s p r e s e t . The b i r d s are maintained i n a s i n g l e f l o c k by r e q u i r i n g t h a t whenever a b i r d takes f l i g h t , the remaining b i r d s must j o i n ( INTF = 0 ) . F l o c k s i z e s from 1 to 16 were t e s t e d over the same range of food dumpings as before. During.these experiments i t became apparent that an a d d i t i o n a l v a r i a t i o n i n prey d i s t r i b u t i o n was necessary. While a l a r g e f l o c k may e x c e l at l o c a t i n g clumps c f prey, i f there are few prey i n the clump, aggregation i n l a r g e f l o c k s may be disadvantageous. Consequently, the above experiments were repeated, r e p l a c i n g the t h i r t y - s i x clumps of t w e n t y - f i v e prey each with twelve clumps of s e v e n t y - f i v e prey each. Note that the t o t a l food supply remains constant. R e s u l t s The r e s u l t s of these experiments are presented i n F i g u r e s 7-10 (see Appendix B f o r the same r e s u l t s i n t a b u l a r form). F i g u r e 7 (mean cap t u r e s ; t h i r t y - s i x clumps, twenty-five prey each) r e v e a l s the f o l l o w i n g main p o i n t s . 1. For a given f l o c k s i z e the mean capture r a t e i n c r e a s e s with clumping, reaches a maximal value, and then decreases (as 65 Flock Size Figure 7: Mean captures In 20 minutes with 36 clumps oE 25 prey each. The diagrams are a three dimensional plot In perspective of moan c a p t u r e * (heif.ht of peaks) and a contour man of t he sane plot. The x - a x l s gives flock s i z e , the y-axls fitves the decree of clump inp, i n C ("in 16-C, where C is Lloyd's Index of mean crowding). The values on the contour map arc numher of prey capture Jn 20 pi nu tes. 66 a l s o i n d i c a t e d i n F i g u r e 4). Thus, f o r a f l o c k of f i x e d s i z e , there i s a 'best* prey environment i n terms of prey clumping. • , -2. For a given prey clumping a small range of f l o c k s i z e s are most s u c c e s s f u l . Thus, f o r a given prey clumping, f l o c k s of the 'best* s i z e might be expected to form (when f l o c k s i z e i s otherwise u n c o n s t r a i n e d ) . 3. For low v a l u e s of prey clumping l a r g e r f l o c k s are more s u c c e s s f u l , while f o r high degrees of prey clumping smaller f l o c k s are more s u c c e s s f u l . 4. Except f o r extreme clumping, there i s always some f l o c k s i z e which y i e l d s g r e a t e r success than does s o l i t a r y f o r a g i n g . 5. The highest capture r a t e i s f o r p a i r s of b i r d s f e e d i n g on moderately clumped food. F i g u r e 8 (mean c a p t u r e s ; twelve c l u m p s , s e v e n t y - f i v e prey each) shows t h a t i n c r e a s i n g the number of food items per clump a l t e r s s e v e r a l of the p a t t e r n s . 1. The p a t t e r n f o r f i x e d f l o c k s i z e s t i l l h o l d s , but the peaks f o r l a r g e r f l o c k s have s h i f t e d to higher values of clumping. 2. The p a t t e r n f o r f i x e d prey clumping s t i l l h o l d s , but net as s t o n g l y . , . 3. fl simple r e l a t i o n s h i p between prey clumping and 'best* f l o c k s i z e i s no l o n g e r apparent. 67 Flock Size Figure 8:. Mean captures in 20 minutes with 12 ciumpii of 75 preyench. The diagrams are a three dimensional plot in perspective of r.ean captures (h^tfiht of peaks) and a contour man of the same plot. The x-nxl? F.lves flock size, the .y-axla j?,lvos the decree of clump-ing l h C' ( a l n 16'C, where C la Lloyd's index of mean crowding).' The values on the contour map arc number of ptey capture in 20 minutes. 68 4. Again there i s g e n e r a l l y some f l o c k s i z e which does b e t t e r then the s o l i t a r y b i r d , 5. Again the h i g h e s t capture r a t e i s f o r p a i r s cf b i r d s on moderately clumped food. The major d i f f e r e n c e here seems to be the g r e a t e r success of l a r g e r f l o c k s a t higher degrees o f . p r e y clumping. In the f i r s t t e s t a s i n g l e patch provided only about 1.5 prey per b i r d f o r a l a r g e f l o c k (16) r . w h i l e i n the second s e t of t e s t s a patch provided the same f l o c k with about 4.5 prey per b i r d . Thus, these r e s u l t s suggest t h a t f o r a f l o c k of s i x t e e n , the i n c r e a s e d d i f f i c u l t y of l o c a t i n g a patch when the number of patches was reduced was s m a l l compared with the i n c r e a s e d value of the patch. The p a t t e r n s of r i s k are r a t h e r d i f f e r e n t from those f o r suc c e s s . From F i g u r e 9 we see that r i s k i n c r e a s e s with prey clumping f o r s i n g l e b i r d s . In a d d i t i o n r i s k g e n e r a l l y d e c l i n e s with an i n c r e a s e i n f l o c k 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 come with l a r g e f l o c k s (12-16) and moderate t c high clumping ( C = 2-5). Smaller f l o c k s (2-6) e x h i b i t a s m a l l e r but noteworthy r i s k ; r e d u c t i o n f o r moderate degrees of clumping (C 1 = 2-4). When the food supply i s d i v i d e d i n t o fewer clumps (see Figure 10), we f i n d much the same p a t t e r n . In both cases r i s k decreases f o r l a r g e f l o c k s i n the moderate clumping range ( C =2-4) and then r i s k i n c r e a s e s s h a r p l y f o r a l l f l o c k s i z e s i n the high clumping range , (C 1 .= ,4-6). The major c o n c l u s i o n 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 each. The d l n i ; r a r a 3 are a three dimensional plot In perspective and a con-tour map of the sane p l o t . The x-axts RIVCS flock s i z e s , the y-axls gives the decree, of clumping In C (--In l f , c , wlicre C Is Lloyd's Index of mean crowding). The values on the contour r.ap are per-centage of times a bird can expect to make no captures In 20 ntnutcs. 71 here i s that l a r g e f l o c k s (up to 16) s u b s t a n t i a l l y reduce the r i s k of o b t a i n i n g no food over a s h o r t time horizon (20 minutes). £l§cussion The r e s u l t s of the s i m u l a t i o n experiments show that f l o c k i n g does i n c r e a s e the e f f e c t i v e n e s s with which i n d i v i d u a l s e x p l o i t food r e s o u r c e s . However, while previous workers have s t r e s s e d t h a t f l o c k i n g might r e s u l t i n an i n c r e a s e i n the r a t e of food capture, these r e s u l t s suggest that t h i s may be a l e s s important consequence of f l o c k i n g than reducing the r i s k of d o i n g , badly. In an u n p r e d i c t a b l e environment, minimizing r i s k may be a more a p p r o p r i a t e measure of f i t n e s s than maximizing e f f i c i e n c y . In the f i x e d f l o c k s i z e experiments, these f l o c k s i z e s y i e l d i n g the g r e a t e s t average capture r a t e were not g e n e r a l l y those y i e l d i n g the s m a l l e s t r i s k . Thus i n order to e v a l u a t e which f l o c k s i z e i s " b e s t " f o r a given prey d i s t r i b u t i o n , we must s i m u l t a n e o u s l y account f o r r i s k and capture r a t e . One way to look a t r i s k i s to c o n s i d e r the p r o b a b i l i t y t h a t an i n d i v i d u a l s f o r a g i n g success w i l l be below some c r i t i c a l l e v e l f o r a day. For example, i f the b i r d s feed f o r 10 hours a day and i f we assume t h a t the mean capture r a t e f o r a 20 minute p e r i o d (the time u n i t of the model) i s r e p r e s e n t a t i v e of the capture r a t e throughout the day, we can work out the 72 p r o b a b i l i t i e s t h at a b i r d w i l l f a l l below v a r i o u s t h r e s h o l d f e e d i n g r a t e s when alone and when i n a f l o c k (Table VI). Thus, f o r c r i t i c a l f e e d i n g r a t e s around 2.5 prey per hour, we expect f l o c k s of 16 to be favoured over s i n g l e b i r d s , while f o r c r i t i c a l r a t e s above 5.0 prey per hour s o l i t a r y b i r d s are favoured (neith e r i s favoured a t around.4 prey per hour). For the s i z e of prey used i n the model, 2.5 prey/hour would seem to be a reasonable r a t e f o r s u r v i v a l . The average capture rate has to be around 4 prey/hour over a 10 hour day. In the model, the advantages of f l o c k i n g are l i k e l y to have r e s u l t e d from i n c r e a s e d e f f i c i e n c y i n l o c a t i n g food (because of more eyes s e a r c h i n g and a tendency to copy ne i g h b o r s ) . The disadvantage of f l o c k i n g comes from, i n t e r f e r e n c e between i n d i v i d u a l s when patches of food are found. In t h i s system, the advantage of f l o c k i n g should be most apparent when clumps of food are l a r g e enough to feed the f l e c k and dense enough to i n c r e a s e the f e e d i n g success of b i r d s s e a r c h i n g i n the patch. These trends are born out i n the s i m u l a t i o n r e s u l t s by: 1. The b e t t e r success of b i g f l o c k s (e.g. 16) when there are more prey per patch (although fewer patches) ; 2. The i n c r e a s e with i n c r e a s e d prey clumping of the d i f f e r e n c e i n success between f l o c k i n g and s o l i t a r y b i r d s (provided we s e l e c t the " r i g h t " f l o c k s i z e ) . In a d d i t i o n we would expect r i s k r e d u c t i o n with an i n c r e a s e i n f l o c k s i z e , Copying w i l l tend to be more b e n e f i c i a l , when 73 Feeding r a t e Flock s i z e S o l i t a r y Flock of 16 0.0/h 0.00 0.00 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 Table VI]. P r o b a b i l i t i e s t h a t a b i r d ' s average d a i l y f e e d i n g r a t e w i l l be l e s s then v a r i o u s l e v e l s (the assumption i s made that the capture r a t e f o r 20 minute p e r i o d s i s approximately normally d i s t r i b u t e d ) . 74 there are more b i r d s to copy and when the b i r d s are c l o s e together. S i n c e copying mainly b e n e f i t s the u n s u c c e s s f u l b i r d , t h i s e f f e c t should produce t r e n d s i n r i s k r e d u c t i o n as f l o c k s i z e i n c r e a s e s and as,prey clumping i n c r e a s e s . These trends are borne out except f o r very high degrees of clumping where i n t e r f e r e n c e p l a y s a dominant r o l e . The c o s t of s h a r i n g food should be most severe when patches cover a s m a l l area,and the number of prey per patch i s s m a l l . Thus, f o r high values of prey clumping, b i r d s i n f l o c k s should be l e s s s u c c e s s f u l . than those feeding a l o n e . The r e s u l t s suggest that t h i s point comes between C* = 5 - 6 when patches have 75 prey each and between C* = 4 - 5.when patches have only 25 prey each. A c o n s i d e r a t i o n of p h y s i c a l i n t e r f e r e n c e suggests that while a f l o c k may exhaust a patch r e g a r d l e s s of the area i t covers, when the patch i s too s m a l l to hold the f l o c k , some b i r d s w i l l be l i k e l y to make no c a p t u r e s . V a r i a t i o n between i n d i v i d u a l s i n the f l o c k These experiments w i l l examine the e f f e c t s cn prey capture of a d m i t t i n g d i f f e r e n c e s between the b i r d s i n t o the model. The v a r i a t i o n was i n two parameters: p r o b a b i l i t y of prey d e t e c t i o n without prey p r e f e r e n c e , ±10%; and p r o b a b i l i t y of prey d e t e c t i o n with prey p r e f e r e n c e , ±20%. These two were chosen s i n c e a) b i r d s are l i k e l y t o d i f f e r i n p e r c e p t u a l a b i l i t y , and b) prey capture r a t e was e a r l i e r found to be s e n s i t i v e to changes i n 75 p r o b a b i l i t y of prey d e t e c t i o n without prey p r e f e r e n c e (hence we should see some s i g n i f i c a n t changes i f v a r i a t i o n between b i r d s has a major impact). These s i m u l a t i o n experiments were performed f o r both f l o c k i n g and n o n f l o c k i n g b i r d s f o r a range of f i v e values of prey prey clumping. R e s u l t s Tables VII and. VIII present the r e s u l t s of these experiments. The major c o n c l u s i o n s are: 1. V a r i a t i o n had no e f f e c t upon mean capture r a t e or mean r i s k f o r f l o c k i n g 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 range of prey prey clumping. 2. No s i g n i f i c a n t d i f f e r e n c e i n mean capture rate c r mean r i s k was found between the above-average and below-average b i r d s . 3. Given i n d i v i d u a l v a r i a t i o n , f l o c k i n g d i d not enhance capture r a t e of the b e t t e r b i r d s at the expense c f the l e s s able b i r d s nor v i c e versa. Neither was r i s k s i g n i f i c a n t l y r e a p p o r t i o n e d . 4. When f l o c k s i z e was f i x e d , f l o c k s of 8 -16 were more s u c c e s s f u l than s m a l l e r f l o c k s . 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 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 Table VIIj. Capture r a t e f o r f l o c k i n g b i r d s with i n d i v i d u a l v a r i a t i o n , n o n f l o c k i n g b i r d s with i n d i v i d u a l v a r i a t i o n , f l o c k i n g b i r d s without i n d i v i d u a l 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 without i n d i v i d u a l v a r i a t i o n . 77 C F-Iv J f - I v 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 Risk of making no captures i n twenty minutes expressed as a percentage. The f o u r columns r e s p e c t i v e l y are f o r f l o c k i n g b i r d s with i n d i v i d u a l v a r i a t i o n , n o n f l o c k i n g b i r d s with i n d i v i d u a l v a r i a t i o n , f l o c k i n g b i r d s without i n d i v i d u a l 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 without i n d i v i d u a l v a r i a t i o n . 78 I t i s a somewhat s u r p r i s i n g r e s u l t t h a t d i f f e r e n c e s i n prey d e t e c t i o n a b i l i t y made l i t t l e d i f f e r e n c e i n the b i r d s ' success. The most p l a u s i b l e e x p l a n a t i o n i s t h a t the v a r i a t i o n i n success c r e a t e d by v a r i a t i o n i n prey d e t e c t i o n a b i l i t i e s was small i n comparison with the v a r i a t i o n i n success i n h e r e n t i n the search process. The g r e a t e r success of l a r g e f l o c k s i n comparison to s m a l l ones suggests t h a t as a r e s u l t of copying, the average of the prey d e t e c t i o n a b i l i t i e s f o r b i r d s i n a l a r g e f l o c k i s e f f e c t i v e l y g r e a t e r than the average of the i n d i v i d u a l a b i l i t i e s . P c l ^ m c r ^ h i c Prejj P o p u l a t i o n s Before p r e s e n t i n g the experiments, a b r i e f review i s given of p r e v a i l i n g i d e a s on the r o l e of polymorphism i n n a t u r a l p o p u l a t i o n s . In r e c e n t years i t has been argued t h a t ^ p r e d a t o r s may s e l e c t f o r v i s u a l polymorphism i n t h e i r prey s p e c i e s . T h i s form of s e l e c t i o n , known as a p o s t a t i c s e l e c t i o n (Clarke 1962), occurs when the predator takes common morphs i n g r e a t e r p r o p o r t i o n than t h e i r . o c c u r r e n c e i n the p o p u l a t i o n and r a r e morphs i n lower frequency than they occur i n the p o p u l a t i o n . T h i s phenomenon has been invoked, f o r example, to e x p l a i n the maintenance of polymorphisms i n s e v e r a l s p e c i e s cf s n a i l s (Cain 5 Sheppard 1954; C l a r k e 1960,69; Owen 1965). A p o s t a t i c s e l e c t i o n on a r t i f i c i a l b a i t s has been demonstrated i n 79 l a b o r a t o r y s t u d i e s f o r v a r i o u s b i r d s p e c i e s (Allen & C l a r k e 1968; A l l e n 1972; Manly, M i l l e r 6 Cook 1972). Using n a t u r a l prey K e t t l e w e l l (1955) demonstrated d i s p r o p o r t i o n a t e l y low p r e d a t i o n on a c r y p t i c morph of the moth B i s t e n b e t u l a r i a , while Murton's (1971) f i e l d experiments with pigeons gave l i t t l e i n d i c a t i o n t h at they d i s p r o p o r t i o n a t e l y favored the commoner prey. Confounding .the s i t u a t i o n are the d i f f i c u l t i e s of accounting f o r density-independent p r e f e r e n c e s of the. predator (Manly e t . a l . 