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Behav Ecol Sociobiol (1986) 19:409~:~15 Behavioral Ecologyand Sociobiology9 Springer-Ve rlag 1986

P a t c h c h o i c e u n d e r p e r c e p t u a l c o n s t r a i n t s :a c a u s e f o r d e p a r t u r e s f r o m a n i d e a l f r e e d i s t r i b u t i o nM a r k V . A b r a h a m sBiologica l Sc iences Depar tme nt , S imon F raser Univers i ty , Burnaby, Br i ti sh Colum bia , V5A 1$ 6 , C a n a d aReceived Janu ary 25 , 1986/A ccepted Ju ly 20 , 1986

Summary. A review of tests of ideal free distribu-tion (IFD) theory reveals a characteristic bias:patches with a small proportion of the resourcesare relatively overused and patches with a largerproportion of the resources are relatively under-used. A model is developed to examine how ani-mals with limited abilities to perceive differencesin patch quality affect an IFD by foragers. Thisconstraint produces the observed bias, a bias thatis exaggerated as the num ber o f patches increases.Sutherland (1983) has developed a model basedon interference which can also explain the observedbias. The two models can be differentiated in thatonly the perception limit model (this paper) is sen-sitive to absolute changes in overall resource avail-ability and to increases in the number of animals.Additionally, Sutherland's model predicts thatwhen deviations away from an IFD occur therewill be no differences in intake rates, while theperception limit model predicts that intake ratesshould vary between patches.

IntroductionThe ideal free distribution (1FD) th eory was devel-oped by Fretwell and Lucas (1970) and Fretwell(1972) to describe how animals should distributethemselves within an environment between patchesof varying suitability. It patch suitability declinesas the density of animals in that patch increases,the IFD predicts that the proportion of animalsin a patch will equal the proportion of resourcesavailable in that patch. For such a situation tooccur, all animals must be 'ideal' in their abilityto select a patch which maximizes fitness (theymust have perfect information about the environ-ment) and be 'free' to enter and use any patch

on an equal basis with other residents (all animalsmust be identical with respect to their competitiveabilities).In systems where patch suitability can be deter-mined, IFD theory can be used to predict animaldistributions. Published studies have fallen intotwo major categories: animals competing forbreeding opportunities (Parker 1978; Davies andHalliday 1979; Cou rtney and Parker 1985; Sargentet al. 1986) and animals competing for food (Mi-linski 1979, 1984; Zwarts and Drent 1981 ; Harper1982; Sutherland 1982; Talbot and Kramer 1986;Godin and Keenleyside 1984; Power 1984; Gillis1985). These experiments record data in either spa-tial or temporal form. The spatial experiments re-cord the proportion of resources in a patch andthe proportion of animals using those patches. Inthese studies a characteristic bias has been ob-served: patches which have a small proportion ofthe resources have disproportionately more indi-viduals using them and conversely, patches witha large proportion of the resources are underused(Table 1). In addition, Sutherland (1982) noted inhis field observations of oystercatchers (Haemato-pus ostralegus) that too many individuals usedpoor patches than would be predicted by an IFD.The temporal experiments record the distribu-tion of animals through time between two non-depleting patches. These experiments also exhibita bias towards overuse of poor patches (Table 2).Further, since all experiments contained only twopatches the most profitable patches had fewer ani-mals than predicted.Some authors have observed that their animalswere not all of equal competitive ability and hy-pothesized that the observed deviation resultedfrom a violation of the 'free' assumption (calledan ideal despotic distribution, Fretwell 1972).However, Milinski (1984) could find no effect of