1972 present a method of handling t h i s d i f f i c u l t y ) and the apparent disadvantage of polymorphism to the prey s p e c i e s at high d e n s i t i e s ( A l l e n 1972; Greenwood 1969; H o l l i n g 1965) and p o s s i b l y a t very low d e n s i t i e s as well (Greenwood 1969). F i n a l l y , we note t h a t i n Croze's (1970) study the combined a t t a c k r a t e on a t r i m o r p h i c p o p u l a t i o n was s m a l l e r than on any of the t h r e e monomorphic p o p u l a t i o n s ( o v e r a l l d e n s i t y being kept c o n s t a n t ) . Accepting t h a t polymorphism may be an e f f e c t i v e s t r a t e g y a g a i n s t an avian predator, one may ask what c c u n t e r s t r a t e g i e s are a v a i l a b l e to the b i r d s . In t h i s s e c t i o n we i n v e s t i g a t e one such p o s s i b i l i t y , examining the value o f . f l o c k i n g as a means to i n c r e a s e the b i r d s ' f e e d i n g success on a polymorphic prey 80 p o p u l a t i o n . 6 S i n c e the modelled b i r d s form s h o r t term prey p r e f e r e n c e s , we can. expect that they w i l l be l e s s e f f e c t i v e p r e d a t o r s on a polymorphic prey p o p u l a t i o n than cn a mcnomorphic one whenever the prey encounter r a t e exceeds the reinforcement r a t e r e q u i r e d to maintain the prey p r e f e r e n c e . When the encounter r a t e i s too low, prey p r e f e r e n c e s are r a r e l y formed, and polymorphism i s u n l i k e l y t o c o n f e r any advantage to the prey. However, given t h e i r a b i l i t y to copy, a b i r d i n a f l o c k may copy a nearby b i r d ' s a c t i o n s , thus more r a p i d l y l e a r n i n g prey and prey s i t e c h a r a c t e r i s t i c s . T h e r e f o r e , b i r d s i n f l o c k s have the p o t e n t i a l t o . s w i t c h from a t t a c k i n g one prey morph to another more r a p i d l y than do s o l i t a r y b i r d s . , The s i m u l a t i o n experiments which f e l l o w were intended to answer s e v e r a l q u e s t i o n s . , 1. To what extent does the p a r t i t i o n i n g cf a s i n g l e prey p o p u l a t i o n i n t o two d i s t i n c t types (morphs) decrease ; the mean capture r a t e and how i s t h i s e f f e c t i n f l u e n c e d by f l o c k i n g and i n d i v i d u a l v a r i a t i o n (as d e f i n e d p r e v i o u s l y ) ? 2. Does the p a r t i t i o n i n t o two prey types i n c r e a s e r i s k (to the b i r d s ) and how do f l o c k i n g and i n d i v i d u a l v a r i a t i o n 6 Since the d i s t i n c t i o n i n the model i s between prey 'types', these types 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 prey s p e c i e s r a t h e r than morphs o f . a s i n g l e s p e c i e s . Hence, t h i s 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 separate prey s p e c i e s as a r e s u l t of d e n s i t y dependent p r e d a t i o n (e.g. Murdoch 1969). In a d d i t i o n the 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 than on v i s u a l d i f f e r e n c e s . Throughout the t h e s i s the term polymorphism w i l l be used to r e f e r to any of these cases. 81 a l t e r t h i s e f f e c t ? 3. Does f l o c k i n g a l t e r the r a t i o of a t t a c k s upcn the two prey types? 4. To what extent does i n d i v i d u a l v a r i a t i o n c r e a t e d i f f e r e n c e s i n capture r a t e and r i s k between above-average b i r d s and below-average ones, and does f l o c k i n g m i t i g a t e cr enhance the d i f f e r e n c e ? . 5. Do the b i r d s a t t a c k the commoner of two prey types d i s p r o p o r t i o n a t e l y o f t e n ? The f o l l o w i n g s e t s of s i m u l a t i o n experiments were performed to examine these q u e s t i o n s . In the f i r s t s e t the two prey types were equal i n number, s i z e (same handling time) and d i s t r i b u t i o n ( i . e . , the a c t u a l d i s t r i b u t i o n of each prey type was determined by the same p r o b a b i l i t y d i s t r i b u t i o n ) . The t o t a l prey d i s t r i b u t i o n s were i d e n t i c a l with the d i s t r i b u t i o n s used i n the one-prey s i m u l a t i o n experiments, with t h i r t y - s i x clumps of tw e n t y - f i v e prey each (900 t o t a l p r e y ) . Five v a l u e s of clumping were used, ranging from random (C=0.07) to h i g h l y clumped (C=9.48) prey. At each of these, f i v e r e p l i c a t e s ( d i f f e r e n t random number sequences) were run. The s i m u l a t i o n experiments modelled twenty b i r d s f o r a g i n g f o r twenty minutes, t e s t i n g the fou r combinations: with f l o c k i n g and i n d i v i d u a l v a r i a t i o n ; with f l o c k i n g , without i n d i v i d u a l v a r i a t i o n ; without f l o c k i n g , with i n d i v i d u a l v a r i a t i o n ; without f l o c k i n g or i n d i v i d u a l v a r i a t i o n . In the second s e t of s i m u l a t i o n experiments only the prey 82 d e n s i t y was changed; the two prey types were at twice t h e i r p revious d e n s i t i e s (1800 t o t a l p r e y ) . A t h i r d s et cf s i m u l a t i o n experiments examined, the e f f e c t of f i x e d f l o c k s i z e with i n d i v i d u a l v a r i a t i o n present. The prey d e n s i t y used was i d e n t i c a l to t h a t i n the f i r s t s e t (900 t o t a l p r e y ) . In a f o u r t h s e t of s i m u l a t i o n experiments the prey were presented i n a 9: 1 r a t i o (900 t o t a l p r e y ) , and i n the. l a s t s e t the prey were of t h r e e types i n a 1:1:1 r a t i o (900 t o t a l p r e y ) . These l a s t two s e t s of s i m u l a t i o n experiments were conducted only f o r the case with i n d i v i d u a l v a r i a t i o n between the b i r d s . Table IX summarizes these experimental designs. Re s u l t s Mean prey capture rates, f o r the f i r s t two s e t s of s i m u l a t i o n experiments are presented i n Table X along with the corresponding r e s u l t s f o r the one-prey case and the three-prey case. The p a r t i t i o n i n t o two types .caused a s i g n i f i c a n t r e d u c t i o n i n mean capture r a t e (p<.0001; unless otherwise noted, a l l hypotheses were, evaluated with the randomization t e s t , S i e g e l 1956). T h i s r e d u c t i o n , while small f c r .randomly d i s t r i b u t e d prey, was quite l a r g e f o r h i g h l y clumped prey. Examining each o f the f o u r cases s e p a r a t e l y ( f l o c k i n g , n o n f l o c k i n g , i n d i v i d u a l v a r i a t i o n , nc i n d i v i d u a l v a r i a t i o n ) , we f i n d the r e d u c t i o n s t i l l s i g n i f i c a n t ( s i g n i f i c a n c e l e v e l s from .002 to .006). 83 Experiment # prey prey t o t a l f l o c k case s examined s e t types r a t i o prey s i z e F F N f N f f i x e d Iv Niv Iv Niv 1 2 1 :1 900 No * * * * 2 2 1:1 1800 NO * * * * 3 2 1:1 900 Yes * 4 2 9:1 900 No * * 5 3 1 :1:1 900 Wo * Table IX_: Design of the f i v e s e t s of experiments. Under cases, examined the a b b r e v i a t i o n s are as f o l l o w s : F - f l o c k i n g ; Nf n o n f l o c k i n g ; Iv - with i n d i v i d u a l v a r i a t i o n ; Niv - without i n d i v i d u a l v a r i a t i o n . 84 Clumping 0.07 0.74 1.48 5.24 9.48 2 prey types F -Iv 1.12 2.56 2.22 2.42 2.21 t o t a l c f 900 F -Niv 1.08 2.20 2.13 2.52 1.39 Nf-Iv 1 .06 2.15 1.86 1.83 1. 14 Nf-Niv 1.13 1.93 1.75 1.81 1 .34 1 prey type F - I v 1 .28 2.44 2.73 3.02 2.49 t o t a l of 900 F -Niv 1.30 2.23 2.20 3.44 2.35 Nv-I v 1 .42 1.92 2.34 2.40 2.57 Nf-Niv 1.08 2.16 2.63 3.58 2.31 2 prey types F - I v 2. 16 3.79 4.15 4.99 3.26 t o t a l of 1800 F -Niv 1 .99 3.26 4.51 3.41 2.82 Nf-Iv 2. 14 3.84 3.33 4.28 2.31 Nf-»iv 2.06 3.14 2.91 3.63 3.08 3 prey types F -Iv 0.95 2.34 2.40 „ 1.50 1 .54 t o t a l c f 900 Nf-Iv 1.25 2.04 2.21 1.29 1.30 Table X_: Hean capture r a t e s a t f i v e values of prey clumping and f o r each combination of f l o c k i n g and n o n f l o c k i n g , with and without i n d i v i d u a l v a r i a t i o n , where F = f l o c k i n g Nf = n o n f l o c k i n g Iv = i n d i v i d u a l v a r i a t i o n 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 prey r e s u l t s are f o r equal numbers, s i z e and d i s t r i b u t i o n of the prey types. 85 Whereas i n the s i n g l e prey s i m u l a t i o n experiments f l o c k i n g was found to have no e f f e c t upon mean capture rate (see Table X) , with two prey types, f l o c k i n g does s i g n i f i c a n t l y i n c r e a s e mean capture r a t e (p<.00T); e s p e c i a l l y when prey are clumped. A d d i t i o n a l l y , i n d i v i d u a l v a r i a t i o n has.greater impact with twc prey than i n the s i n g l e prey case. T h i s d i s t i n c t i o n i s most prominent a t extreme values of clumping (C=9.48), where f l o c k i n g b i r d s with i n d i v i d u a l v a r i a t i o n d i d s u b s t a n t i a l l y b e t t e r than those i n the other three cases.. The , s m a l l s e t of t r i m o r p h i c experiments s u b s t a n t i a t e the above r e s u l t s , with capture r a t e s reduced from the monomorphic case and b i r d s i n f l o c k s feeding more s u c c e s s f u l l y than,those alone. At the d e n s i t i e s used here i t appears that the .prey's marginal gain i n p r o t e c t i o n i s s m a l l e r f o r the a d d i t i o n of a t h i r d morph to twc than i t i s f o r the a d d i t i o n of a second morph to one. Also polymorphism p r o v i d e s g r e a t e r p r o t e c t i o n f o r clumped prey than randomly d i s t r i b u t e d prey. Another way to view these r e s u l t s -is... t c examine the magnitude of the r e d u c t i o n i n capture r a t e caused by the p a r t i t i o n i n t o two prey types (see Table XI). T h i s r e d u c t i o n i s s m a l l e r f o r b i r d s i n ; f l o c k s than f o r s o l i t a r y f o r a g e r s (p=.066). Again i n d i v i d u a l v a r i a t i o n appears d e s i r a b l e (net s i g n i f i c a n t , .10<p<.20). When f l o c k s i z e s are f i x e d (see Table X I I ) , we again f i n d f l o c k i n g paying o f f r a t h e r more than i n the s i n g l e prey case. 86 Clumping 0.07 0.74 1.48 5.24 9.48 F -Iv F Niv Nf-Iv Nf-Niv .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 r e d u c t i o n of capture r a t e by the p a r t i t i o n of 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 pes. Values are the d i f f e r e n c e i n mean capture r a t e between the one prey and two prey r e s u l t s when both are at the same t o t a l 87 Clumping Flock s i z e 0 .07 0. 74 1. 48 5 .24 9.48 1 1 .06 2. 15 1. 86 1 .83 1.14 2 .90 2. 00 1. 25 .95 1.65 4 1 .20 1. 70 3. 13 2 .20 1 .95 8 .90 1. 46 2. 54 1 .41 1.35 16 1 .18 2. 11 1. 94 1 .95 2.37 Table XII,: Mean capture 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 prey are of two types with t o t a l d e n s i t y of one per s i x t e e n sguare yards. 88 In p a r t i c u l a r we f i n d ..large f l o c k s (16) c o n s i s t e n t l y egual to or above the s o l i t a r y b i r d s . Risk, the percentage p r o b a b i l i t y of making no capture i n twenty simulated minutes, i s presented i n Table X I I I . . I t i s apparent that r i s k - i n c r e a s e s with the p a r t i t i o n of the prey p o p u l a t i o n (p<.00005); and that f l o c k i n g c o n t i n u e s , as i n the one prey case, to be an e f f e c t i v e means of r i s k r e d u c t i o n (p<.006) and p<.001), e s p e c i a l l y when prey are h i g h l y clumped and there i s v a r i a t i o n between the b i r d s . These r e s u l t s held f o r the t r i m o r p h i c p o p u l a t i o n as w e l l , with the a d d i t i o n of the t h i r d morph g e n e r a l l y i n c r e a s i n g r i s k . F l o c k i n g b i r d s remain a t an advantage. For f i x e d f l o c k s i z e (see Table XIV), as with the s i n g l e prey case, l a r g e f l o c k s produce the g r e a t e s t r i s k r e d u c t i o n , and r i s k i n c r e a s e s l i t t l e as clumping i n c r e a s e s f o r the l a r g e s t f l o c k s i z e t e s t e d ( f l o c k s of 16). To answer the t h i r d , q u e s t i o n r e g a r d i n g the d i s t r i b u t i o n of a t t a c k s on the two prey types, l e t us suppose the two types are captured i n . t h e p r o p o r t i o n s R:S, where R<S and R*S=1. Since the prey are equal and l i m i t e d i n number, over a long p e r i o d of fe e d i n g i n a s i n g l e a r e a . R.and S w i l l be n e a r l y equal. But i n gene r a l r e a l b i r d s . w i l l move on to new feed i n g grounds well before the prey are s u f f i c i e n t l y r a r e to ensure t h a t a t t a c k s have been e q u a l l y d i s t r i b u t e d (birds ..are known to leave a f o r a g i n g area before exhausting the food supply ,e.g. Gibb 89 Clumping 0.07 0.74 1.48 5.24 9.48 2 prey types F -Iv 39 46 44 47 54 t o t a l c f 900 F -Niv 41 48 49 51 75 Nf-Iv 43 46 64 73 82 Nf-Niv 40 49 63 68 83 1 prey type F - I v 30 29 34 52 49 t o t a l of 900 F -Niv 29 38 47 45 56 Nf-Iv 31 47 57 68 73 Nf-Niv 32 45 53 59 73 2 prey types F - I v 18 21 31 20 37 t o t a l of 1800 F -Niv 15 24 32 42 46 Nf-Iv 16 39 41 55 67 Nf-Niv 17 33 44 61 63 3 prey types F -Iv 43 42 42 72 65 t o t a l of 900 Nf-Iv 34 52 67 80 86 Table XIIIj. RiskT "the percentage p r o b a b i l i t y t h a t a b i r d w i l l go twenty minutes without c a p t u r i n g any prey. The n o t a t i o n i s the.same as i n Table X.. 90 Clumping Flock s i z e 0.07 0.74 1.48 5.24 9.48 1 43 46 64 73 82 2 45 65 60 70 80 4 33 45 45 55 60 8 36 46 52 59 61 16 29 32 26 35 31 Table XIVi Risk, the percentage p r o b a b i l i t y c f a b i r d ' s g c i n g twenty minutes without c a p t u r i n g any prey, 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)). 7 Thus, i t i s . o f i n t e r e s t to observe the degree t o which a t t a c k s are concentrated on one prey type or the other during a s i n g l e f e e d i n g episode (over a s e r i e s of f e e d i n g bouts the a t t a c k r a t i o must converge to the prey r a t i o of 1:1 s i n c e the s i m u l a t i o n model pro v i d e s no mechanism f o r developing long term p r e f e r e n c e s ) , and how t h i s i s a f f e c t e d by f l o c k i n g and the prey clumping of the prey d i s t r i b u t i o n . Values f o r R are presented i n Table XV. Two f e a t u r e s are e v i d e n t . f i r s t , the value c f R decreases as prey clumping i n c r e a s e s (p<.00U). Thus, when prey are c o n c e n t r a t e d , a t t a c k s tend to be concentrated on one of the two types. Second, R i s l a r g e r when the b i r d s f l o c k (p=.031). That i s , b i r d s i n f l o c k s d i s t r i b u t e t h e i r , a t t a c k s more evenly than do b i r d s f e e d i n g by themselves. Thus, the f l e c k i n g b i r d s more e f f e c t i v e l y maintain the i n i t i a l prey r a t i o . Next, l e t us examine the apportionment of prey c a p t u r e s and r i s k between those b i r d s above and those below the average i n prey d e t e c t i o n a b i l i t i e s . T a b les XVI and XVII summarize the r e s u l t s f o r the two prey case with 900 t o t a l prey. I n d i v i d u a l v a r i a t i o n has only ..a.small e f f e c t on capture r a t e . Thus, the capture r a t e f o r above-average b i r d s i s g r e a t e r than i n the nc v a r i a t i o n case (p=.010); but the belcw-average b i r d s dc nc worse than i n the no v a r i a t i o n case. . Furthermore, f l o c k i n g i n c r e a s e s t o t a l captures f o r both groups of b i r d s (p=.003 and p=.013 f o r 7 D e p l e t i o n r e p r e s e n t e d no problem i n the s i m u l a t i o n experiments s i n c e never more than 11% of the prey were taken i n the twenty simulated minutes. , 92 Clumping 0.07 0.74 1.48 5.24 9.48 f l e c k i n g .435 .407 .357 .347 .291 n o n f l o c k i n g .432 .398 .290 .267 .256 Table XVi The average f r a c t i o n of a t t a c k s on the l e s s attacked F^ey p o p u l a t i o n f o r twenty minute f o r a g i n g episodes; two prey types at t o t a l d e n s i t y of one prey per s i x t e e n square yards. 