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41 0T a b l e 1 . D e p a r t u r e s f r o m a n I F D f o r s p a t i a l d a t a , w h i c h i l l u s -t r a t e t h e c h a r a c t e r is t i c o v e r u s e o f p o o r p a t c h e s a n d u n d e r u s eo f g o o d p a t c h e s . A g o o d p a t c h c o n t a i n s a n a b o v e a v e r a g e p r o -p o r t i o n o f re s o u r c e s a n d a b a d p a t c h c o n t a i n s a b e l o w a v e ra g ep r o p o r t i o n o f t h e r e s o u r c e s . A s t u d y i s c o n s i d e r e d t o s h o we v i d e n c e o f a b i a s w h e n g r e a t e r t h a n o n e h a l f o f t h e o b s e r v a -t i o n s d e v i a t e i n o n e d i r e c t i o n f r o m t h e p r e d i c t e d r e s u l t .N u m b e r s i n p a r e n t h e s e s r e p r e s e n t t h e s a m p l e s iz eD a t a s o u r c e a n d s p e c i es P r o p o r t i o n P r o p o r t i o no f o b s e r v a t i o n s o f o b s e r v a t i o n sg r e a t e r t h a n l es s t h a ne x p e c t ed o n e x p e c t ed o np o o r p a t c h e s g o o d p a t c h e sCo ur tn ey and Park er (1985) 0 .76 (37) 0 .77 (35)F i g s . 5, 9 a n d r a w d a t aTarueus theophrastusDavies and Hal l i day (1979) 0 .60 (5 ) 0 .75 (4 )Fig. 13Bufo bufoGill is (1985)Fig. 3 30 fis h 0.33 (6) 0.33 (3)Fig. 3 60 fis h 0.83 (6) 0.67 (3)Fig . 3 120 fish 1.00 (6) 1.00 (3)Fig. 3 240 fish 0.67 (6) 1.00 (3)Braehydanio rerioPark er (1 978 ) 1 .00 (2 ) 1 .00 (3 )Fig. 8.3Scatophaga stercorariaSargent et al . (1986)F ig . 2a 1981 da t a 0 .80 (10) 0 .73 (11)F ig . 2b 1982 da ta 0 .67 (6 ) 0 .50 (6 )Oncorhyneus kisutehTa lbo t an d Kr am er (1986) 0 .73 (15) 1 .00 (10)F ig . 2Poeeilia reticulataZ w a r t s a n d D r e n t ( 1 9 81 )F ig . 6 Low dens i ty 0 .86 (7 ) 0 .88 (8 )Fig. 6 Hig h den sity 0.83 (6) 0.88 (8)Haematopus ostralegusT o t a l g r e a t e r t h a n 0 . 5 1 1 1 0T o t a l 1 2 1 2P r o p o r t i o n 0 . 92 0 .8 3

u n e q u a l c o m p e t i t i v e a b i l i t i e s o n a n I F D i n s t i c k l e -b a c k s (Gasterosteus aculeatus). G o d i n a n d K e e n -l e y s i d e ( 1 9 8 4 ) a l s o o b s e r v e d n o s i g n i f i c a n t d i f f e r -e n c e s i n in t a k e r a t e s o f a n i m a l s w i t h i n a p a t c hw h i c h m i g h t h a v e i n d i c a t e d d e s p o t i c b e h a v i o u ra n d a v i o l a t io n o f t h e ' f r e e ' a s s u m p t i o n . T h e r e -f o r e, b a s i n g e x p l a n a t i o n s o f d e v ia t i o n s f r o m a nI F D u p o n d e s p o ti c b eh a v i o u r s m a y n o t a l w a y s b ew a r r a n t e d .U n e q u a l c o m p e t e t i v e a b i li ti es m a y c a u s e d e v i a -t io n s f r o m a n I F D b y a m e c h a n i s m o t h e r t h a nd e s p o t i c b e h a v i o u r . S u t h e r l a n d a n d P a r k e r ( 1 9 8 5)

T a b l e 2 . D e p a r t u r e s f r o m a n I F D f o r t e m p o r a l d a t a , w h i c hi l l u s t r a t e o v e ru s e o f p o o r p a t c h e s . T h e s e s t u d i e s e x a m i n e s i n g led i s t r i b u t i o n s w i t h m u l t i p l e o b s e r v a t i o n s a t p r e d e t e r m i n e dp o i n t s i n t i m e . A g a i n , a n o v e r b i a s o c c u r s w h e n g r e a t e r t h a no n e h a l f o f t h e o b s e r v a t i o n s a r e g r e a t e r t h a n p r e d i c t e d b y a nI F D . D a t a r e p r e se n t t h e m e a n r e s u l ts o f m u l t i p l e o b s e rv a t i o n st a k e n o n l y w h e n t h e d i s t r i b u t i o n r e a c h e d a n e q u i l i b r i u m . I fa d y n a m i c s t a g e e x i s t e d i n t h e d i s t r i b u t i o n , i t w a s e x c l u d e db y f i t t i n g a s t r a i g h t l i n e t o t h e i n i t i a l p o i n t s o f t h e d i s t r i b u t i o n .T h e p o i n t a t w h i c h o b s e r v a t i o n s d i d n o t c o n f o r m t o t h i s l i n ew a s c o n s i d e r e d t o b e t h e b e g i n n i n g o f t h e e q u i l i b r i u m d i s t r i b u -t i o nD a t a s o u r c e a n d s p e c ie s D a t a f r o m l e a s t p r o f i t a b l e p a t c h