93 Clumping 0.07 0.74 1.48 5.24 9.48 F -Iv above 65 58 126 49 104 4 7 121 5 0 127 5 7 below 47 4 2 130 S 1 118 5 3 121 5 0 94 4 3 F -Niv above 54 50 100 4 5 94 4 4 128 5 1 65 4 7 below 54 SO 120 55 119 5 6 124 4 9 74 S 3 Nf -Iv above 45 4 2 102 4 7 108 5 8 91 5 0 79 6 9 below 61 58 113 53 78 42 92 5 0 35 3 1 Nf -Niv above 50 4 4 88 46 89 5 1 89 4 9 58 4 3 below 63 56 105 54 86 4 9 92 5 1 76 5 7 Table XVJ.I Each p a i r of numbers r e p r e s e n t s the apportionment of prey c a p t u r e s 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 are the c o n t r o l group). The f i r s t number i n the p a i r i s an absolute measure cf capture r a t e , the second number i s the percentage of the captures going to the given group. 94 Clumping 0.07 0.74 1.48 5.24 9.48 F -Iv above 32 4 1 42 4 6 38 43 44 4 7 46 4 3 below 46 59 50 54 50 5 7 50 5 3 62 5 7 F -Niv above 40 49 46 4 8 40 53 46 4 S 70 4 7 below 42 5 1 50 5 2 36 4 7 56 5 S 80 5 3 Nf -Iv above 44 5 1 50 54 62 4 8 72 4 9 80 4 9 below 42 4 9 42 46 66 52 74 5 1 84 5 1 Nf -Niv above 38 4 8 54 55 64 5 1 68 5 0 80 4 8 below 42 52 44 4 5 62 4 9 68 5 0 86 52 Table Mill Each p a i r of numbers r e p r e s e n t s the apportionment of r i s k between above average and below 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 are the c o n t r o l group).. The f i r s t number i n the p a i r i s an a b s o l u t e measure of r i s k , the second number i s the percentage of the r i s k assumed by the given group. 95 above-average and below-average b i r d s r e s p e c t i v e l y ) , with the gains a c c r u i n g to the above-average b i r d s net s i g n i f i c a n t l y g r e a t e r than those a c c r u i n g to the below-average b i r d s . . Also, the p o r t i o n of the capture going to the above-average (or below-average) b i r d s does not vary s i g n i f i c a n t l y with the clumping of the prey nor with f l o c k i n g . However, i n d i v i d u a l v a r i a t i o n does a f f e c t r i s k on a polymorphic p o p u l a t i o n . F l o c k i n g reduces r i s k s i g n i f i c a n t l y f o r the above-average b i r d s but net f o r the below average b i r d s (p=.031 and .125 r e s p e c t i v e l y ) , and t h i s r e d u c t i o n i s s i g n i f i c a n t l y g r e a t e r (p=.031) f o r the above-average b i r d s . F i n a l l y , i n c o n t r a s t to the case with prey capture r a t e s , i n d i v i d u a l v a r i a t i o n does r e a p p o r t i o n the r i s k amongst the b i r d s as measured by the percentage of r i s k i n each group (p=.065), with the above-average b i r d s r e d u c i n g t h e i r r i s k r e l a t i v e tc the below-average b i r d s . R e s u l t s o f , experiments performed to t e s t whether the simulated b i r d s would.in f a c t e x e r t , a p o s t a t i c s e l e c t i o n cn a polymorphic, prey ' p o p u l a t i o n are presented i n Tables XVIII to XXI. Eighteen clumps of f i f t y prey each were d i s t r i b u t e d with the r a t i o s between morphs being 1: 1, 9:1,and 1:0. Polymorphism was again seen to provide p r o t e c t i o n to the prey as a t o t a l p o p u l a t i o n , and t h i s p r o t e c t i o n (reduction i n capture rate) was g r e a t e r when the prey are 1:1 than 9:1 i n r a t i o . Furthermore, the b i r d s were more e f f e c t i v e i n f l o c k s than f o r a g i n g s e p a r a t e l y . On prey i n equal p r o p o r t i o n s the a t t a c k s were g e n e r a l l y i n a 1:1 r a t i o , though the e f f e c t of prey preference 96 Clumping 0.07 0.74 1.48 5.24 9.48 f l o c k i n g 1: 1 88 81 70 83 89 9:1 91 94 90 89 98 n o n f l o c k i n g 1: 1 75 68 48 76 51 9:1 78 86 72 80 80 Table X V I I I i T o t a l prey capture r a t e as a percentage of the capture r a t e cn an i d e n t i c a l l y d i s t r i b u t e d monomorphic prey p o p u l a t i o n . The two morphs are i n the p r o p o r t i o n s 1:1 and 9:1. 97 f l o c k i n g n o n f l o c k i n g Clumping 0.07 0.74 1.48 5.24 9.4.8 1:1 54.5 56.6 59.0 51. 2 42. 1 s i g . NS .034 .008 . NS .02 0 9:1 92.2 99.0 96.4 97.0 98.8 s i g . NS <10~ 6 <10~* .0002 . <10-s 1:1 49.1 45. 1 56.5 51.4 43. 9 s i g . SS NS .076 NS NS 9: 1 90.1 95.3 96.8 97.9 93. 2 s i g . SS .002 <10"* .0001 .074 Table XIX:. The percentage of a t t a c k s on prey type 1 i n a dimorphic p o p u l a t i o n . I f a t t a c k s are i n the same p r o p o r t i o n as prey, then f o r prey r a t i o 1:1 we expect 50% a t t a c k s on prey type 1, and f o r prey r a t i o 9:1 we expect 90% a t t a c k s on type 1. S i g n i f i c a n c e values d e r i v e d from the normaLl approximation t o the b i n o m i a l d i s t r i b u t i o n (minimum number of samples was 106). 98 Clumping f l o c k i n g 0.07 0.74 1.48 5.24 9.48 t y p e 1 1:1 96 92 83 85 75 t y p e 2 1:1 80 70 57 81 103 c o H o n morph 9:1 93 103 96 96 108 r a r e morph 9:1 71 9 32 27 12 n o n f l o c k i n g t y p e 1 1:1 74 61 54 78 45 t y p e 2 1:1 76 75 42 74 57 common morph 9:1 78 91 77 87 83 r a r e morph 9: 1 77 40 23 17 54 Table XXj. Capture r a t e per i n d i v i d u a l prey on each morph i n a' dimorphic p o p u l a t i o n , expressed as percentage of the capture r a t e on an e g u i v a l e n t l y d i s t r i b u t e d monomorphic p o p u l a t i o n . The e n t r i e s are p a r t i t i o n s by morph of the values i n Table XVIII. 99 Clumping 0.07 0.74 1.48 5.24 9.48 f l o c k i n g 1: : 1 39 46 44 47 54 9: : 1 31 35 37 68 66 1 j :0 30 29 34 52 49 n o n f l o c k i n g 1: ; 1 43 46 64 73 82 9: :1 34 44 54 80 86 1: :0 31 47 57 68 73 Table XXIj. Risk, the percentage p r o b a b i l i t y of a b i r d ' s g c i n g twenty minutes without making a capture, 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 runs on a s i n g l e mcrph) s h i f t e d the r a t i o on some o c c a s i o n s . However, when.the prey were i n a 9:1 r a t i o , the more common prey 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 more a t t a c k s except when the prey were d i s t r i b u t e d randomly (in which case the a t t a c k s were i n a 9:1 r a t i o ) . Thus, s h o r t term prey pr e f e r e n c e formation was found s u f f i c i e n t to produce a p c s t a t i c s e l e c t i o n . In a d d i t i o n , f l o c k s appear to s e l e c t more s t r o n g l y a g a i n s t the high frequency prey than do s o l i t a r y b i r d s . These r e s u l t s are i l l u s t r a t e d perhaps more c l e a r l y by the p r e s e n t a t i o n i n Table XX, which, gives the a t t a c k . r a t e cn each morph per i n d i v i d u a l of that morph., The r a r e morph r e c e i v e d f a r fewer a t t a c k s per i n d i v i d u a l than d i d the common mcrph. P r e d a t i o n by n o n f l o c k i n g b i r d s on the,common morph i s at a lower r a t e than on a monomorphic p o p u l a t i o n , whereas f o r b i r d s i n f l o c k s the pre d a t i o n r a t e on the common morph was not reduced by the presence o f the second morph. Thus, f l e c k i n g appears to reduce or e l i m i n a t e the l e s s i n pr e d a t i o n on common prey experienced by s o l i t a r y b i r d s ; a l o s s which a r i s e s from o c c a s i o n a l attempts to concentrate on the r a r e and u n p r o f i t a b l e morph. F i n a l l y , r i s k on polymorphic prey was lpwer f o r b i r d s i n f l o c k s r e g a r d l e s s of the r a t i o between the prey (p<.002) and polymorphism i n c r e a s e s r i s k (p<.0003), e s p e c i a l l y when prey were clumped. D i s c u s s i o n There i s some evidence i n the l i t e r a t u r e that polymorphism i n prey may c o n s t i t u t e an e f f e c t i v e s t r a t e g y a g a i n s t a v i a n 101 p r e d a t i o n (e.g. C l a r k e 1969; Croze 1970; Manly et a l . . 1972). In t h i s s e c t i o n we have reexamined t h i s a s s e r t i o n and extended the i n v e s t i g a t i o n to explore whether f l o c k i n g by b i r d s i n c r e a s e s the a t t a c k r a t e on polymorphic prey. The s i m u l a t i o n experiments l e n t a d d i t i o n a l support to the h y p o t h e s i s t h a t polymorphism causes a s i g n i f i c a n t r e d u c t i o n i n p r e d a t i o n . While . t h i s r e d u c t i o n i s s m a l l f o r randomly d i s t r i b u t e d prey, i t becomes more s i z a b l e . a s prey prey clumping i n c r e a s e s . Polymorphism seems to reduce the e f f e c t i v e n e s s of l e a r n i n g processes (e.g. formation of .search images). The b e n e f i t s of l e a r n i n g are s m a l l when prey are randomly d i s t r i b u t e d but i n c r e a s e with prey clumping, s i n c e the l e a r n i n g process depends upon the chances of f r e q u e n t encounters with prey to provide r e i n f o r c e m e n t f o r a prey preference. These chances are s m a l l f o r randomly d i s t r i b u t e d prey and l a r g e f o r clumped prey. When two morphs were presented i n unequal p o r t i o n s , the s i m u l a t i o n experiments r e v e a l e d t h a t the b i r d s a t t a c k e d the common morph more f r e q u e n t l y than i t s occurence i n the p o p u l a t i o n . Such a p o s t a t i c s e l e c t i o n i s to be expected whenever a predator forms short-term prey p r e f e r e n c e s (Krebs 1973b), r a r e morphs being p r o t e c t e d by being r a r e . F l o c k i n g was found to enhance the b i r d s ' l e a r n i n g e f f i c i e n c i e s i n polymorphic prey environments, hence s i g n i f i c a n t l y improving mean capture r a t e s f o r b i r d s . T h i s improvement i s not, however, s u f f i c i e n t to o f f s e t the advantages 102 of polymorphism to the prey. That i s , the combined capture r a t e s by b i r d s i n f l o c k s on a l l the morphs of a polymorphic prey p o p u l a t i o n were l e s s than the capture r a t e cn an e g u a l l y abundant monomorphic prey p o p u l a t i o n . The advantage of f l o c k i n g i n the model appears to l i e i n the o p p o r t u n i t y i t provides f o r copying. Thus, the b i r d s more r a p i d l y focus t h e i r search on the l o c a l l y more common morph ( t h i s would i n c l u d e s w i t c h i n g from one mcrph to another when th e ; f i r s t became r a r e through heavy p r e d a t i o n ) . A d d i t i o n a l experiments, using f i x e d f l e c k s i z e s , i n d i c a t e d t hat l a r g e r f l o c k s are a t more of an advantage over s m a l l e r f l o c k s f e e d i n g on polymorphic prey than en monomorphic prey. The i n t r o d u c t i o n of i n d i v i d u a l v a r i a t i o n i n t o the model d i d net a l t e r s i g n i f i c a n t l y ,: the r e s u l t s c i t e d above. It d i d , however, seem t o r e a p p o r t i o n r i s k among b i r d s , with above-average b i r d s reducing t h e i r r i s k r e l a t i v e to below-average b i r d s . Compared t o the monomorphic prey case i t appeared that with a polymorphic prey p o p u l a t i o n s e l e c t i o n a g a i n s t . t h e below-average b i r d s i s more pronounced. Nonetheless, the below-average b i r d s s t i l l reduced t h e i r r i s k by j o i n i n g a f l o c k . To o b t a i n some i n d i c a t i o n s as to the e f f e c t s en p r e d a t i o n of i n t r o d u c t i o n of f i n e r p a r t i t i o n s of prey (adding morphs), experiments with t r i m o r p h i c prey p o p u l a t i o n s were compared with the monomorphic and dimorphic cases. As expected, a d d i t i o n of a t h i r d morph f u r t h e r reduced p r e d a t i o n ( p r i m a r i l y when prey 103 clumping was f a i r l y h i g h ) , but with d e c r e a s i n g marginal e f f e c t (at l e a s t under the experimental c o n d i t i c n s modelled). C o n s i d e r a t i o n s of how a predator should s e l e c t i t s d i e t have l e d to the hypothesis t h a t the. predator should ignore u n p r o f i t a b l e prey items (such as r a r e morphs) and to concentrate i t s search.on the more v a l u a b l e ones (e.g. T u l l c c k 1971; Charnov 1973). In so f a r as pre d a t i o n on a rare morph i n t e r f e r e s with the capture of more common ones, prey of a r a r e morph are l e s s p r o f i t a b l e than are ones c f a common morph. Proper d i e t s e l e c t i o n ("optimal f o r a g i n g " ) should lead to a p o s t a t i c s e l e c t i o n , m a i n t a i n i n g polymorphism i n the prey p o p u l a t i o n . Flocking.was found i n the s i m u l a t i o n experiments to i n c r e a s e the i n t e n s i t y of a p o s t a t i c s e l e c t i o n and to reduce or sometimes e l i m i n a t e the l o s s i n predation on common prey experienced by s o l i t a r y b i r d s ; a l o s s a r i s i n g from the tendency of s o l i t a r y f o r a g i n g b i r d s to o c c a s i o n a l l y focus t h e i r search on the r a r e and u n p r o f i t a b l e morph. By s e l e c t i n g mere s t r o n g l y i n fav o r of the commonest prey type, f l o c k i n g b i r d s reduce r i s k and i n c r e a s e capture r a t e , on a l l but randomly d i s t r i b u t e d prey r e g a r d l e s s of the abundance r a t i o s among prey t y p e s . 101 S ujmary. F i n a l l y , one must note t h a t s i m u l a t i o n experiments are intended to examine p l a u s i b l e i m p l i c a t i o n s of primary hypotheses and t o suggest p o t e n t i a l l y p r o f i t a b l e d i r e c t i o n s f o r f i e l d and l a b o r a t o r y experimentation. , The r e s u l t s given here suggest t h a t i t would be p r o f i t a b l e to t e s t under n a t u r a l c r l a b o r a t o r y c o n d i t i o n s s e v e r a l hypotheses r e g a r d i n g f l o c k i n g . 1. B i r d s a d j u s t t h e i r giving-up times on the b a s i s of recent feeding s u c c e s s . 2. B i r d s i n f l o c k s capture n e i t h e r .more nor fewer prey on average than do s o l i t a r y b i r d s when there i s a s i n g l e prey type. 3. B i r d s i n f l o c k s capture more prey when a t t a c k i n g a polymorphic prey p o p u l a t i o n than do the same b i r d s f o r a g i n g alone. 4. B i r d s i n f l o c k s have a s m a l l e r v a r i a t i o n i n capture r a t e f o r a f i x e d f o r a g i n g p e r i o d than do b i r d s f o r a g i n g alone. 105 CHAPTER SIX! MARKOV MODELS M§ikodoiogy There are four b a s i c elements to the m u l t i - l e v e l modelling procedure d e s c r i b e d i n t h i s c h a p ter. They are: 1. C o n s t r u c t i o n o f a d e t a i l e d mechanistic s i m u l a t i o n ; 2. Choice of a mathematical u n d e r l y i n g s t r u c t u r e ; . 3. D e r i v a t i o n of a "black box" a s s o c i a t i v e mathematical model; 4. Transformation and manipulation of the. model t o provide a l t e r n a t i v e f o c i and p e r s p e c t i v e s of the process. In the f i r s t stage of the procedure a r e s e a r c h problem i s s t a t e d and the r e l e v a n t system i s d e f i n e d . In t h i s stage b a s i c r e l a t i o n s h i p s are modelled by analogy to the r e a l world, i . e . i n the model one can d i r e c t l y i d e n t i f y corresponding mechanisms of behaviour to those found i n the r e a l world (mechanistic approach). T e s t s of both components of behaviour as well as h o l i s t i c p a t t e r n s i n terms of correspondences t o r e a l world data are necessary to ensure v a l i d i t y of the model. In the second stage, an a n a l y s i s of model s t r u c t u r a l a t t r i b u t e s must be conducted to i d e n t i f y p o s s i b l e c h o i c e s cf a n a l y t i c s t r u c t u r e compatible with the s i m u l a t i o n and t h e , d e s i r e d focus. When a t e n t a t i v e d e t e r m i n a t i o n of a compatible mathematical model has been made, one f a c e s an i n f e r e n c e problem; how to estimate the parameters of the mathematical (black box) model. The problem i s o f t e n compounded by the t e c h n i c a l and economic r e s t r i c t i o n on 106 sample s i z e of s i m u l a t i o n runs which are a v a i l a b l e f o r the e s t i m a t i o n of parameters. One must note a l s o t h a t t r a d e - o f f r e l a t i o n s h i p s e x i s t between the ease of parameter e s t i m a t i o n , second l e v e l model v a l i d a t i o n , and the r i c h n e s s of p o s s i b l e a n a l y t i c m a nipulation p e r m i s s i b l e with, the model. . The f i n a l stage employs v a r i o u s mathematical techniques t o manipulate the second l e v e l model (black box model) to o b t a i n a l t e r n a t i v e m a g n i f i c a t i o n s and f o c i of the process s t u d i e d . In t h i s study the mathematical s t r u c t u r e chosen to c a s t the s i m u l a t i o n model i s a Markov p r o c e s s . 8 t h i s choice was d i c t a t e d by the s t o c h a s t i c nature of the s i m u l a t i o n , the res e a r c h problem which focussed upon questions of long run s t a b i l i t y and convergence, and the number of powerful a n a l y t i c procedures which are a v a i l a b l e f o r an examination of both long-term and short-term behaviour of a Markov model. In the f o l l o w i n g s e c t i o n s the procedures by which a Markov model,can be o b t a i n e d from a s i m u l a t i o n (or the " r e a l system") are developed and a p p l i e d . 8 A Markov process i s any parametric s t o c h a s t i c process { X(T),T>0 } such t h a t f o r any s e t o f n time p o i n t s TKT2<...<Tn i n the index s e t of the process, 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 given v a l u e s of p r i o r X's, depends only cn X(Tn-1), the most recent known value ; i . e . f o r a n y , r e a l numbers X1,...,Xn belonging to 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 T1)=Xn-1 ]. I n t u i t i v e l y , t h i s means that f u t u r e values depend only upon the c u r r e n t s t a t e (independent of the past h i s t o r y l e a d i n g to the c u r r e n t s t a t e ) . For a thorough treatment of Markov processes see F e l l e r (1957) or Parzen (1962). 107 The Develo£ment Of B i r d F l o c k i n g Markov Models , In order to model a system (in t h i s case a s i m u l a t i o n model) as a Markov process we proceed i n the f o l l o w i n g way. 1. I d e n t i f i c a t i o n of the s t a t e s S i . In g e n e r a l more than one s e t of s t a t e s might be s e l e c t e d as r e p r e s e n t i n g the system. In f a c t i t may be v a l u a b l e to c r e a t e s e v e r a l d i f f e r e n t Markov models by using d i f f e r e n t s t a t e s and comparing the r e s u l t s . In some cases i t may be that a model using very few s t a t e s i s s u f f i c i e n t to capture the e s s e n t i a l p o i n t s cf the system while i n other cases a simple model may be inadequate. 2. E s t i m a t i o n of the t r a n s i t i o n p r o b a b i l i t i e s P i j . These estimates may be based upon e i t h e r theory or data. For t h e o r e t i c a l ( s t r u c t u r a l ) reasons some t r a n s i t i o n s may be impossible (p=0) or c e r t a i n (p=1) . Others may be d i c t a t e d by the s t r u c t u r e of the system, or s e v e r a l may be r e q u i r e d to be equal. The remainder are e s timated from data by means d i s c u s s e d i n a l a t e r section.,, 3. V a l i d a t i o n o f the model. One must f i r s t demonstrate that the t r a n s i t i o n counts are not generated by a random process which i s independent of the c u r r e n t s t a t e - i . e . t h a t a Markov model i s necessary. Then one must demonstrate t h a t the t r a n s i t i o n counts are independent of the system h i s t o r y preceding the c u r r e n t s t a t e - i . e . t h a t a Markov model i s s u f f i c i e n t . In a d d i t i o n our purposes w i l l r e q u i r e us to show a) that . no c o n s i s t e n t b i a s i n e s t i m a t i o n of t r a n s i t i o n p r o b a b i l i t i e s i n t o a given s t a t e e x i s t s - i . e . t h a t the model 108 c o r r e c t l y p r e d i c t s mean s t a t e r e s i d e n c e time; and b) that the t r a n s i t i o n p r o b a b i l i t i e s do not change ever time. (time homogeneity) f o r the.time h o r i z o n i n v o l v e d i n the s i m u l a t i o n (20 simulated minutes). 4. 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 of the v a l i d a t i o n t e s t s may be used to decide upon the degree of s i m p l i f i c a t i o n which i s p o s s i b l e 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 establishment of t r a n s i t i o n p r o b a b i l i t i e s . Given an .adequate Markov model, one can o b t a i n v a r i o u s a n a l y t i c r e s u l t s , i n c l u d i n g long-term r e s i d e n c e times, the b i a s introduced i n t c residence times by p a r t i c u l a r s t a r t i n g s t a t e s , and s e n s i t i v i t y a n a l y s i s on t r a n s i t i o n p r o b a b i l i t i e s and other parameters. I ^ s n t i f i c a t i o n Of S t a t e s Our major i n t e r e s t i s i n l e a r n i n g more about the f e e d i n g success f o r v a r i o u s prey d i s t r i b u t i o n s , so the Markov model w i l l not i n c l u d e any e x p l i c i t c o n s i d e r a t i o n c f the b i r d ' s movements and l o c a t i o n s . As the s i m u l a t i o n model c o n s i d e r s a c t i v i t i e s on a 15 second b a s i s , we w i l l use t h a t time i n t e r v a l here. The s i m p l e s t p o s s i b l e Markov model then i s a, two s t a t e model: S1, the b i r d searched without su c c e s s ; S2 the b i r d searched and captured a prey. In the s i m u l a t i o n the f o l l o w i n g d i s t i n c t i o n s are made: 1. Lea r n i n g - a b i r d which has l e a r n e d something of the c h a r a c t e r i s t i c s of the prey has an improved a b i l i t y a t l o c a t i n g such a prey, a "prey p r e f e r e n c e " (e.g., search image) - two 109 s u c c e s s i v e c a p t u r e s to form a prey p r e f e r e n c e ; 2. Memory span - a prey p r e f e r e n c e must be r e i n f o r c e d with s u f f i c i e n t frequency t o be r e t a i n e d - prey preference l o s t i n 8 p e r i o d s . i f not r e i n f o r c e d ; 3. Copying - a b i r d i n a f l o c k may ccpy another, reducing the reinforcement necessary to form a prey p r e f e r e n c e - o n e capture while copying i s s u f f i c i e n t t o form a prey p r e f e r e n c e ; 4. Handling time - a b i r d which captures.a prey must spend time h a n d l i n g i t before resuming the search f o r mere prey - two p e r i o d s . Taking a l l f o u r d i s t i n c t i o n s i n t o account, we o b t a i n the s t a t e space given i n Table XXII. In the case of birds< not f l o c k i n g s t a t e E i s e l i m i n a t e d . Smaller models can be c o n s t r u c t e d by i g n o r i n g some of p o i n t s 1-4. These models may be i n s t r u c t i v e i n he l p i n g tc e s t a b l i s h the r e l a t i v e importance of the d i s t i n c t i o n s . Thus, b e s i d e s the 2-s t a t e and the 15-state models a l r e a d y given, we could develop a 4-st a t e model (1), 5 r s t a t e model (1,3), 11-state mcdel (1,2,3), 4-state model ( 4), 8-state model (1,4), and 9-state model (1,3,4). In t h i s paper we.will c o n s i d e r j u s t two models, a l a r g e one (15 s t a t e s ) f o r f l o c k i n g b i r d s (14 f o r no n f l o c k i n g ) and a s m a l l one, the 5-state model (4 f o r n o n f l o c k i n g b i r d s ) i n which c o n s i d e r a t i o n s of handl i n g time and memory span are dropped -l e a v i n g s t a t e s A,B,C,D,E. Comparing these twc models may o f f e r 1 10 State A B i r d s e a r c h i n g without prey preference (PP) f a i l e d to make a capt u r e . B1,...,B7 B i r d i n s t a t e Bn was se a r c h i n g with PP and f a i l e d to capture a prey f o r n s u c c e s s i v e p e r i o d s . The ei g h t h f a i l u r e r e s u l t s i n l o s s of PP and r e t u r n to s t a t e A. C0,C1,C2 B i r d s e a r c h i n g without PP captured a prey (CO), handled i t the f i r s t p eriod (C1), the second p e r i o d (C2) . D0,D1,D2 B i r d s e a r c h i n g with PP captured a prey c r handled i t (as above) . E B i r d s e a r c h i n g without PP made no capture but i s copying another b i r d f o r the next p e r i o d . Table XXII.: S t a t e space f o r the Markov model. 111 some i n s i g h t i n t o the importance of the time lags c r e a t e d by the handling time and memory span. Table XXIII g i v e s the p o s s i b l e t r a n s i t i o n s f o r the l a r g e model and any known bounds and t h e i r p r o b a b i l i t i e s . A s i m i l a r t a b l e may be d e r i v e d from t h i s one f o r the s m a l l model by simply e l i m i n a t i n g the d i s t i n c t i o n s w i t h i n s t a t e s B,C,D. The t a b l e g i v e s t h i r t y p o s s i b l e t r a n s i t i o n s of which four are c e r t a i n . The remaining twenty-six must be estimated from s i m u l a t i o n output. The next s e c t i o n w i l l d e a l with that problem. E s t i m a t i n g T r a n s i t i o n P r o b a b i l i t i e s Host a p p l i c a t i o n s of Markov processes to date have assumed that the t r a n s i t i o n p r o b a b i l i t i e s of the process are e x a c t l y known. However, a more r e a l i s t i c approach should express knowledge about t r a n s i t i o n p r o b a b i l i t i e s i n the form of p r o b a b i l i t y d i s t r i b u t i o n s over unknown parameters of the process from which p o i n t e s t i m a t e s can be o b t a i n e d . S t a t i s t i c a l i n f e r e n c e concerning Markov processes has been i n v e s t i g a t e d s i n c e the e a r l y 1950»s. A review of e a r l y s t u d i e s i s given by B i l l i n g s l e y (1961). The need to c o n s i d e r d e c i s i o n s made on the b a s i s of s m a l l samples and to u t i l i z e p r i o r i n f o r m a t i o n about the s t u d i e d process has l e d to the a p p l i c a t i o n c f . t h e Bayesian approach to the e s t i m a t i o n of parameters of Markov chains ( S i l v e r 1963; Martin 1967; Lee, Judge and Z e l l n e r 1968; Roussas 1965; Vickson and v e r t i n s k y 1974). 112 T r a n s i t i o n P r o b a b i l i t y Comments A —>A P1 P 1 8 ° = 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 run. A —>C0 P2 P2C016 based upon c h a r a c t e r i s t i c s of the s i m u l a t i o n and the prey d e n s i t i e s used. A — > E P3 P3 should vary c o n s i d e r a b l y with the degree of prey clumpedness. B1—>B2 PU B1—>D0 P5 We expect P5>P2 B2~>B3 P6 B2—>D0 P7 We expect P5>P7>P2 B3—>BU P8 B3—>D0 P9 We expect P7>P9>P2 BU—>B5 P10 B4 — >D0 P11 We expect P9>P11>P2 B5~>B6 P12 B5—>D0 P13 We expect P11>P13>P2 B6—>B7 P l l B6~>D0 P15 We expect P13>P15>P2 B7~>A ' P16 B7—>D0 P17 We expect P15>P17>P2 B7—>E P18 We expect P18>P3 CO—>C1 P19=1 Only p o s s i b l e t r a n s i t i o n . C1—>C2 P20=1 Only p o s s i b l e t r a n s i t i o n . C2~>A P21 C2—>D0 P22 We expect P22>P5 C2-->E P23 We expect P23>P18 Do—>D1 P24=1 Only p o s s i b l e t r a n s i t i o n . D1—>D2 P25=1 Only p o s s i b l e t r a n s i t i o n . D2~>B1 P26 D2~>D0 P27 We expect P27>P22; P27 shculd vary c o n s i d e r a b l y with the prey d i s t r i b u t i o n . E —>A P28 E ~>D0 P29 We expect P29>P5 E ~>E P30 We expect P30>P23 Table 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 (15 seconds) and bounds on the t r a n s i t i o n p r o b a b i l i t i e s . 113 For the c r e a t i o n of a second l e v e l Markov model cf a complex s i m u l a t i o n , the Bayesian approach i s e s p e c i a l l y a p p r o p r i a t e . In the Bayesian approach one assumes seme p r i o r knowledge about the unknown parameters expressed i n the form of p r o b a b i l i t y d i s t r i b u t i o n s over these parameters, which are considered as random v a r i a b l e s (Savage 1962). The bases for s u c h . d i s t r i b u t i o n s are many. For example, on the b a s i s of a n a l y s i s of the mathematical e x p r e s s i o n s embedded i n the s i m u l a t i o n model, i t i s o f t e n p o s s i b l e t o . o b t a i n some bounds on v a r i o u s t r a n s i t i o n p r o b a b i l i t i e s . S i m i l a r l y , bounds may be i n f e r r e d from b i o l o g i c a l considerations.„ These p r i o r p r o b a b i l i t y d i s t r i b u t i o n s are modified through the use of Eayes' r u l e 9 when o b s e r v a t i o n s of the process are made (in cur case a sample of s i m u l a t i o n r u n s ) . As a consequence of Eayes' r u l e a p o s t e r i o r d i s t r i b u t i o n i s o b t a i n e d . Using the p o s t e r i o r d i s t r i b u t i o n and a s p e c i f i e d l o s s f u n c t i o n , p o i n t e s t i m a t o r s can be i n f e r r e d f o r the . t r a n s i t i o n p r o b a b i l i t i e s sc as to minimize the l o s s f u n c t i o n . The c h o i c e of a l o s s f u n c t i o n depends on the p a r t i c u l a r problem. For example, choosing the l e a s t square d e v i a t i o n c r i t e r i o n , one o b t a i n s an unbiased estimator which minimizes the l i k e l i h o o d of l a r g e d e v i a t i o n s frcm the e s t i m a t o r on e i t h e r s i d e . In c o n t r a s t , some asymmetrical T o s s f u n c t i o n s can be be used t o compensate f o r suspected over or under 9 P (A | B)=P (B | A) *P (A)/P (B) where P(A|B) i s the c o n d i t i o n a l p r o b a b i l i t y of A given the occurrence of B. 114 r e p r e s e n t a t i o n s of a p a r t i c u l a r t r a n s i t i o n i n an observed sequence of t r a n s i t i o n s . In e s t i m a t i n g the t r a n s i t i o n p r o b a b i l i t i e s from the s i m u l a t i o n model, a mul t i d i m e n s i o n a l beta d i s t r i b u t i o n was assumed as a p r i o r d i s t r i b u t i o n over the t r a n s i t i o n p r o b a b i l i t i e s of a s i n g l e row cf the t r a n s i t i o n matrix. The i n t e r v a l s over which the d i s t r i b u t i o n was defined depended on bounds i n f e r r e d from the a n a l y s i s of the s i m u l a t i o n model or p r i o r b i o l o g i c a l knowledge. In a d d i t i o n , we have i n c o r p o r a t e d p r i o r knowledge on order r e l a t i o n s h i p s among c e r t a i n p r o b a b i l i t i e s i n the t r a n s i t i o n matrix. We s h a l l c o n s i d e r now i n d e t a i l s e v e r a l examples of d e r i v i n g l e a s t square Bayesian e s t i m a t o r s which were used i n our modelling procedure. In g e n e r a l Bayesian e s t i m a t o r s are derived as f o l l o w s . We wish to estimate V = (P1,.., ,Pr) . We have c e r t a i n p r i o r e x p e c t a t i o n s f o r P which may be expressed by 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 F (P). We make N. o b s e r v a t i o n s , [N] = ( L 1 , . . . , L r ) , where I i i s the number of o b s e r v a t i o n s P i r *** 8 {Li} = N. *** i=1 From these we d e r i v e a p o s t e r i o r d e n s i t y f u n c t i o n H(P|[N]) = K([N])G (L1,...,Lr|P)F{P) 115 r *** where g = 6 { P i * * L i J *•* i=1 and K ([N ]) i s a n o r m a l i z a t i o n c o n s t a n t . F i n a l l y , we must s e l e c t a l o s s f u n c t i o n L(P,P) which expresses the l o s s of o b t a i n i n g P f o r a true value cf P. Then the Bayesian estimate P i s found by minimizing V(P;[N]) = J L (P,P)H (P|[N])dF. I n our example: a) the P i ' s w i l l be the t r a n s i t i o n p r o b a b i l i t i e s ; b) the l o s s f u n c t i o n w i l l be the sum of the sguared e s t i m a t i o n e r r o r s , i . e . , r *** L (P,P) = « { (Pi - P i ) z } . *** i=1 To minimize V(P;[N]) i t i s necessary and s u f f i c i e n t that av (P; [N ] ) / 3 P i = 0 and t h a t a l l the r o o t s of the determinant D (x) be p o s i t i v e , where 116 D(x) = V11-X V12 V21 V22-X V1r V2r |vr1 vr2 ... Vrr-x and V i j = a*V (F;[ N ])/aPi3Pj. Since V ( P ; [ N ] ) / a P i = J ... J* {2 (Pi - Pi) H (P | [ N ]) d p i . . . dPr }, V i j = 0 f o r i # j = 2 f o r i= j . Thus, D (x) = ( 2 - x ) * * r , whose only r o o t (x=2) i s p o s i t i v e . Now av (P;[N ] ) / 3 P i = 0 i m p l i e s P i = . . . J * {PiH(P|[N]) dPl...dPr} = E ( P i ) , the expected value of P i f c r p o s t e r i o r d e n s i t y f u n c t i o n H(P|[N]). The o n l y problem which remains i s to s e l e c t the p r i o r d i s t r i b u t i o n and perform the necessary i n t e g r a t i o n s . Since the P i are elements i n a Markov matrix, there w i l l be a number of i n t e r r e l a t i o n s between them (row sums = 1). The P i ' s may be s p l i t i n t o groups such that each P i i s independent of those P»s i n other groups and i s not independent of those P's i n i t s own group. Then 117 K ***** F (P1,.., ,Pr) = * * { Fi[P1 ( i ) , . . . ,PKi (i) ]}, * * i=1 where K i s the number.of groups, there are Ki members i n group i ( i = 1 , . . . , K ) l a b e l l e d P1 ( i ) , . . . ,PKi (i) with F i the p r i o r d e n s i t y f u n c t i o n f o r group i . Thus, estimates of each P i depend on l y upon r e s u l t s w i t h i n i t s group. Choices f o r F i were r e s t r i c t e d to the f o l l o w i n g : 1. F i = 1. T h i s may be con s i d e r e d the complete ignorance case. 2. F i = C f o r (P1 ( i ) , . .. ,PKi (i)} & R i £ [0,1 ]**Ki = 0 f o r (P1 (i) ,... ,PKi (i) ) ? B i , where C i s the i n v e r s e of the volume of R i . This case allows 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 . 