R a t i o P r o p o r t i o n no f o b s e r v a t i o n sgrea te rt h a n e x p e c t edG o d i n a n d K e e n l e y s i d e (1 9 84 )Fig. 2 2:1 0.61 28Fig. 2 5:1 i .00 25Aequidens curvicepsHarper (1982)F ig . 2 2 :1 0 .50 12Anas pIatyrhynchosM il insk i (1979)Fig. 1 5 : 1 0.71 28Fig . 2 2 :1 1.00 15Fig. 2 2:1 1.00 13Gasterosteus aculeatusM il insk i (1984)F ig . 3 2 :1 0 .44 16Fig. 3 2:1 0.38 16Gasterosteus aculeatusT o t a l g r e a t e r t h a n 0 . 5 5T o t a l 8P r o p o r t i o n 0 . 6 2 5

a n d P a r k e r a n d S u t h e r l a n d ( 1 9 8 6 ) a r g u e t h a t i ne x p e r i m e n t s w i t h a c o n t i n u o u s - i n p u t o f r es o u r c es ,u n e q u a l c o m p e t e t i v e a b i l i ti e s w il l r e s u l t in a n i m a l sd i s tr i b u t in g t h e m s e l v es s u c h t h a t t h e s u m o f c o m -p e t e t i v e a b i l i t i e s w i l l c o n f o r m t o t h e p r e d i c t e d d i s -t r i b u t io n o f an I F D . I f t h e a n i m a l s a r e n o t a l lo f e q u a l c o m p e t e t i v e a b i l i ty , t h e d i s t r i b u t i o n o f a n -i m a l s w i l l n o t n e c e s s a r i l y c o n f o r m t o a n I F D . T h i sm e c h a n i s m d o e s n o t p r e d i c t a s p e ci f ic d e v i a t i o nf r o m a n I F D a n d a s s u c h w o u l d n o t p r e d i c t t h ec o n s i s t e n t o v e r b i a s o b s e r v e d i n t h e l i t e r a t u r e .S u t h e r l a n d ( 1 9 8 3 ) h a s s u g g e s t e d i n s i t u a t i o n sw h e r e t h e r e i s n o c o n t i n u o u s - i n p u t , a s i n m o s t f i e ldo b s e r v a t i o n s , d e v i at i o n s f r o m a n I F D c o u l d b e d u et o i n t e r f e r e n c e . I n S u t h e r l a n d ' s m o d e l , i n t e r f e r e n c eis c o n s id e r e d t o b e a n y k i n d o f i n te r a c t io n b e t w e e np r e d a t o r s w h i c h r e d u c e s s e a r c h i n g e f f i c i e n c y . T h i si s d i s t i n c t f r o m d e s p o t i c b e h a v i o u r i n t h a t a l l a n i -

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41 1

o~

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PL . . . . .

NN u m b e r o f A n i m a l s i n t h e E n v i r o n m e n t

F i g . 1 . T h e r e l a t i o n s h i p b e t w e e n t h e d i f f e r e n c e i n p a t c h s u i t a b il -i t y a n d t h e n u m b e r o f a n i m a l s i n t h e e n v i r o n m e n t . I f a l i m i t e da b i l i t y e x i s t s t o r e s o l v e d i f f e r e n c e s b e t w e e n p a t c h s u i t a b i l i t i e s( P L ) t h e n o n l y a l i m i t e d n u m b e r o f a n i m a l s ( n ) w i l l b e a b l et o c h o o s e w h i c h p a t c h t o e n t e r. A l l a n i m a l s b e y o n d t h i s p o i n tw i l l b e u n a b l e t o r e s o l v e d i f f e r e n c e s i n p a t c h s u i t a b i l i t y a n dw i l l h a v e t o c h o o s e a p a t c h a t r a n d o m ( " g u e s s " )