3. An e x t e n t i o n of case 2 i s to allow F i t c take cn s e v e r a l d i f f e r e n t p o s i t i v e constant values corresponding to d i f f e r e n t r e g i o n s of the p r o b a b i l i t y space. For example, we might think that P23 ( r e f e r r i n g to the 15-state model) i s most probably between .2 and .8, but a l s o t h a t i t might.be outside that range. Then a p o s s i b l e c h o i c e f o r F would be F = .25 0<P<.2 = 1.5 .2<P<.8 = .25 .8<P<1.0. 118 4. another s p e c i a l , case which extends case 2 i s the admission of p r i o r order estimates on the p r o b a b i l i t i e s , i . e . , P1>P2. Case 1: We w i l l f i r s t c o n s i d e r the b i n o m i a l case i n which there are j u s t two p r o b a b i l i t i e s i n a row, c a l l them P and Q (P+Q=1) with 1 o b s e r v a t i o n s of P and n - 1 of Q. Cp** (1*1 )*Q** ( n - 1 ) dP I (1+1, n - 1 ) Then P = E (P) = f l P * * l * Q * * ( n - 1 ) dP 1 ( 1 , n - 1 ) Now I ( l , n - l ) = J \ > * * 1 * (1-P) ** (n - 1 ) dP = 1 ! ( n - 1 ) !/(n+1) ! = B ( l , n - l ) So 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) . r . *** For the m u l t i n o m i a l case with 9 {Pi} = 1 *** i=1 we again have I (L1 .+ 1 ,L2,... ,1s) P = •— I(L1,L2, ...,LS) with I (L1,. . . ,Ls) - X 1 r1-P1 1-P1-.. .-Ps-1 dP1*P1**L1 f ... f dPs-1*Ps-1**Ls-1*Ps**Ls 0 J 0 J 0 and Ps = 1 - P1 - ... - Ps-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, P i = ( L i + 1)/ (n + S) . Case 2: Again we begin with the b i n o m i a l case. F(P) = 1/(b-a) 0<a<P<b<1 and a*b = 0 otherwise. Then 1(1,n-1) = | P * * l * (1-P) ** (n-1) *dP a = B(l,n-1) n-1 *** *** i=0 so P = I (1 + 1, n-1) 1(1, n-1) *P** (l+i+1)* (1-P) ** ( n - l - i ) n-1 *** / n+1 B(l,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 For the mul t i n o m i a l case with a s i n g l e r e s t r i c t i o n a<P1<b, 120 f i n d the same formula as above with n r e p l a c e d by n+S-2. With more r e s t r i c t i o n s on the P i , we o b t a i n a more complex region of i n t e g r a t i o n , but the same p r i n c i p l e s h old. Case 3: Consider the case with a s i n g l e r e s t r i c t i o n F = C1 0<PKa = C2 a<Pl<t = C3 b<P1<1. Then c a l l i n g n-L1+S-2 = j and dropping the s u b s c r i p t from L1, we f i n d j L+1 ***/n + S\ P1 = {C3 + (C2-C3) 8 j |*b** (L + i + 2) * (1-b) ** ( j - i ) n+S * * * \ j - i / i=0 *** + (C1-C2) ® *** i=0 n+S *a** (L+i+2) * (1-a) ** ( j - i ) )/ 3 - i 1 ***/n+S-1' {C3 + (C2-C3) 8 ( |*b** (L + i+1) * (1-b) ** ( j - i ) *#* i=0 ( \J j ***/n+S-1\ + (C1-C2) « | *a«* (l+i + 1)* (1-a) * * . ( j - i ) }. ***\ j - i i=0 Case 4: Consider 1 o b s e r v a t i o n s of P n-1 " of 1-P 121 k » of Q m-k " of 1-Q. Let us f u r t h e r r e g u i r e that P<Q. Then the p o s t e r i o r d e n s i t y f u n c t i o n i s H = C[ p * * l * (1-P) ** (n-1) ][ Q**k* (1-0) ** (m-k) ] f o r 0<P<Q<1 = 0 otherwise. J dP*P** (1 + 1)* (1-P) ** (n-1) P dQ*Q**k* (1-Q)** (m-k) 0 J P p -• — J1 ap*p**l* (1-P) ** (n-1) f dQ*Q**k* (1-Q) ** (m-k) 0 m-k * * * i m+1 \ B (1+1,n-1) - « B(l+k+i+2, n+m-l-k-i) *( *** \m-k-ij i=0 p . . m-k *** . m+1 \ B ( l , n - 1 ) -s » B(l+k+i + 1,n+m-l-k-i) * j *** Im-k-i i=0 In our model the f o l l o w i n g p r i o r d i s t r i b u t i o n was used. Each r e l a t i o n s h i p given below was deemed to hold 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 a d d i t i o n we know th a t P2<1/64. To apply the above r e l a t i o n s h i p s i n c a l c u l a t i n g each P i was co m p u t a t i o n a l l y 122 untenable. Consequently, we r e s o r t e d to the f o l l o w i n g approximations. A l l the P i were estimated a p p l y i n g none of the above r e s t r i c t i o n s . Then those P i which v i o l a t e d a r e s t r i c t i o n were recomputed t a k i n g the v i o l a t e d r e s t r i c t i o n s i n t o account. These new estimates of the P i ' s were used as p r i o r s , and the steps were repeated u n t i l the P i ' s converged (two s u c c e s s i v e estimates d i f f e r e d by l e s s than .001 i n any P i ) . Estimates of the t r a n s i t i o n p r o b a b i l i t i e s were made from 10 runs of 80 p e r i o d s each f o r f l o c k s of 20 b i r d s and fcr.200 runs of 80 periods each f o r s o l i t a r y b i r d s . Thus, a t o t a l of 16,000 t r a n s i t i o n s were used t o estimate the t r a n s i t i o n , p r o b a b i l i t i e s f c r each of the four models — 15-state (with f l o c k i n g ) , In-s t a t e (without f l o c k i n g ; s t a t e E removed), 5-state (with f l o c k i n g ) , and 4 - s t a t e (without f l o c k i n g ; s t a t e E removed). S e v e r a l methods are a v a i l a b l e to t e s t the v a l i d i t y of the Markov model as an approximation to the system being modelled. F i r s t one must demonstrate t h a t the t r a n s i t i o n c c u n t s are not random - i . e . that i n g e n e r a l N i j * Ni.*N.j 123 where Ni. = r *** ® 3=1 {Nij}, N.j = • r *** ® *** i=1 {Nij 3 and N = r *** © *** {Ni. } r *** ® *** { N . j } ( i , j=1,2 r ) . i=1 j=1 T h i s i s s i m i l a r t o an r by r contingency t a b l e with s u i t a b l e m o d i f i c a t i o n to e l i m i n a t e those t r a n s i t i o n s which are impossible (Fienberg 1972). R e s u l t s of t h i s a n a l y s i s are presented i n Table XXIV (see Appendix C f o r a worked example). An a l t e r n a t i v e method of v a l i d a t i o n u s i n g an i n f o r m a t i o n measure i s presented by C h a t f i e l d and lemon (1972). The second, part of the v a l i d a t i o n i n v o l v e s demonstrating that a Markov model with constant t r a n s i t i o n p r o b a b i l i t i e s i s s u f f i c i e n t to account f o r 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 above. I n t u i t i v e l y , t h i s means t h a t we must show t h a t the c u r r e n t s t a t e i s s u f f i c i e n t t o p r e d i c t the next s t a t e . T h i s i s a two stage process. F i r s t the data must be t e s t e d f o r time homogeneity, i . e . t e s t the h y p o t h e s i s . t h a t P i j i s constant over time. T h i s was. done by s p l i t t i n g the data i n q u a r t e r s and comparing the P i j f o r a l l q u a r t e r s of the time u t i l i z i n g a. c h i -squared t e s t ( B i l l i n g s l e y 1961). Results of these t e s t s i n d i c a t e d that the t r a n s i t i o n p r o b a b i l i t i e s were constant over time i n a l l cases (p>.10) 12a Clumping 0.07 0.74 1.a8 5. 2a 9.a8 f l e c k i n g X* df=5 28.7 (<10~*) 443 (<10-S) 932 (<10 - s ) 573 (<10-5) 2019 (<10 - s ) n o n f l o c k i n g .186 (>.1) 3.35 (<.070) 24.5 (<10-5) 31.0 (<10-5) 16.7 (<10-*) X2 df = 1 Table XXIV,: 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 l e v e l s t e s t i n g 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; i e . P(A|B) = P(A), u n l e s s c o n s t r a i n e d to be 0. The t e s t s have very few degrees of freedom s i n c e so many of the t r a n s i t i o n s are i m p o s s i b l e . 125 except f o r t r a n s i t i o n s i n t o s t a t e E which i n c r e a s e d r a p i d l y i n the f i r s t f i v e minutes and remained constant ( s t a t i s t i c a l l y ) t h e r e a f t e r . T h i s e f f e c t i s n o . s u r p r i s e s i n c e i m i t a t i o n (entry i n t o s t a t e E) i s i m p o s s i b l e u n t i l a -bird*s.neighbor . (within 15 f e e t ) has made a capture. R e p l a c i n g the assumption of constancy i n t r a n s i t i o n s i n t o . E with a l i n e a r i n c r e a s e from zero to the a p p r o p r i a t e constant f o r the f i r s t p eriod and constancy f o r the second, t h i r d and f o u r t h g u a r t e r s ( f i v e simulated minutes each) y i e l d e d s a t i s f a c t o r y r e s u l t s . To t e s t whether a given Markov model i s s u f f i c i e n t , twc methods are a v a i l a b l e . In both cases, a Markov model i s generated from a s e t of data (set 1). T h i s model's p r e d i c t i o n s are then compared. with a second set of data (set 2)., A c h i -squared t e s t i s performed to compare t r a n s i t i o n counts, i n data se t 2, with those, p r e d i c t e d by the Markov model, and a second chi-sguared t e s t i s performed to compare. ,residence times f o r each s t a t e . i n data s e t 2 with.those p r e d i c t e d by the model (in the f o l l o w i n g s e c t i o n . a method w i l l be presented f o r o b t a i n i n g expected r e s i d e n c e .times f o r a given .sequence length and given s t a r t i n g s t a t e ) . R e s u l t s of these t e s t a r e . given below i n T a b l e s . XXV . (compares, t r a n s i t i o n p r o b a b i l i t i e s ) and XXVI (compares r e s i d e n c e t i m e s ) . I t i s noteworthy t h a t s e v e r a l of the models which appeared to be adequate i n accounting f o r t r a n s i t i o n f r e q u e n c i e s (Table XXV) were u s e l e s s f c r p r e d i c t i n g s t a t e residence> times (Table XXVI). T h i s apparent discrepancy arose from the p o s i t i v e c o r r e l a t i o n s between the t r a n s i t i o n 126 Clumping 0.07 0.74 1.48 5.24 9.48 f l o c k i n g 14. 3 9.0 7.4 27.0 22.6 15x15 df=15 (.54) (.88) (.95) (.03) (.09) n o n f l o c k i n g 2.67 13.7 17.1 19.9 15.2 14x14 df=10 (.99) (.18) (.07) (.03) (. 12) f l o c k i n g 20. 2 5.5 5.9 22.4 18.7 5x5 df=11 (.04) (.90) (.88) (.02) (.06) n o n f l o c k i n g 2.47 6.2 -12.0 15.6 4.40 4x4 df=6 (.87) M 0 ) (.06) (.02) (.62) Table XXV.: Comparison of expected number of t r a n s i t i o n s as given by the Markov model (from data set 1) with the t r a n s i t i o n counts from data set 2. Numbers i n the t a b l e are chi-sguared values, with a s s o c i a t e d s i g n i f i c a n c e l e v e l s below i n parentheses, Prey clumpedness i s measured with L l o y d ' s index of mean crowding. 127 Clumping 0.07 0.74 1. 48 5. 24 9.48 f l e c k i n g 1. 27 9. 43 1. 96 14 .1 12.4 15x15 df=14 (>.99) (.80) (> .99) (. 45) (.54) n o n f l o c k i n g 1. 50 16.9 14 .2 17 .9 3.37 14x14 df=13 (>.99) (.20) (. 36) (. 16) (>.99) f l o c k i n g 70.8* 174.7* 77 .6* 44 .8*. 11.5 5x5 df=4 (<.00001) (<. 00001) (< .00001) (< .00001) (.021) n o n f l o c k i n g .554 11.77* 11 .51* 14 .13* .194 4x4 df=3 (.90) (.008) (. 009) (. 003) (.98) Table XXVIj. Comparison of expected s t a t e r e s i d e n c e times as given by the Harkov model (from data s e t 1) with the re s i d e n c e times from data set 2, g i v i n g c h i -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 l e v e l s , i n parentheses. S t a r r e d values i n d i c a t e t h a t the 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 f r e q u e n c i e s i n t o a s i n g l e s t a t e j , such that the p r e d i c t i o n s of t r a n s i t i o n s i n t o j are a l l biased in., the same d i r e c t i o n . Although these models.must be r e j e c t e d f o r f u r t h e r a n a l y s i s (the s t a r r e d e n t r i e s i n Table XXVI), i t i s i n s t r u c t i v e t c examine the causes f o r f a i l u r e : 1. In the s m a l l models (5x5) with f l o c k i n g , the e l i m i n a t i o n of handling time and prey preference memory r e s u l t e d i n a s u b s t a n t i a l . o v e r e s t i m a t e of i m i t a t i o n (state E) 2. In the. in t e r m e d i a t e ranges of prey clumping, the lack of handli n g time i n the s m a l l models (4x4 and 5x5) r e s u l t e d i n an overestimate of the. prey capture by a b i r d with a prey p r e f e r e n c e ( s t a t e 0). A n a l y s i s Of The Markov Models In t h i s s e c t i o n , we w i l l present a b r i e f , mathematical d i s c u s s i o n of Markov processes and some a n a l y t i c procedures which are u s e f u l . i n a n a l y s i n g a Markov model. The l a t t e r w i l l be i l l u s t r a t e d using the 4x4 model with uniform prey d i s t r i b u t i o n . For our purposes, we w i l l c o n s i d e r a Markov process with the f o l l o w i n g p r o p e r t i e s : 1. A d i s c r e t e time parameter . (t=0,1,2,...) ; 2. A d i s c r e t e s t a t e . s p a c e S=[S1,S2,...,Sr ]; 3. S t a t i o n a r y t r a n s i t i o n p r o b a b i l i t i e s , i . e . P[ Xn=Sj ,X (n-1)=Si] = P i j = constant (n=1,2,..,). 129 r *** Note t h a t « { P i j ) = 1. *** j=1 4. Ergodic s t a t e s -.- each s t a t e can be reached from each other s t a t e i n at most r t r a n s i t i o n s . Such a Markov p r o c e s s . i s commonly termed a Markov c h a i n . Let us write X(n) as a vec t o r X (n) = (X1,X2,.. .,Xr) (n) where X i = P[X(n).= S i ] and S i i s the r - v e c t o r with 1 i n column i and O's elsewhere. r *** Thus, ® { Xi} = 1. *** i=1 The vector X(n) expresses the occupancy p r o b a b i l i t i e s f o r each of the s t a t e s a t time t=n. l e t t i n g P = [ P i j ] , the matrix of t r a n s i t i o n p r o b a b i l i t i e s , we have X(n) = X(n-1)P = X (n-2)P**2 = X(0)P**n. Just as P i s the matrix of one step t r a n s i t i o n p r o b a b i l i t i e s , P (n) i s the matrix of n-step t r a n s i t i o n p r o b a b i l i t i e s . The s m a l l model without f l o c k i n g has the f o l l o w i n g t r a n s i t i o n matrix: 130 A = .9855 0 .0145 0 .1029 .8330 0 .0641 .3270 0 .6670 .0060 .. 0 .3150 - 0 .6850 where s t a t e s 1-4 correspond to s t a t e s A-D. The matrix P can be decomposed i n t c a sum c f con s t a n t matrices each m u l t i p l i e d by an eige n v a l u e of P. That i s , • r r *** *** P•= © {Li*¥i} and P <n) = © { L i * * n * Y i } . *° *** ~ *** ~ i=1 i=1 where L i are the eigenvalues c f P ( s o l u t i o n s cf the equation Vi*P = L i * P ) , and Y i are a s s o c i a t e d matrices with Yi*Y j = 0 f o r i # j = Y i f o r i = j . The Y i are given by 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 a column vector and a row vector) are s o l u t i o n s of the equations p * U i = P * L i and Vi*P = L i * P and C i - 1 = V i * U i (the inner product cf Oi and V i ) . 1 0 I f zero i s an eigenvalue of m u l t i p l i c i t y s, i . e . (I1 = L2=.. ,=Ls=0;Ls+ 1, . .. ,Lr#0) , . r. . . . . . . then P(n) = 0 {Li * * n Y i } + Zn, *** ~ i=s+1 where Zn's are constant matrices f o r n=0,1,...s-1 and Zn=0 f o r n>s. For the case where a nonzero eig e n v a l u e has m u l t i p l i c i t y g r e a t e r than one see Appendix D. 131 ( F e l l e r , 1957 pp. 380-84) New f o r P a Markov matrix we have 1. |Li|<1 (i=1,2 , . . . , r ) ; 2. L i = 1 f o r p r e c i s e l y one value cf i , (i=1 t • • • i r) • Without l o s s of g e n e r a l i t y we can say 11 = 1. Then P(n) = Y1 + r *** © *** {Lk**n*Yk}. k=2 I f |Lk|<1 (k=2 ,... ,r) , then P (n) — > Y1 as n ~ > i n f i n i t y . Thus, Y1 g i v e s the steady s t a t e t r a n s i t i o n p r o b a b i l i t i e s f c r each s t a r t i n g s t a t e . For an e r g o d i c process ( t r a n s i t i o n from any s t a t e i to any s t a t e j i n r steps or l e s s has p o s i t i v e p r o b a b i l i t y ) , the rows of Y1 are i d e n t i c a l . Each element Y i j of Y1 may a l s o be i n t e r p r e t e d .as the long-term p r o b a b i l i t y of f i n d i n g the system i n s t a t e S j given t h a t the system began i n S i . , The terms Lk**n*Yk (k=2,...,r) r e p r e s e n t t r a n s i t o r y behavior f o r |Lk|<1 and c y c l i c behavior f o r |Lk|=1. In the l a t t e r case, rows of Y1 s t i l l r e p r e s e n t mean r e s i d e n c e times; however, there i s no.longer a steady s t a t e but r a t h e r c y c l e s of p e r i o d m, where m i s the s m a l l e s t i n t e g e r such that Lk**m=1. The decomposition of our example matrix proceeds as f o l l o w s . F i r s t we s o l v e Vi*A = L i * A . In g e n e r a l t h i s eguation 132 has r s o l u t i o n s where A i s an r b y r m a t r i x . 