m a l s w i t h i n a p a t c h a r e a f f e c t e d e q u a l l y ; d e s p o t i cb e h a v i o u r r e s u l t s i n u n e q u a l u s e o f r e s o u r c e s b yi n d i v i d u a l s w i t h i n a p a t c h . S u t h e r l a n d ' s ( 1 9 83 )m o d e l a s s u m e s t h a t i n t e r f e r e n c e w i l l c a u s e p a t c hs u i t a b i l i t y t o d e c l i n e m o r e r a p i d l y w i t h i n c r e a s e si n a n im a l d e n s i ty t h a n w o u l d b e p r e d i c te d b y a nI F D . T h e r e f o r e , t h e d e v i a t io n f r o m a n I F D m a yb e d u e s o l e ly t o e r r o r s i n p e r s p e c t iv e , w i t h t h e o b -s e r v e r o v e r e s t i m a t i n g p a t c h s u i t a b i l i t i e s a t h i g hd e n s i t i e s i n c o m p a r i s o n t o t h e i r a c t u a l v a l u e t ot h e an i m a l . W h a t a p p e a r s t o t h e o b s e r v e r a s a de -p a r t u r e f r o m a n I F D m a y a c t u a l l y b e a p e r f e c tI F D f r o m t h e a n i m a l s ' p e r sp e c ti v e .

T h i s p a p e r p r e s e n t s a n a l t e r n a t i v e m o d e l t o e x -a m i n e d e p a r t u r e s f r o m I F D ' s b a s e d o n l i m it e d p e r -c e p t u a l a b i l i t i e s . I t e x a m i n e s t h e p r e d i c t e d r e s u l t so f a n I F D e x p e r i m e n t w h e n a n i m a l s h a v e a l im i t e da b i l i t y t o r e s o l v e d i f f e r e n c e s i n p a t c h s u i t a b i l i t yw h i l e f o r a g i n g f o r f o o d .

Me t h o d sI n d e s i g n i n g t h i s m o d e l I a s s u m e d t h a t a n i m a l s a s s e s s p a t c hs u i t a b i l i t y f r o m p e r c e i v e d i n d i v i d u a l i n t a k e r a t e s , n o t f r o m t h et o t a l a m o u n t o f f o o d a v a i l a b le i n th e p a t c h ( se e H a r p e r 1 9 8 2;M i l i n s k i 1 9 8 4) . H e n c e , t h e q u a l i t y o f a p a t c h i s e q u a l t o t h et o t a l a m o u n t o f f o o d a v a i la b l e ( F ) d iv i d e d b y t h e to t a l n u m b e ro f a n i m a l s i n t h e p a t c h ( n) p lu s t h e a n i m a l w h i c h i s m a k i n gt h e d e c i s i o n w h e t h e r o r n o t t o e n t e r o r s t a y i n t h a t p a t c h ,i .e . p a t c h s u i t a b i l it y e q u a l s F / ( n + 1 ).

A c o n s e q u e n c e o f a n i m a l s a s s e s s in g p a t c h s u i t a b i li t y i n th i sm a n n e r i s t h a t t h e d i f f e r e n c e b e t w e e n a n y t w o p a t c h e s w i l lb e g r e a t e s t a t t h e l o w e s t t o t a l d e n s i t ie s o f p r e d a t o r s . T h e e f f e c to f a n i m a l s p r e f e r e n t i a l l y g o i n g t o t h e b e s t p a t c h e s , t h e r e b y r e -

no

I ESTABLISH INITIAL CONDITIoNs]number of patchesfood distributionanimal perception limits ,

l ONE NI MAL ENTERS ENVIRONMENT

U S S E S S Q H A L Y O F A LL .A B T A T S Iequals individual intake ratesF/(N+I)

ENTER A HABITAT BASED ON DECISION RULErandomly choose a habitat f rom tbosewhose quality is bounded by the qualityof the bes t habitat and the besthabitat minus the perception limit