1 1 The s o l u t i o n s are: L1 = 1 .0000 V1 = (" 13.119, - .033, -- .571, -.018) L2 = .6533 V2 •= (" 1.086, - . 150, 1.150, .0 86) L3 = .5984 V3 = (- 2.620, .808, .055, -.602) L4 = .9188 V4 = ( 1.474, -1.225, .085, -.334) 0728 -.0361 - .0048 .0032 V - i = • 0728 -.0081 .3310 -.5951 *™ • 0728 .8282 .1282 -.0149 ™" • 0728 .0809 -1.2040 -.8019 F i n a l l y .9547 .0024 .0415 .0013 A(n) = .9547 .0024 .0415 .0013 .9547 .0024 .0415 .0013 .9547 .0024 .0415 .0013 .0393 .0054 -.0416 -.0031 +.6533**n .0088 .0012 -.0094 -.0007 .8992 -.1242 .9526 .0709 -.0878 -.0121 .0931 .0069 .0013 -.0039 -.0003 .0029 +.5984**n -.0866 .2674 .0183 -. 1991 -.0335 . 1036 .0071 -.0771 .3149 -.9726 -.0666 . 7243 .0048 -.0040 .0003 -.0011 +.9188**n -.8770 .7290 -.0505 . 1985 -.0220 .0183 -.0013 .0050 •1.1818 .9823 -.0681 .2675 The above decomposition of P may be used to f i n d the expected r e s i d e n c e times f o r each s t a t e i n a sequence of s t a t e s of l e n g t h n. Let Tij(n)=E[number t r a n s i t i o n s i n t o S j i n n steps | X0=Si] 1 1 See fo o t n o t e 1 0 and Appendix D f o r the cases when there are l e s s than r independent s o l u t i o n s . 133 and T (n) = [ T i j ]. n *** Then T(n) = ® {P**k}. *** k=1 With L1=1, T (n) = n r *** *** 8 8 { L i * * k Y i ) *** *** ~ k=1 i=1 r *** 9 *** i=1 n *** 8 { L i * * k Y i } k=1 ,r *** = nYl + 8 { L i * Y i ( 1 - L i * * n ) / ( 1 - L i ) }. *** ~ ~ i=2 For our example we f i n d t h a t T(80) = 76.507 .153 3.250 .090 66.240 8.833 2.768 2.059 74.383 .319 5. 119 . 179 63.323 9.823 2.635 4.219 Since the s i m u l a t i o n g e n e r a t i n g data set 2 was s t a r t e d i n s t a t e A, we used the f i r s t l i n e of the above matrix to compare with the residence times found i n data s e t 2. Resu l t s The Markov second l e v e l model was used t o examine the dynamics and long run behaviour of f e e d i n g success. These 134 phenomena co u l d not be e c o n o m i c a l l y or adequately answered d i r e c t l y from the s i m u l a t i o n . By employing •. the h i e r a r c h i c a l modelling . method d e s c r i b e d above, a d d i t i o n a l i n s i g h t s were obtained i n t o the value and c o s t of f l o c k i n g f o r d i f f e r e n t given prey d i s t r i b u t i o n s . Whereas .in the previous chapter average d a i l y f e e d i n g rates... and v a r i a n c e i n f e e d i n g among b i r d s were used to compare s t r a t e g i e s , the a n a l y s i s presented here p r o v i d e s an a l t e r n a t i v e vantage by c o n s i d e r i n g e x p l i c i t l y the path of search and f e e d i n g . success and f a i l u r e , through time. T h i s p e r s p e c t i v e permits e x p l i c i t a n a l y s i s of l e a r n i n g r a t e s and long run p o t e n t i a l s . T r a n s i t i o n matrices f o r the f l o c k i n g and n o n f l o c k i n g case were estimated f o r a range of f i v e v a l u e s . o f prey clumping. As i n d i c a t e d above, s i m p l i f i c a t i o n . t o f o u r or f i v e s t a t e s was not g e n e r a l l y p o s s i b l e , so. only r e s u l t s f o r t h e . l a r g e r models (14 and 15.state) w i l l be g i v e n . Examining the t r a n s i t i o n matrices (given i n appendix E) s e v e r a l f e a t u r e s are apparent. As prey clumping i n c r e a s e s : a) the p r o b a b i l i t y of c a p t u r i n g a prey while s e a r c h i n g with a prey, p r e f e r e n c e decreases;, and b) the p r o b a b i l i t y of c a p t u r i n g two prey on s u c c e s s i v e attempts (C2 — > DO and D2 —•> DO) i n c r e a s e s . : These are c o n s i s t e n t with the s p a t i a l e f f e c t of prey clumping. . Comparisons of matrices f o r f l o c k i n g with n o n f l o c k i n g b i r d s r e v e a l s . t h a t . t h e p r o b a b i l i t y of capture f o r a b i r d without prey preference i s s m a l l e r f o r the f l o c k i n g b i r d s , an i n d i c a t i o n of i n t e r f e r e n c e . 135 Repeating the s i m u l a t i o n experiments with a prey p o p u l a t i o n of two t y p e s 1 2 a s i m i l a r s et of r e s u l t s was.found. Again only the l a r g e models were s a t i s f a c t o r y , and the patterns i n the t r a n s i t i o n matrices were, the same, as, those given atove. However, important d i f f e r e n c e s were rev e a l e d i n the a n a l y s i s which f o l l o w s . • When e i t h e r i n d i v i d u a l b i r d s . o r a f l e c k of b i r d s f l i e s to a new area to feed, we,find that the chance o f . a b i r d ' s c a p t u r i n g a prey i n c r e a s e s over time.up to some asymptotic l i m i t . T h i s asymptote, the long run capture r a t e , may be considered to be the f u l l r e a l i z a b l e f e e d i n g p o t e n t i a l of. the b i r d when a l l l e a r n i n g processes ..are completed. The r a t e of. i n c r e a s e i n capture r a t e up to t h i s l i m i t may t h e r e f o r e be con s i d e r e d an i n d i c a t o r of l e a r n i n g , and the curve graphing the approach t o the asymptote a l e a r n i n g curve. The .g e n e r a l form of the l e a r n i n g curve i s a monotonically i n c r e a s i n g . f u n c t i o n beginning at zero at time zero, and bounded above. ..In ;our example we found that an accurate r e p r e s e n t a t i o n of t h i s curve was the. f u n c t i o n f (t) =? a (1 - | L | * * t ) , . where a - i s the asymptote d e r i v e d from the steady, s t a t e p r o b a b i l i t i e s and L i s the . second, . l a r g e s t (dominant) eigenvalue of .the t r a n s i t i o n , matrix . (the l a r g e s t e i g e n v a l u e i s always equal to 1). Examples cf these l e a r n i n g 1 2 Two d i s t i n c t types.were used, n u t r i t i o n a l l y 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 with prey p r e f e r e n c e 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 of type 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 d i s t r i b u t i o n was as i n the previous experiments. 136 curves are graphed i n F i g u r e s 11, 12 and 13. Examination of these F i g u r e s and of Tables XXVII and XXVIII r e v e a l s the f o l l o w i n g p o i n t s : 1.. When there i s a s i n g l e major prey type, f l o c k i n g reduces the values o f both a and L; t h a t i s , by f l o c k i n g the b i r d s reduce t h e i r maximum r e a l i z a b l e f e e d i n g , r a t e , but i n c r e a s e t h e i r l e a r n i n g r a t e . The r e s u l t s of t h i s t r a d e o f f are v i s i b l e i n the F i g u r e s , where we see f l o c k i n g b i r d s i n i t i a l l y f e e d i n g a t a higher average r a t e than n o n f l o c k i n g b i r d s , t u t l a t e r being passed by the n o n f l o c k i n g b i r d s . The d i f f e r e n c e i n l e a r n i n g r a t e i s most prominent a t high values of prey clumping, where e f f e c t i v e sampling o f . t h e environment and convergence on food c o n c e n t r a t i o n s i s important. Conversely, f o r low values of prey clumping there i s l i t t l e gain even f o r s h c r t p e r i o d s of time i n f l o c k i n g so long as only search e f f i c i e n c e s are considered. In a d d i t i o n i t should be noted t h a t the value of r a p i d l e a r n i n g d e c l i n e s as increased, prey d e n s i t y makes i t p o s s i b l e to remain f e e d i n g i n the same area f o r longer periods of time. , 2. When there are two major prey, types, we f i n d a very d i f f e r e n t s e t of r e s u l t s . Again flocking.enhances the l e a r n i n g r a t e , but i n a d d i t i o n , i t i n c r e a s e s the maximum feed i n g r a t e , e s p e c i a l l y f o r high v a l u e s of prey clumping. Thus f l e c k i n g i s a p a r t i c u l a r l y e f f e c t i v e s t r a t e g y when prey p o p u l a t i o n s are polymorphic, clumped and at low d e n s i t y . I t i s noteworthy that the s i m u l a t i o n r e s u l t s do not themselves r e v e a l that the d i f f e r e n c e i n capture r a t e s between f l o c k i n g and no n f l o c k i n g 13 7 4CH Time (minutes) Figure 11: Capture rate per bird as a -function of time for very highly . clumped prey (09.48) . Labels £ and nf identify tine curves for flocking and nonflocking birds" respectively. Latels 1 and 2 identify the curves for 1 and 2 prey types. 138 T ime (minutes) Figure 12: Capture rate per bird as a function of time for highly clumped prey (G=5.24). Labels as i n Figure 11. 139 Figure 13: Capture rate per bird as a function of tixne for randomly distributed prey (C=.07), Labels.as in Figure 11. n o Clumping 0.07 0.74 1 .48 5.24 9.48 f l o c k i n g 1 prey . 057 . 125 182 . 156 .176 n o n f l o c k i n g type .059 , . 137 198 . 222 . 183 f l o c k i n g 2 prey .056 . 142 129 . 135 .154 n o n f l o c k i n g types .052 .118 106 .088 .077 Table XXVIII Asymptotic capture r a t e s expressed i n prey per b i r d per minute ( i n the s i m u l a t i o n model the prey are considered to be l a r g e , approximately 1/2 gram each) . 14 1 Clumping 0.07 0.74 1.48 5.24 9.48 f l o c k i n g 1 prey .717 .915 .933 .918 .957 n o n f l o c k i n g type .859 .943 .944 .959 . 971 f l o c k i n g 2 prey . 702 .930 .932 .921 .934 n o n f l o c k i n g types .933 .942 .947 .961 .961 Table XXVIII:. Dominant ei g e n v a l u e s - the e i g e n v a l u e which determines the r a t e of convergence to the steady s t a t e c o n d i t i o n f o r the two l a r g e models (15x15 and 14x14) at f i v e value of prey clumping. Note t h a t s m a l l e r values of the dominant eigenvalue i n d i c a t e f a s t e r l e a r n i n g r a t e s . 142 b i r d s i s i n f a c t a long-term as w e l l as a short-term phenomenon. 3. A m u l t i p l i c i t y of d i s t i n c t types i n the. prey p o p u l a t i o n depresses the l e a r n i n g curves (providing some p r o t e c t i o n f o r the p r e y ) , but . t h i s r e d u c t i o n i n prey s u s c e p t i b i l i t y i s smaller when the b i r d s are f l o c k i n g than when they are s o l i t a r y . Thus, f l o c k i n g may be favored when prey p o p u l a t i o n s are h i g h l y polymorphic. In a d d i t i o n t o t r a n s i e n t and long-term capture r a t e s , the Markov.model y i e l d s long-term v a r i a n c e e s t i m a t e s . 1 3 Table XXIX presents these as v a r i a n c e over mean capture r a t e corresponding t c the f i r s t r i s k measurement i n , t h e s i m u l a t i o n experiments. For the s i n g l e prey case the long-term r i s k i s s i m i l a r to the short-term r i s k r e p o r t e d f o r the s i m u l a t i o n study. In both cases r i s k i s about- equal f o r f l o c k i n g and n o n f l o c k i n g b i r d s when prey are d i s t r i b u t e d randomly;, .but. as prey clumping i n c r e a s e s , r i s k i n c r e a s e s much.more r a p i d l y f o r the n o n f l o c k i n g b i r d s . In a d d i t i o n long-term. r i s k i s somewhat higher than short-term r i s k at ..high values of clumping. With two prey we f i n d s u b s t a n t i a l l y the.same p a t t e r n i n long-term r i s k f o r the n o n f l o c k i n g b i r d s . However, f o r f l o c k i n g . b i r d s there appears to be a c r i t i c a l l e v e l . o f . p r e y clumping beyond which r i s k , d o e s not i n c r e a s e s u b s t a n t i a l l y . , One must i n t e r p r e t , t h i s phenomenon c a u t i o u s l y s i n c e the variance over mean measures spread around the average r a t h e r than measuring the r i s k c f being below 1 3 For mathematical development see Appendix F. 143 Clumping 0 .07 0. 74 1. f8- 5.24 9.48 f l o c k i n g 1 prey 0 .91 3. 18 4. 32 4.73 10.76 n o n f l o c k i n g type 0 .96 4. 37 5. 55 9.66 15. 95 f l o c k i n g 2 prey 0 .93 4. 06 4. 41 5.35 5.54 n o n f l o c k i n g types 0 .95 2. 28 5. 48 12.01 12. 24 Table XXIX:. Variance over mean capture r a t e (for f o r f i v e values of prey clumping. twenty minutes) or " r i s k " 144 average. In t h i s case the major c o n t r i b u t i o n to the measure comes from b i r d s f e e d i n g below the average r a t e where f o r the s i n g l e prey c a s e . b i r d s f e e d i n g above the average r a t e accounted f o r about h a l f of the variance.,, 145 CHAPTER SEVEN! CONCLUSIONS This study began with a d i s c u s s i o n of f o r a g i n g and some of i t s b e h a v i o u r a l components. To examine whether f l e c k i n g might be c o n s i d e r e d as a f o r a g i n g s t a t e g y , a s i m u l a t i o n model was c o n s t r u c t e d of b i r d s f o r a g i n g i n f l o c k s . From the s i m u l a t i o n experiments with t h a t model and i t s subsequent.analysis i n terms of a Markov model s e v e r a l c o n c l u s i o n s can be drawn. 1. F l o c k behaviour can be modelled as the sum of i n d i v i d u a l behaviours. 2. Prey d i s t r i b u t i o n has a major e f f e c t upon f e e d i n g s u c c e s s . 3. Prey d i s t r i b u t i o n .requires , at l e a s t three parameters to de s c r i b e i t adequately: e.g. average d e n s i t y , average within clump d e n s i t y , average number of prey items (mass) per clump.. . 4. Giving-up time has a,major e f f e c t upon success. As i t i s l i k e l y to.be under the b i r d s ' c o n t r o l , i t should vary with d i f f e r e n t 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 . . 5. F l o c k i n g i s advantageous i n severe weather because f l o c k i n g reduces the r i s k . t o a b i r d of i n s u f f i c i e n t f e e d i n g f o r maintainence, p a r t i c u l a r l y when prey are clumped. 6. F l o c k i n g , n e i t h e r i n c r e a s e s nor decreases the mean capture r a t e when prey are of a s i n g l e type or prey are randomly d i s t r i b u t e d ; but f l o c k i n g does i n c r e a s e mean capture rate when prey are polymorphic ( s e v e r a l types) and clumped. However, i f the twenty minute peri o d cf the s i m u l a t i o n 146 experiments i s much l e s s than the time that r e a l b i r d s remain f o r a g i n g i n one small area (e.g. clumps of prey are l a r g e and r i c h ) , then the l a s t of the above c o n c l u s i o n s must be modified. . . . 7. When b i r d s ( e i t h e r i n d i v i d u a l l y or i n f l e c k s ) begin f e e d i n g i n a new a r e a , t h e i r capture r a t e begins at zero, i n c r e a s i n g t o some p l a t e a u . T h i s ' l e a r n i n g * rate i s greater f o r b i r d s i n f l o c k s than f o r those alone. However, the p l a t e a u i s higher f o r f l o c k i n g b i r d s only, when prey are polymorphic. When prey are monomorphic, i n d i v i d u a l f o r a g e r s have a higher long-term capture r a t e . I f the average stay i n a p a t c h . i s . s h o r t y f l o c k s w i l l be favored; i f the average s t a y i s long, s i n g l e i n d i v i d u a l s w i l l be favored . . The c r o s s - o v e r p o i n t seems to be i n the range of 10-30 minutes, depending upon prey d i s t r i b u t i o n and d e n s i t y (at l e a s t ) . Now we must c o n s i d e r how a p p l i c a b l e these c o n c l u s i o n s are to r e a l b i r d s . F i r s t , . t h e s e r e s u l t s concerning f l o c k i n g should apply to b i r d s w h o s e . c h a r a c t e r i s t i c s match these cf the model. In p a r t i c u l a r , t h e model . should be a p p l i c a b l e (in winter) to small b i r d s which: a) overwinter i n c o l d c l i m a t e s ; . b) are s u b j e c t t o l i t t l e p r e d a t i o n during t h i s p e r i o d ; c) form short-term prey p r e f e r e n c e s ; 147 d) feed on marginal resources, i . e . prey d e n s i t y i s lew; e) copy each others f o r a g i n g p r e f e r e n c e s . Examples of such b i r d s are the great, t i t which winters i n England and the black-rcapped chickadee which winters i n many p a r t s of North America. One should note that even i n these cases, . s e v e r a l of the c r i t i c a l behaviours (prey-preference and copying) have>been demonstrated under l a b o r a t o r y c o n d i t i o n s , but not as yet under f i e l d c o n d i t i o n s . The p r e d i c t i o n s f o r these b i r d s are, t h a t i f prey are clumped a n d . i f the winter i s severe or t h e r e are s e v e r a l important prey types, or both, then the b i r d s w i l l f orage . i n f l o c k s . I f prey.are.net clumped or the winter i s m i l d , the model p r e d i c t s . no advantage to f l o c k i n g . Under .these circumstances, one would expect ether s e l e c t i o n p ressures to take precedence. The a n a l y s i s u sing Markov models suggests t h a t i f p r e y . d e n s i t i e s are unusually high and mainly of one type (or s u f f i c i e n t l y . h i g h t h a t a b i r d can a f f o r d to ignore a l l but one . type) ,.. then., f l o c k i n g , by... reducing the. average capture r a t e , ..will be s l i g h t l y disadvantageous. F i n a l l y , while the model has not provided any simple s u c c i n c t statements of when f l o c k i n g has s u r v i v a l value f o r . b i r d s , i t has aided to c l a r i f y the issue.of,when f l o c k i n g may have s u r v i v a l value i n terms of f o r a g i n g success., 1'* 1 4 i t i s p o s s i b l e , of course, that f l o c k i n g nc longer 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 behaviour may be extant simply because animal behaviours are o f t e n slow t o evolve (Geist 1974 ) . 