IARE ALL ANIMA LS IN THE ENVI RONMENT? Iye s

I S T B T O N I

I D o Y M OV E ONE I I V I D U A L FR OMA I T A T I

[C H OO S E ONE I T A T A S A BOV E

[RECORD D I S T R I B U T I O N ll o o p I 0 0times

[CALCULATE MEAN AND VARIANCE OF DISTRIBUTION]F i g . 2 . F l o w c h a r t i l l u s t r a t in g m o d e l o p e r a t i o n . S e e te x t f o r d e -t a i l sd u c i n g t h e i r s u i t a b i l i t y , w i l l l o w e r t h e a b s o l u t e d i f f e r e n c e b e -t w e e n p a t c h e s . A s a r e s u l t, w i t h a f i x e d f o o d s u p p l y t h e d i f f e r -e n c e s b e t w e e n p a t c h e s w i l l d e c r e a s e a s y m p t o t i c a l l y a s t h en u m b e r o f a n im a l s i n t h e e n v i r o n m e n t in c r e a s e s ( F ig . 1 ).I f a n i m a l s a r e ' i d e a l ' t h e y w i l l b e u n a f f e c t e d b y t h i s d e c a yf u n c t i o n . H o w e v e r , i f t h e r e i s s o m e f i n i t e l e v e l b e y o n d w h i c hd i f f e re n c e s i n p a t c h s u i t a b i li t y c a n n o t b e r e s o l v e d ( a p e r c e p t i o nl i m it ) , th e n t h o s e a n i m a l s e n t e r i n g a f t e r t h i s p o i n t w i l l b e u n a b l et o g a i n s u f f i c i e n t i n f o r m a t i o n t o c h o o s e t h e m o s t p r o f i t a b l epa tch (F ig . 1 ) .

T h e i n i t i a l c o n d i t i o n o f t h e m o d e l ( F i g . 2) w a s a n e n v i r o n -m e n t c o n t a i n i n g a n u n d e p l e t a b l e f o o d s u p p l y a v a i l a b l e a t ar a t e o f 2 0 f o o d u n i t s p e r u n i t t i m e , d i s t r ib u t e d b e t w e e n t w o ,t h r e e , o r f o u r p a t c h e s . S i m u l a t i o n s w e r e p e r f o r m e d i n s i t u a t i o n s

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41 22 P a t c h e s 3 P a t c h e s 4 P a t c h e s"1-

o t / t / t.7 5 / ' J " . < "EL 0.5 ~ ~Z 0 . 2 5 / / ; " " "~=_ _

f . O ' ' ' / '...l

< 0 5 1 s - w "ZOI -O 0.75 , _ ~ = _ , 1 " / "

0225 0 ' . 5 0;75 0;25 0[5 0.75

l PL = 2

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PL = 3

0'.25 o' .s 0' .75PROPORTION OF RESOURCES IN PATCH

Fig. 3 . Resul t s of s imula t ions forthree d i f f e rent pe rcept ion l imi t s(PL) of anima l s in ane nv i r onm e n t w i t h t he r e s ou r c e sdi s t r ibuted be tween two, three orf ou r pa t c he s . T he d ia g o n a l l in esr e p r e s e n t t he d i s t r i bu t i onp r e d i c t e d by I FD , t he c u r v e d l i n esa r e i n t e r po l a t i ons o f t he s i m u l a t edd i s t r i bu t i ons

w he r e t he f ood w a s d i v i de d be tw e e n pa t c he s i n s uc h a w a yt ha t no pa t c h c on t a i ne d l es s t ha n t w o f ood un i t s .T he m ode l op e r a t e d by s e que n t i a ll y e n t e r ing 10 a n i m a l sin to the envi ronment . Eve ry anima l was ident i ca l , i . e . , t heyw e r e e a c h c a pa b l e o f c ons um i ng a l l t he f ood by t he m s e l ve s ,t he y ha d t he s a m e pe r c e p t i on l i m i t (t he y w e r e a b l e t o pe r c e i vedi f f e rences in pa tch qua l i ty of one , two, o r t hree uni t s of foodpe r un i t t i m e p e r i nd i v i dua l ) , a nd t he y a l l u s e d t he s a m e de c i s ionrule . The dec i s ion ru l e was s imply to go to th e bes t pa t ch w hent ha t c ou l d be pe r c e i ve d . I f t he be s t pa t c h c ou l d no t be r e s o l ve df rom th e next be s t pa t ch d ue to the d i f f e rence in sui t abi l i t ybe ing l es s than the pe rcep t ion l imi t , one o f the indi s t ingu i shablepa t c he s w a s c hos e n a t r a ndom . I n a m u l t i - pa t c h s y s t e m , a sthe pe rcep t ion l imi t i nc rease s , or pa tch sui t abi l i t i e s becom em or e s i m i l a r , t he a n i m a l m us t c hoos e r a ndom l y be t w e e n a ni nc r ea s i ng num be r o f pa t c he s . T h i s de c i s i on r u le c a n m o s t e a s i l ybe e xp l a i ne d by a num e r i c a l e xa m pl e . C ons i de r a n a n i m a l e n t e r -i ng a n e nv i r onm e n t w i t h f ou r pa t c he s w h i c h ha ve pa t c h s u i t a b i -l i t i e s of 3 .5 , 2 , 1 .4 and 1 . I f t he anima l ha s a pe rcept ion l imi tof one i t wi l l cor r ec t ly ente r t he pa tch wi th the h ig hes t su i t abi l i -ty . How eve r , i f t he pe rcep t ion l imi t is two the an ima l wi l l beunab le to pe rce ive d i f f e rences be tween 3 .5 and anyth ing grea te rt ha n 1 .5 a nd w i l l ha ve t o c hoos e r a ndo m l y b e t w e e n the t w obe s t pa t c he s. A s t he pe r c e p t i on l i m i t i ncr e a se s t he p r oba b i l i t yi nc re a s es tha t a n a n i m a l w i l l ha ve t o c hoos e a pa t c h r a n dom l y(hence for th r e fe r r ed to a s gues sing) .A f t e r a n i n t i a l d i s t r i bu t i on w a s de t e r m i ne d , t he a n i m a l swere a l lowed to r edi s t r ibute . Thi s i s ana logous to swi t ching,w h i c h ha s be e n o bs e r ve d i n I FD e xpe r i m e n t s ( M i l i n s k i 1979 ;pe r s ona l obs e r va t i on ) . I nd i v i dua l s w e r e c hos e n r a ndo m l y t ova c a t e a pa t c h ( c a us i ng a n a d j u s t m e n t i n t he r a t e o f f ood a va i l -a b i l it y ) a nd t o r e c hoos e a pa t c h u s i ng t he o r i g i na l de c is i onr u l e. O ne hund r e d r e s a m pl i ngs w e re a l l ow e d .D a t a w e r e r e c o r de d on t he pos i t i on o f t he a n i m a l s f o re a c h o f t he 100 r e s a m pl i ng pe r i ods a nd t he f i na l r e s u l t ob t a i ne dwas the ov e ra l l ave rage . The s im ula t ion was run 10 t imes fore a c h d i s t r i bu t i on o f f ood a nd pe r c e p t i on l i m i t a nd t he m e a na nd va r i a nc e o f t he d i s t r i bu t i ons w a s r e c o r de d .T he de v i a t i ons f r om a n I FD p r oduc e d by t h i s m ode l w e r ec om pa r e d t o t he p r e d i c t i ons o f Su t he r l a nd ' s ( 1983 ) m ode l t o

de t e r m i ne i f t he y c ou l d be d i s t i ngu i s he d qua n t i t a t i ve l y . T hep r e d i c t e d I FD d i s t r i bu t ions f r om Su t he r l a nd ' s m od e l a r e ge ne r-a t ed by the equa t ion B~=ca i 1/m where B i i s t he pro po r t ion ofanima l s in the i t h pa tch , a~ i s t he pr op or t io n o f r e sources inthe i t h pa tch , m i s t he degree of in t e r fe rence and c is a norm a l i z -ing cons tan t such tha t t he B~ va lues sum to u ni ty .

R e s u l t s

T h e r e s u l t s o f t h e s i m u l a t i o n a r e i l l u s t r a t e d inF i g . 3. T h e m o s t s t r i k in g r e s u l t o f t h i s s i m u l a t i o ni s t h a t a n y d e v i a t i o n f r o m p e r f e c t i n f o r m a t i o n w i l lp r o d u c e o n l y o n e t y p e o f b i a s , a n u n d e r u s e o f g o o dp a t c h e s a n d a n o v e r u s e o f p o o r p a t c h e s .