148 .An. attempt was made to compare .model p r e d i c t i o n s of feeding r a t e s with those found by Smith (1974) f o r thrushes f e e d i n g on a r t i f i c i a l b a i t s . S e v e r a l parameters had to be . adjusted s i n c e the thrushes had a s m a l l e r handling time f o r the prey and moved f a s t e r . , S i m u l a t i o n r e s u l t s were obtained at two prey d e n s i t i e s - .064 prey per sg. metre (nearly the same as t h a t used i n the previous s i m u l a t i o n experiments), and . 299 prey per s g . metre. In both . cases the prey were d i s t r i b u t e d randomly. The p r e d i c t i o n s were r e a s o n l b l y good a t the lower prey d e s i t y - .45 prey per minute compared t o t h e . r e a l b i r d s ! .59 prey per minute; while f o r the higher d e n s i t y the model.predicted 1.79 prey per minute as compared to the r e a l b i r d s ' 2.30 prey per minute. However, the r e s u l t s concerning the value of f l o c k i n g do not have u n i v e r s a l a p p l i c a b i l i t y . The r e s u l t s are r e s t r i c t e d to c o n d i t i o n s of. l o w p r e y d e n s i t i e s , when b i r d s are f e e d i n g near s u b s i s t a n c e l e v e l by devoting 90-100% of waking... hours to f o r a g i n g a c t i v i t i e s . The r e s u l t s are a l s o r e s t r i c t e d to cases when the prey are of approximately the s i z e and d i f f i c u l t y to d e t e c t . p o s t u l a t e d i n . the model., Experiments i n c r e a s i n g the d e t e c t i o n chances i n d i c a t e no s u b s t a n t i a l - c h a n g e f o r comparative r e s u l t s beween f l o c k i n g and s o l i t a r y . f o r a g i n g . I f prey are c c n s i d e r a b l e y harder to f i n d f o r b i r d s both with and without prey p r e f e r e n c e , ..then, the e f f e c t i s as i f the p r e y . d e n s i t y were reduced. But, i f . p r e y are harder to f i n d o n l y f o r b i r d s without prey p r e f e r e n c e , i t i s not c e r t a i n that these r e s u l t s a p p l y . 149 L i k e l y such a s i t u a t i o n would tend to give an advantage to the f l o c k i n g b i r d s . , The model i s not a p p l i c a b l e to the breeding season. I t may.also,be i n a p p l i c a b l e when, prey d e n s i t i e s are high enough to support a t e r r i t o r i a l system, s i n c e nc s e l e c t i o n pressures which might-favor t e r r i t o r i a l i t y are embedded i n the model. A comparison with a t e r r i t o r i a l . s y s t e m i s p o s s i b l e by comparing the success-of say .15 b i r d s f o r a g i n g i n f l o c k s over 15 separate f e e d i n g areas with the success of 15 s i n g l e b i r d s each f o r a g i n g over one of the 15 s i t e s . I n , a d d i t i o n , the model i s not a p p l i c a b l e t o , s i t u a t i o n s w h e r e . s u b s t a n t i a l changes i n prey abundance occur by means other-than t h e . b i r d s ' f o r a g i n g . For example, when.other predators take a s i g n i f i c a n t p o r t i o n . o f the prey,. .when the p r e y . p o p u l a t i o n ( s ) . a r e h i g h l y mobile cr when the prey are reproducing the r e s u l t s may be i n a p p l i c a b l e . F i n a l l y , the .model.is n o t . d i r e c t l y a p p l i c a b l e to mixed s p e c i e s f l o c k s i n which there i s s i g n i f i c a n t . n i c h e d i f f e r e n t i a t i o n . i n terms of f e e d i n g . Krebs (1973a).suggests t h a t a c t u a l l y many m u l t i - s p e c i e s f l o c k s have c o n s i d e r a b l e f e e d i n g niche . o v e r l a p and that i n t e r s p e c i f i c copying.takes p l a c e (at l e a s t i n the l a b o r a t o r y ) . Thus, the model may help e x p l a i n some moderate s i z e d mixed-s p e c i e s f l o c k s of the temperate zone. Those c o n c l u s i o n s , which do - not concern t h e . value of f l o c k i n g are l i k e l y to have wider a p p l i c a b i l i t y -than t h a t p o s t u l a t e d f o r t h i s model. On the b a s i s . o f t h e . g e n e r a l p a t t e r n of movements of : the f l o c k , .the model , appears tc give a reasonable d e s c r i p t i o n of f l o c k behaviour, at l e a s t at a 1 5 0 s u p e r f i c i a l l e v e l . . T h i s , i n d i c a t e s , . h o p e f u l l y , that no beh a v i o u r a l parameters of c r u c i a l - i m p o r t a n c e are missing from .the model. I t should, be r e i t e r a t e d that i n p u t t c the model concerns i n d i v i d u a 1 b i r d b e haviours, while t h e o u t p u t p r o v i d e s f l o c k , behaviours; hence,- t h e . c o n c l u s i o n s are net cr e a t e d by a c i r c u l a r - reasoning process. A d d i t i o n a l evidence sup p o r t i n g the v a l i d i t y of the model, i s th a t . i t s p r e d i c t i o n that f e e d i n g s u c c e s s , r e g a r d l e s s of whether b i r d s . f l o c k , i s very s e n s i t i v e t o givi n g - u p time. T h i s . p r e d i c t i o n , has been made elsewhere, based upon e n t i r e l y d i f f e r e n t p r i n c i p l e s . (Charnov 1973). The c o n c l u s i o n t h a t attack r a t e v a r i e s w i t h , prey d i s t r i b u t i o n (average prey d e n s i t y held constant) i s not a_new r e s u l t (see Chapter 2 f o r r e f e r e n c e s ) . I t has not., been g e n e r a l l y r e c o g n i z e d , however, ..that ..a two-parameter d e s c r i p t i o n of a prey p o p u l a t i o n my be i n s u f f i c i e n t t o c h a r a c t e r i z e i t f o r comparing f o r a g i n g s t a t e g i e s . S e v e r a l p o s s i b i l i t i e s are . a v a i l a b l e , f o r c o n t i n u i n g t h i s study. One.is to expand the model c a p a b i l i t i e s , s p a t i a l l y ; i . e . to i n c r e a s e the area.and number of prey c o n s i d e r e d . A second i s to expand the .model ,,temporally.. .This .might r e q u i r e c o n s i d e r a t i o n of hunger, energy requirements (temperature) and m o r t a l i t y (e.g. b i r d s below a c e r t a i n weight.die; d a i l y feeding and temperature determine weight f o r each day). Comparisons of the weight d i s t r i b u t i o n s and chance of s u r v i v a l o f the b i r d s over a p e r i o d of about 10 days f o r d i f f e r e n t weather c o n d i t i o n s , d i f f e r e n t i n i t i a l 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 might lead t c 151 more p r e c i s e hypotheses. One might a l s o examine whether there are c o n d i t i o n s (e.g.. higher prey d e n s i t y ) under which s o l i t a r y i n d i v i d u a l s have, higher capture r a t e s . A n a l y s i s using the Markov models suggested that high prey d e n s i t i e s and l a r g e patches might be such a s e t of c o n d i t i o n s . A second set of p o s s i b i l i t i e s i s to..make a more.thorough i n v e s t i g a t i o n of the importance of giving--up time. F i r s t . . o n e might, e x p l o r e , the r e l a t i o n s h i p between giving-up time; ,prey .density and prey d i s t r i b u t i o n . More i n t e r e s t i n g i s t h e _ p o s s i b i l i t y t h a t f l o c k i n g may a i d i n the "adjustment" of giving-up times. To measure giv i n g - u p time as c r e a t e d by f l o c k behaviour, one could c r e a t e s e v e r a l areas with d i f f e r e n t prey d e n s i t i e s . and determine the " b e s t " g i v i n g - u p time f o r s o l i t a r y , f o r a g e r s . Then, measure whether f l o c k s of b i r d s whose gi v i n g - u p time parameters d i f f e r from t h i s "best" v a l u e tend toward t h i s " b e s t " value? However, b e f o r e , c o n t i n u i n g to use such a model, i t should be determined e x p e r i m e n t a l l y j u s t how u s e f u l the model has been i n making p r e d i c t i o n s about r e a l b i r d s . The . f o l l o w i n g hypotheses which were.generated by t h i s model .might be t e s t e d . . 1. B i r d s . a d j u s t t h e i r giving-up times on the b a s i s of r e c e n t f e e d i n g success. 2. B i r d s , i n f l o c k s capture n e i t h e r more nor.fewer prey on average than d o , s o l i t a r y b i r d s when there i s a s i n g l e , prey type. ... . .. • 3. B i r d s . i n f l o c k s capture more . prey when a t t a c k i n g a polymorphic prey p o p u l a t i o n than do the same b i r d s f o r a g i n g 1 5 2 alone. 4. B i r d s i n f l o c k s have a s m a l l e r v a r i a t i o n i n capture r a t e f o r . a f i x e d f o r a g i n g p e r i o d than do b i r d s f o r a g i n g alone.: , 5. 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Diagram 6 FLOW CHARTS OF MAJOR MODEL COMPOI^NTS :Subroutine TMSTP Subroutine' FLY Subroutine FLOGG Subroutine EAT Subroutine LMTATE Main Program Write out tin* time period CM,!, 'KAT' _y CAI,h 'n.CX'.G' once every fonr periods, and vliciirvor a bird Hew in tho. previous, period Write current stntus of each bird C A M , 'Kl-Y j A cti vo bi rds move Movement t tati r, t ic r. computed Print map of b ird locations if a b ird flew or if the last map witt' pr inlcd N M A i * ( pc j iod s a j;o y I n i s h S U H K O U T I N T T I . Y ' 168 (A) Have A l l l . i n l s l u ' m chert. . .! (or ( l i f .M: Yr » No No 11a;; b i r d ' s M ' . \ i i h i u ( ; IIKIVC:; I ( l u - y o n d vdy e o f l i m ' ' i1 r.ts car lie d Oid hi i d f l y in t i n s or p i cvions period? © Is bi rd hand I inr. food? , Yr j No No © Il ird f l irs v/ilh prob.vhilily P(fly) -•• ~ N if N>G - 0 If U&G where: N = nc-iiods t . inrc l> j rd a I r l.t st prry C - mi'iiriuiin r iv ing up I line Doc K bi rrl fly? Jtss. Have .il l turds been c ln.-c keel for fi'.l Inwi up © No Is .bird in i a m c flock as leader''? Yes No Dor s bird fly in tl.i c d ? V -© Set de lay <• d follow . s w i t c h on ' J h t B hlrd may f w l l u w A n y t i m e i n t h e n e x t m i n u t e D i a g r a m 2 169 Diil Ui it) fly l:.M i iod? I «t bi r'll II.IIHU i p icy '.' II ni:iy f„l|,,w in Mic iw\t 's - |n-i iixl.-. ]<il<l follows W i l l i probability N- y J'(follow) - I N where: K^ - a threshold Docs l,i|:<] follow? Jlircl follow... Icicle r l l ; i v » : c i u . u j . l i b i r i l t l l d w n u . t n i ^ ' e r . M l . i n t e r , I . I t e i l f l i l ; h t ? (ulcerated [|i-|;lit takes place AW l> i rd i in s a m join tl,r; intrj e flor k .15 Ic.nlc r 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,,.? I"ir\ish If ili.l.iycl follnv. j. w i ; , I, „„ „ v i. r 3 (•(: rio.lt, «il off I" II-: I'i M l ' . . l e h . y , .1 r,,l|„-„ „wil, I. t,n? 170 No Is l,i III ha nil 1 i n|- prey ' l)i<] l i in l (ly tlii i. or jnevious period? No Dili I b i s bird's f k . e k | , . i v r an integrated f lir.ht in the l,i K .< pi' ri.ul s ? Hird f o l l o w s w i t h p r o b a b i l i t y ' N-K ; ... ]>(follow) = -•' N whr rc: 'K = a I l l I r 5 liolil No ® Dors b i i il follow? 11 i rrl fol lows leade r Y e s Delayed follow switch of( 1 7 1 MH.KOUTIUK 'I'l.OCC. NO NO / N O NO Al l l.lnl:i i-i;11r • I lo flock 7t ro ••• 0 • Test 11 i l d s j, j = 1, .... NIllllllS lave all birds been Irs rd? Is bird j already assigned to a flock? KK = i ; i ; 1 1. Assifoi bird j to flock KK vi: s • •i--,fii F i n i s h Have a l l birds been tested? Tect birds i , t = 1, . . . , .NIMH MS ~ \ ^ Yl.. r Has bird i ;il ready bo en ossil'ncd to the same floe k a s bii d j? NO NO Is bin! i in the same, ti r e as hi r cl. j? A i t b i r d * i A m i j l u i . i than PFI.OK feet .'[.ait? Is bi id i A 1 re in! y a i; s if*ne cl to a f 1 nek' A H s i[;n bi nl' i to the . e.uiu Hock a n binl j. Y ICS Y l ' S Oonibi ne Ike bird* 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 . S U W I O U T I N F: 'I:AT' •172 NO ll . i v e all birds k e n checked? Y i : NO Is \> i i d li.'md liiij', prey Ibis period? Y i:s Kind which trrc the bird is in This bird will fly tli i s pe r i od NO KO NO NO Is the bi rd in a tree? Docs bird have .1 searching image? Translate bird's coordinate location into g rid loi.a t ion Docs bird have a temporary search image? Hfvd te arches only for prey corresponding to thU image (i) Probability of success on piey type. I Is G i l pe r pi e. y in t lie grid c e l l Have all the members of p i r y type i (n the g rid ( ell bee :. te :.U.d? YKS YKS < . I) r A v/ i*a nd oui numbc r Doe 0 bi rd c a | ii r e tills p 1 e y ? Z>—© ]) i;i|;) ,;in ,i A 1 7 3 NO ® 'Ul r t ! t.r. J n:\tr t- for ; i 11 | i r r y ly pr a findinf, piry type j v.-illi p I'ldi.i hi I ity )' 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 in the f; r id t e l l be r n te :de d V Y KS l lraw random number NO Does bird capture this prey? N O Tills bird.fed successfully, Remuvc from thr £rid cr 11 the captured prey. Assign handling time to the hire] cor re '.pond\n[[ tu the prey type (NSTA'l ) "Dues bird have search imaf-.e? YI-TS S e t r . f i i r c h i m a g e c q o n l t o I c m p o i .11 y D e . i r c h i r . ' . v ; e ( i f . m y ) a n d U i.-t 0 1 ;ir•/ c h J i M . - i t j i : 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 . ® 1 7 4 NO ® The bird failed to fr c <1 this prriod. 5;ct tf i i i j iu t J ry search im.i[;c lo zero. ll.l s the 1 ii rd f. iled (o feed for MIIUN periods? Set search imaee to 7.e ro Y l\S NO Have p.l 1 birds been tested? - — K . n l . h NO Is the bird ha:idlih|; prey? "~1 The bird attempts to Imitate, anothe r bird by invoking jubrouline IM I AT K Y I : S Doer, bird have a search ima|;e? i Kit 11 it tun: INK 1IMTA TV.' N O i l r i l I attempts In imitate nearest hnc ( r:'. M i l l iM'i -.' Id i a u r ll.i vc .ill l)inls been .-h.-c U'il 7 Yl'S c—.-N O I>id bird j capture prey this period? N O A r c birds i and j in thr same tree Y-K: Is distance between birds i and j less than the maximum imitation distance? N O / Is bird j n i J i r r ti. b ud I than any birds previously tested'' IM11IHI) N O Did bird i find a bird to tmiUtc? IUrd i. obtains temporary search imajfc for th*: prey typo he ohu-rvrd IMI'dlUJ c;.pturiny', l i i I" <1 i o b t a i n s n o ( z e r o ) t e m p o r a r y £ 4..T T C h i l t U | U : )- MU§ F i n i s h M A1N I'UOC'.ll AM 176-H e . 1 d i n ju r . u i i i ' 11-1' a H i - , n l i n l.i r i l a . a n i l I m M l i n i i S v i a M i l t n u i l i i . f ' f t l M T ' Initialize i'lii.ht an.I f a d i n g vaiiables 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 distribute piey v.ia s ul) i mil in r ' }' 1 \' IT 1 Initialise M I K K K H immbci p r ne i a.t'o r s and the x lock J"l !.•; -- 0 1', oducr food maps via subroutine 'FMAP' S " i l birds into flocks v 'PI.OGG' a sub rout inc Produce ma |> of bird locations via nit) rout i or ' I I M A I " Produce statistics d i s t l i hut i on s v i .i s 1 a n.i lysis of prey uh rout im*' ' I'A N A 1,1 JTIMK = JTIM P." f 1 C A D . TM STP NO Is t be ! mi nl.i I inn I , : i , s lied ! (J'l l\< I-.'V/.NI IM 1.) O > • 11 • < • ( f i n a l l e j u l l s v i . i K nb I lul l i ne s ' I I M A |', I M Ai-, I A : ; ,\ 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 for birds foraging on one prey type Table XXXI Capture rate for d i f f e r e n t f l o c k sizes on 36 clumps of 25 prey each Table XXXII Capture rate f o r d i f f e r e n t flock sizes on 12 clumps of 7.5 prey each Table XXXIII Risk for d i f f e r e n t f l o c k sizes on 36 clumps of .25 prey each Table XXXIV Fisk for d i f f e r e n t flock sizes on 12 clumps of 75 prey each 178 Clumping Mean Caps. F Nf F Nf F Nf 0.98 0.88 0.29 0.32 1.52 1.96 0.25 0.30 2.19 2.89 0.34 0.37 3.58 2.42 0.36 0.39 3.70 4.99 0.38 0.44 2.98 4.04 0.46 0.40 3.57 5.80 0.38 0.49 4.20 7.09 0.37 0.53 4.45 9.16 0.47 0.53 7.04 6.27 0.49 0.52 6.14 8.95 0.38 0.63 5.69 8.71 0.46 0.62 5.83 7.83 0.45 0.59 6.62 10.72 0.56 0.73 Var/Mean Caps. Prob. Zero Caps. 0. 07 1. 30 1, 08 0.13 1.55 1.60 0.25 1.27 1.76 0.33 2.12 1.46 0. 46 2.04 1. 85 0. 62 1.40 2.20 1. 01 2.23 2. 16 1.23 3.23 2.93 1.48 2. 20 2.63 1.86 3.02 2.72 2.44 4.28 2.78 3.64 3.34 3.36 5. 24 3.44 3. 58 9.48 2.35 2.31 Table XXX_: Capture 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 prey type f o r a range of values of prey clumping. Each value r e p r e s e n t s an average over 100 s i m u l a t e d f o r a g i n g episodes, each r e p r e s e n t i n g twenty minutes. ' These value correspond to F i g u r e s 4-6 where C = (16*Clumping) and Clumping i s L l o y d ' 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 r a t e s are not s i g n i f i c a n t l y d i f f e r e n t (p>.10) between f l e c k e r s and n o n f l o c k e r s , however both measures of r i s k do d i f f e r s i g n i f i c a n t l y (p<.005) . 179 C = In (16*clumping) F l o c k s i ze 0. 16 1. 39 1. 65 1. 99 2. 29 2.78 3. 38 4. 06 5.00 1 1.08 1 .55 1.65 1.85 1. 50 2. 10 3. 05 2. 78 . 2.25 2 1.05 1. 40 1.00 2.40 2. 55 2.40 7. 50 7. 20 1.60 4 1.90 1. 95 1.75 3.35 2. 25 6.30 3. 85 3. 00 1 .50 6 1.22 1 .67 1.50 2.00 1. 47 2. 13 4. 27 5. 03 2.02 8 1.50 1. 53 2. 50 1. 30 1. 53 2.03 4. 13 3. 83 2.15 12 1.29 2.10 2.63 2.18 1. 62 2.82 2. 67 3. 85 1. 84 16 1. 48 1. 30 1. 50 2.03 2. 91 2.76 2. 53 3. 30 1 .69 Table XXXI:. Mean captures i n 20 minutes when prey are 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. These values correspond t c Figure 7. 180 C» = (16*clumping) Flock s i z e 0. 16 1.38 1 .89 2.53 3. 14 3.72 4. 44 5. 29 6.08 1 1. 08 1. 30 2. 85 2.05 2. 80 3.75 3. 45 3. 00 2. 60 2 1. 05 1.75 2.50 4. 10 7. 85 3.40 4. 25 2. 80 1 .80 3 1. 44 2.10 2.93 1.67 4. 43 3.03 4. 70 3. 40 1. 56 6 1. 22 1. 47 2. 42 2. 68 3. 27 2.03 2. 95 2. 25 2.08 10 1. 40 1.84 2.41 2.75 2. 64 5.25 3. 72 3. 69 1. 34 16 1. 48 1. 44 1. 67 1.68 2. 07 4.07 4. 15 4. 15 2.11 Table XXXII:. Mean captures i n twenty minutes 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. These values correspond t c F i g u r e 8. 181 C» = (16*clumping) Flock s i z e 0.16 1.39 1.65 1.99 2.29 2.78 3.38 4.06 5.00 1 32 30 40 45 50 45 45 70 75 2 25 30 40 20 50 50 30 30 85 4 20 20 30 25 15 10 45 35 75 6 18 20 43 27 37 27 3 13 58 8 28 30 10 40 28 35 28 45 55 12 24 27 7 23 28 10 18 8 52 16 19 28 20 20 11 11 14 5 31 Table XXXIII.: Percentage zero c a p t u r e s i n twenty minutes 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 values correspond to F i g u r e 9. 182 C 1 = {16*Clumping) F l o c k s i z e 0. 16 1. 38 1.89 2. 53 3. 14 3.?2 4.44 5.29 6.08 1 32 30 30 55 65 55 70 75 85 2 25 35 40 50 30 50 55 75 75 3 23 47 27 37 33 50 43 67 87 6. 18 33 20 30 35 63 57 67 68 10 26 26 17 15 31 14 31 38 68 16 19 24 27 29 29 12 18 17 37 Table XXXIVj. Percentage zero captures i n twenty minutes 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. These values correspond to F i g u r e 10., 183 APPENDIX Ci INCOMPLETE CONTINGENCY TABLES When some c e l l s of a contingency t a b l e r e p r e s e n t impossible combinations, the u s u a l a n a l y s i s f o r independence of row and column e f f e c t s i s no longer p o s s i b l e . I t i s p o s s i b l e , however, to analyse, the t a b l e f o r "quasi-independence" (fienfcerg 1972). T h i s a n a l y s i s d i f f e r s s l i g h t l y from the usual one by accounting e x p l i c i t l y f o r the i m p o s s i b l e c e l l s . Let N i j be the number of o b s e r v a t i o n s i n rcw i , column j ; and l e t Ni. = r *** © { N i j } , *** j=1 N. j = r *** ® { N i j ] *** i=1 and N = r *** © {Ni. } *** i=1 r *** 9 {N.j} *** j=1 for i=1,...,r and j=1,...,s. Furthermore, l e t X i j = 0 i f c e l l ( i , j ) must be empty = 1 otherwise. Then we f i t the s e t of simultaneous l i n e a r equations 184 *** N i . = ® { A i * B j * X i j } *** j = 1 r *** N.j = 6 { A i * B j * X i j } , *** i=1 Then E i j = E[ # counts i n c e l l ( i , j ) ] = A i * B j * X i j . *** The sum © { [ N i j - E i j ] 2 / E i j } i s a s y m p t o t i c a l l y c h i - s q u a r e d , *** with ( r - 1 ) * (s-1)-K degrees of freedom, where K = the number of empty c e l l s (those 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 t a b l e . 185 Hi. 1 2 3 20 10 30 40 10 10 10 10 60 30 50 M.j 60 60 20 13=14 0 Then we s o l v e the f o l l o w i n g s e t of eg u a t i c n s . A1*(B1+B2) = 60 B1* (A1+A2+A3) = 60 fl2* (B1+B2 + B3) = 30 B2* (A1+A2 + A3) = 60 A3* (B1+B2 + B3) = 50 B3*(A2+A3) = 20. Th i s system i s underdetermined, but f o r any two s e t s of s o l u t i o n s A i , B j and A i ' , B j ' we have Ai*B j * = Ai»*Bj'. Thus, we may s e t any of the v a r i a b l e s to an a r b i t r a r y nonzero constant and s o l v e f o r the remaining ones. One s o l u t i o n i s A1=1 A2=3/8 A3=5/8 B1=30 B2=30 E3=20. Then f o r the t a b l e of expected values we f i n d 30 30 11.25 11.25 7.5 18.75 18.75 12.5 S c c h i 2 = 20.14 with 3 degrees of f r e e d c 187 APPENDIX Dj. REPEATED EIGENVALUES AND ZERO EIGJNV ALUJS Example jk Repeated Eigenvalues When an NxN matrix has fewer than N d i s t i n c t e i g e n v a l u e s , i t may be im p o s s i b l e to proceed with the matrix decomposition as given i n Chapter 6. For example. .6 . 2 . 2 p = .1 .8 .1 . 1 . 3 . 6 has eigenvalues 1 , .5 and .5. When we s o l v e the equations P*Oi = ,5*P Vi*P = .5*P, we f i n d t h a t the dot product U i * V i = 0. Hence, P**N cannot be represented as a sum of co n s t a n t matrices m u l t i p l i e d by the N'th powers of t h e i r r e s p e c t i v e e i g e n v a l u e s . To s o l v e t h i s problem, we may use the method of generating f u n c t i o n s (or • z - t r a n s f o r i s 1 ; Howard 1960). i n f *** Let P (z) = ® p * * } j * z * * n ( i n f = i n f i n i t y ) . *** n=0 Then, P(z) = I + P (z) *z*P (P**0 = I, the i d e n t i t y m a t r i x ) . Hence, P (z) = [ I - z * P ] - i . Now [ I - z * P ] _ 1 can be decomposed by p a r t i a l f r a c t i o n i n t o a sum of constant matrices Hi times e x p r e s s i o n s of the form ( 1 - L i * z ) * * (-Ri), where the L are the eigenvalues cf P, each of m u l t i p l i c i t y R i . Taking i n v e r s e transforms of these terms, we 188 f i n d *** P**n = e F i ( n ) * L i * * n * M i . *** ~ i where Fi ( n ) . i s a polynomial i n n of degree Ri-1 cr l e s s . For the example matrix we f i n d .20 .56 .24 = .20 .56 .24 .20 .56 .24 .8 -.56-. 16n -. 24+. 16n + .5**n -. 2 .44 + . 04n -.24-. 04n -.2 -.56+. 04n .76-. 04n Example 2\ Zero Eigenvalues, M u l t i p l i c i t y When a l l the eigenvalues are of m u l t i p l i c i t y 1, we may proceed as i n the t e x t , with the understanding t h a t 0**n = 0 f o r n=1,2,... = 1 f c r n=0. Thus, 189 ,6 .6 ,5 .2 .2 .5 .2 .2 .0 **n + (-.2) **n (0) **0 7/12 3/12 2/12 7/12 3/12 2/12 7/12 3/12 2/12 -1/12 3/12 -2/12 -1/12 3/12 -2/12 5/12 -15/12 10/12 1/2 -1/2 0 -1/2 1/2 0 -1 1 0 Example 3_1 Zero E i g e n v a l u e s , M u l t i p l i c i t y S>2 In t h i s case the e i g e n v e c t o r s f o r the zerc eigenvalues are not uniguely d e f i n e d . So, P may be decomposed i n terms of these N-S nonzero e i g e n v a l u e s . Then the Zn (n=0,... ,S-1) may be determined by the d i s c r e p a n c i e s between P**n and the decompostion based upon the nonzero e i g e n v a l u e s . APPENDIX E: .TRANSITION PROBABILITIES FOR THE MARKOV MODELS Table XXXV Transition p r o b a b i l i t i e s for flo c k i n g birds feeding on one prey type . Table XXXVI Transition p r o b a b i l i t i e s f o r nonflocking birds feeding on one prey type 191 Clumping Prob. . 0.07 0.74 1.48 5.24 9.48 1 .9653 .9652 .9695 .9809 . 9836 2 .0164, .0116 .0107 .0054 .0043 3 .0183 .0232 .0198 .0137 .0121 4 .9850 .7600 .7800 .7436 .7917 5 .0150 .2400 .2200 . 2564 . 2083 6 .9850 .8677 .8179 .8127 .7917 7 .0150 .1323 .1821 . 1873 . 2083 8 .9850 .8714 . 8482 .9161 .8327 9 .0150 .1286 .1518 .0839 . 1673 10 .9850 .8786 . 8486 .9552 .9507 1 1 .0150 .1214 .1514 .0448 .0493 12 .9850 .90 37 . 8854 .9653 .9553 13 .0150 .0963 .1146 .0347 .0447 14 .9850 .9483 . 8893 .9691 .9713 15 .0150 .0517 .1107 .0309 .0287 16 .9670 .8597 .8125 .9247 .8563 17 .0150 .0442 .0676 .0168 .0219 18 .0180 .0961 .1199 .0585 . 1218 19 1 1 1 1 1 20 1 1 1 1 1 21 .9004 .6643 .4532 .2216 . 1484 22 .0649 . 2571 .4260 .7210 .7558 23 .0348 .0786 .1208 -.0574 .0958 24 1 1 1 1 1 25 1 1 1 1 1 26 .9850 .7826 .6751 .4937 .2604 27 .0150 .2174 .3249 .5063 .7396 28 .9336 .5433 .4070 .3311 .5727 29 .0302 .1544 .1901 . 2838 . 1453 30 .0362 .3023 .4029 . 3851 .2820 Table XXXVi Estimated values f o r the 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 f l o c k i n g b i r d s f e e d i n g on one prey type. 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 each value of prey clumping. 192 Clumping Prob._ 0.07 0.74 1. 48 5.24 9.48 1 .9850 .9837 .9848 .9906 .9945 2 .0150 .0163 .0152 .0094 .0055 3 NA NA -NA • NA N.A. 4 .9440 .7142 .7373 .7160 .7513 5 .0560 .2858 .2627 .2840 . 2487 6 .9440 .8045 .7945 .7512 .7838 7 .0560 .1955 .2055 .2488 .2162 8 .9440 .8331 . 8333 .8373 .7908 9 .0560 .1669 .1667 . 1627 . 2092 10 .9440 .8518 .8659 .9028 .9358 1 1 .0560 .1482 .1341 .0972 .0642 12 .9440 .8555 .8732 .9194 .9465 13 .0560 .1445 .1268 .0806 .0535 14 .9440 .9051 .8912 .9379 . 9614 15 .0560 .0949 .1088 .0612 .0386 16 .9440 .9506 .9333 .9436 .9614 17 .0560 .0494 .0667 .0564 .0386 18 NA NA NA NA NA 19 1 1 1 1 1 20 1 1 1 1 , 1 21 .9811 .6910 .4503 .2685 . 2940 22 .0189 .3090 . 5497 .7315 .7060 23 NA NA NA NA NA 24 1 1 1 1 1 25 1 1 1 1 1 26 .9440 .6897 .5535 .3321 .2045 27 .0560 .3103 .4465 .6679 .7955 28 NA NA NA NA NA 29 NA NA NA NA HA 3 0 NA . NA NA NA NA Table 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 b i r d s f e e d i n g on one prey type. Since there i s no i m i t a t i o n s t a t e f o r n o n f l o c k i n g b i r d s , c e r t a i n of the t r a n s i t i o n s l i s t e d are not a p p l i c a b l e (NA), 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 s f o r each value of prey clumping. 193 APPENDIX F i DERIVATION OF VARIANCE TO MEAN RATIO To d e r i v e V a r ( j ; n ) , the vari a n c e i n s t a t e r e s i d e n c e time f o r s t a t e j a f t e r n p e r i o d s , we begin with the d e r i v a t i o n of V j 2 , the v a r i a n c e i n recu r r e n c e time f o r s t a t e j i n a f i n i t e Markov c h a i n . F o l l o w i n g F e l l e r (1957), l e t & be the r e c u r r e n t event, Un=P[S occurs at the n f t h t r i a l ] , Fn=P[ 6 occurs f o r the f i r s t time at the n'th t r i a l ] . Also d e f i n e F0=0, 00=1, i n f i n f F(S) *** ® *** n=0 {Fn*S**n} = *** 6 *** n=1 { Fn*S**n} 0(S) i n f *** © *** {Un*S**n}. n=0 f o r n>1 P[ & f i r s t time a t k<n, and again at n] = Fk*Un-k P[8 f i r s t time at n] = Fn = Fn*U0. i n f Thus, Un = *** e *** n=1 { Fk*Un-k}. M u l t i p l y i n g U(S) -Hence, U (S) by S**n and summing from n=1 to i n f i n i t y , we f i n d 1 = F (S).U(.S) . . = 1/(1-F(S)) Now f o r 8 p e r s i s t e n t and n o n p e r i o d i c we have i n f *** ® {Fi}= 1 ; thus F i s a p r o b a b i l i t y d i s t r i b u t i o n , i n f *** with M = ® { i * F i ] = F ' O ) the mean re c u r r e n c e time. *** i n f *** Define Qn = ® {Fn+j} *** j=1 i n f *** Rn = 6 {Qn+j} *** j=1 i n f *** Q(S) = ® {Qn*S**n} *** n=0 i n f *** R(S) = ® {Rn*S**n}. *** n=0 AS S — > 1 , Q (S) --> i n f *** ® *** n=0 {Qn} i n f i n f *** **« 9 8 {Fn+j} 4** *«* n=0 j= 1 / i n f *** 9 {k*Fk} *** k=1 = M ; and B (S) — > i n f 4** © n=0 {Rn] i n f i n f *** *** © ® { Qn+ *** *** n=0 D=1 i n f . i n f i n f *** *** *** ® 9 9 *** *** *** n=0 j=1 i=1 {Fn +j+i} i n f *** ® { k (k-1) *Fk/2 } *** k=2 = [ V 2 + u 2 - u]/2. Thus, Q(S) and R(S) converge f o r |S|<1. 196 i n f n *** *** Now Q(S) = © {S**n [1 - 9 {Fl}]} *** *** n=0 k=0 = 1/(1-S) - {SF1 + S 2(F1+F2) + S3 (F1+F2+F3) . .. ] = 1/(1-S) - F(S) - S{SF1 + S2(F1+F2) + .,.} = 1/(1-S) -F (S){ 1+S+S2 + . . . } = (1-F (s) )/(1-s) ; i n f i n f *** *** and R (S) = 9 { S**n [ 6 {Qn + j}]} *** *** n=0 j=1 i n f n **.* *#• 9 { S**n [M - € {Ok }]} * * * *** n=0 k=0 i n f n *** *** = M/(1-S) - 6 {S**n [ 6 {Ck}]} *** *** n=0 k=0 i n f i n f *** *** = M/(1-S) - « {Qk [ € {S**n}]} *** *** k=0 n=k 197 i n f i n f *** *** = M/(1-S) - 8 {Q*S**k [ » {S**n]]3 * * * * * * k=0 n=0 i n f *** M/(1-S) - 1/{1-S) 9 {Qk*S**k} *** k=0 = (M-Q (S) ) / (1-S) . R (S) M - Q (S) M (1 - F (S) ) Thus = / — MQ(S) (1-S) (1-S) H - Q(S) M (1 - F (S) ) i n f *** Now 9 { (Un - 1/R)S**n} = U (s) - 1/M(1-S) *** n=0 = 1/(1 - F(S)) - 1/M(1-S) = [ H - MS - 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 i n f *** « { Un - 1/M} = Q 2 + M2 - H ] /2H2. * * * n=0 For a f i n i t e Markov c h a i n with t r a n s i t i o n matrix P, f o r a given s t a t e j r *** Un = P n [ j j ] = Y 1 [ j j ] + « [ Ik**n*Yk[ j j ] } *** ~ k=2 and 1/M = 1/Mj = Y 1 [ j j ] . Thus, i n f i n f r **« *** *** € {On - 1/M} © ® {Lk * * n * Y k [ j j ] } *** *** *** ~ n=0 n=0 k=2 *** ® {Yk[ j j ] / | 1 - L k ) } *** k=2 = [ V 2 - M + M 2]/2M 2. r • »** So V 2 = M - M2 + 2M 2* © { Yk£ j j ]/(1-Ik) } *** ~ k=2 [ i n F e l l e r ' s n o t a t i o n Sk/(Sk-1) r e p l a c e s 1/ (1-Lk) ] A l t e r n a t i v e l y , we may sum over n=1,inf, g e t t i n g r *** V 2 = -M + M2 + 2M2* ® { Yk[ j j ] L k / (1-Lk) } k=2 [ i n F e l l e r ' s n o t a t i o n 1/(Sk-1) r e p l a c e s Lk/ (1-Lk) ]. F i n a l l y we have Var(j;n) = n * V 2 j / M 3 j . ( F e l l e r , 1957, p. 310) P u b l i c a t i o n s T h o m p s o n , W . A . and V e r t i n s k y , T. (1975) B i r d f l o c k i n g r e v i s i t e d : p o l y m o r p h i c p r e y p o p u l a t i o n s , J . A n i m . E c o l . ( in p r e s s ) T h o m p s o n , W . A . V e r t i n s k y , I . and K r e b s , J . R . (1974) T h e 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 20 . V e r t i n s k y , T . , T h o m p s o n , W . A . and U y e n o , D . (1974) M e a s u r i n g ' c o n s u m e r d e s i r e f o r p a r t i c i p a t i o n i n 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. and K a n e , J . (1974) K S T M - p o l i c y s i m u l a t i o n : u s e r ' s m a n u a l , T T C C R e v i e w . ( in p r e s s ) . T h o m p s o n , W . A . , V e r t i n s k y , I . and K a n e , J . (1973) C a n a d i a n 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 R a n g e P l a n n i n g D e c e m b e r : 6 6 - 7 3 . K a n e , J . , V e r t i n s k y , I . , and T h o m p s o n , W . (1973) K S I M : a .' m e t h o d o l o g y f o r i n t e r a c t i v e r e s o u r c e p o l i c y s i m u l a t i o n , W a t e r R e s o u r c e s Res .9(1) : 6 5 - 7 9 . i K a n e , J . , T h o m p s o n , W . A . , and 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 e x a m p l e (wa te r p o l i c y f o r B r i t i s h C o l u m b i a ) , I n t e r n a t i o n a l 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 awa , 1: 3 9 - 5 5 . K a n e , J . , T h o m p s o n , W . , and V e r t i n s k y , T. (1972) H e a l t h c a r e d e l i v e r y : a p o l i c y s i m u l a t o r , S o c i o - E c o n . P l a n . S c i . 6: 283-93:. T h o m p s o n , W . A . and V e r t i n s k y , I . (under r e v i e w ) A p p l i c a t i o n of M a r k o v cha in 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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