T h e s iz e o f t h e d e v i a t i o n i s a f f e c t e d b y t h e p e r -c e p t i o n l im i t . A s t h e d e v i a t i o n f r o m a n I F D i n -c r e a s e s , s o d o t h e d i f f e r e n c e s i n i n d i v i d u a l i n t a k er a te s . S i n ce a n i m a l s c h o o s e p a t c h e s u p o n t h e b a s i so f i n d i v i d u a l i n t a k e r a t e s , a n y d e v i a t i o n w h i c h e x -c e e d s th e i r p e r c e p t i o n l i m i t w i l l p r o v i d e e n o u g hi n f o r m a t i o n f o r t h e n e x t a n i m a l t o c h o o s e t h e co r -r e c t p a t c h . T h e r e f o r e t h e p e r c e p t i o n l i m i ts p r o v i d ef e e d b a c k w h i c h l i m i t t h e d i f f e r e n c e in i n d i v i d u a li n t a k e r a t e s a n d t h e m a g n i t u d e o f t h e d e v i a t i o nf r o m a n I F D .

T w o o p p o s i n g f o r c e s a f f e c t t h e s h a p e o f t h ed i s t r i b u t i o n . A s t h e e n v i r o n m e n t b e c o m e s l e s s u n i -f o r m , t h e i n i t i a l d i f f e r e n c e s i n p a t c h s u i t a b i l i t y in -c r ea s e a n d t h e n u m b e r o f a n i m a l s w h i c h m u s tg u e s s is r e d u c e d . T h i s p r o v i d e s a fo r c e l i m i t i n g t h ed e v i a t i o n f r o m a n I F D a t e x t r e m e r e s o u r c e d i s t r i -b u t i o n s . H o w e v e r , s in c e g u e s s i n g d i s t r i b u t e s t h ea n i m a l s i n a u n i f o r m m a n n e r , g u e s s i n g a n i m a l s a r e" m o r e w r o n g " a t l e s s u n i f o r m r e s o u r c e d i s t r i b u -

8/3/2019 Abrahams 1986

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o.1~ 1 ~... . .~... ' "" , ,, ~ > , . . ) . ) ) ) , , . . . , , ,, , " , P L = 3- 0 . 1 "....... ",,,-0 .2 . .. .. . "-- 0 . 3 ~ " ....... " -

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F i g . 4 . T h e b i a s e s p r o d u c e d f r o m a n I F D a s a r e s u l t o f t h ec o m b i n e d e f f ec t s o f v a r y i n g a b i l i t ie s t o p e r c e i v e d i f f e re n c e s i np a t c h s u i t a b il i t y ( P L ) a n d t h e n u m b e r o f a v a il a b l e p a t c h e s.T h e d e v i a t i o n c o r r e s p o n d s t o t h e o b s e r v e d d i s t r ib u t i o n m i n u st h e p r e d i c t e d d i s t r i b u t i o n . S o l i d l i n e s r e p r e s e n t f o u r p a t c h s y s -t e m s , d o t t e d l i ne s t h r e e p a t c h s y s t e m s a n d d a s h e d l i n e s t w o p a t c hs y s t e m s

t i o n s. T h i s p r o v i d e s a f o r c e p r o d u c i n g l e ss d e v i a -t i o n a t a u n i f o r m r e s o u r c e d i s t ri b u t i o n . A c o n s e -q u e n c e o f t h e s e c o n f l i c t i n g f o r c e s is a s m a l l e r d e -v i a t i o n f r o m a n I F D a t e x t r e m e re s o u r c e d is t ri b u -t i o n s w i t h g o o d p e r c e p t i o n l i m i t s ( P L = 1 , F ig . 4).H o w e v e r , t h i s t r e n d d o e s n o t c o n t i n u e a s P L i n -c r e a s e s ( P L = 2 a n d 3 , F i g , 4 ), b e c a u s e t h e l a r g ed i f f e r en c e s i n p a t c h s u i t a b i l it y a r e n o l o n g e r g r e a te n o u g h t o b e p e r c e i v e d a n d t h e n u m b e r o f g ue s se rsi s n o l o n g e r r e d u c e d . T h e r e f o r e t h e i n e f f ic i e n c y o fg u e s s i n g c a u s e s t h e d e v i a t i o n t o i n c r e a s e a t e x t r e m er e s o u r c e d i s t r i b u t i o n s .Perception limit model vs Sutherland's modelS u t h e r l a n d ' s ( 19 8 3 ) i n t e r f e r e n c e m o d e l p r o d u c e sa p p r o x i m a t e l y s i m i l a r d y n a m i c s t o t h i s m o d e lw h e n l a r g e m v a l u e s ( l e ve ls o f i n t e r f e r e n c e ) a r eu s e d ( m a x i m u m m v a l u e f r o m t h e l i t e r a tu r e is 1 .1 3 )

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