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An examination of risk-sensitivity in rufous hummingbirds Paton, Steven Robert 1986

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EXAMINATION OF RISK-SENSITIVITY IN RUFOUS HUMMINGBIRDS by STEVEN ROBERT PATON B.Sc.,McMaster University,Hami1ton,1983 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE in THE FACULTY OF GRADUATE STUDIES (Department of Zoology) We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA July 1986 (5) Steven Robert Paton, 1986 In presenting t h i s thesis i n p a r t i a l f u l f i l m e n t of the requirements for an advanced degree at the University of B r i t i s h Columbia, I agree that the Library s h a l l make i t f r e e l y available for reference and study. I further agree that permission for extensive copying of t h i s thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. I t i s understood that copying or publication of t h i s thesis for f i n a n c i a l gain s h a l l not be allowed without my written permission. Department of ZOOLOGY  The University of B r i t i s h Columbia 1956 Main Mall Vancouver, Canada V6T 1Y3 Date A T T G . 1 3 r IQRfi i i ABSTRACT A number of animal species in several major taxa have been shown to be sensitive to both the mean and the variance of prey p r o f i t a b i l i t y under laboratory conditions. This type of behaviour is referred to as r i s k - s e n s i t i v e foraging because i t is thought to minimize the risk of starvation. This thesis consists of three parts. F i r s t , I experimentally examine which of two r i s k - s e n s i t i v e foraging models best predicts the r i s k -sensitive foraging behaviour of rufous hummingbirds in the laboratory. Second, I use computer simulations to explore the a b i l i t y of r i s k - s e n s i t i v e foraging to decrease the risk of starvation over r i s k - i n s e n s i t i v e foraging, and to examine the s e n s i t i v i t y of the predictions of one r i s k - s e n s i t i v e foraging model to changes in some of i t s assumptions. Third, I consider several possible explanations for my observations of r i s k -sensitive foraging behaviour in rufous hummingbirds, even though my computer simulations indicate that such behaviour probably does not s i g n i f i c a n t l y reduce the risk of starvation. There are two main classes of models describing how animals should forage when the quality or frequency of food items i s variable. I test rufous hummingbird preferences between two alternatives to determine whether rufous hummingbird r i s k -sensitive foraging preferences are independent of the absolute value of the mean food value. The results contradict the variance discounting models' prediction of constant r i s k -aversion, and support the z-score model's prediction of decreasing risk-aversion. The z-score model accurately predicts q u a l i t a t i v e behaviour, but does not explain several aspects of individual behaviour. I use computer simulations to examine the advantage of ri s k - s e n s i t i v e foraging, in terms of decreasing the risk of starvation, by comparing the predicted preferences and estimated p r o b a b i l i t i e s of starvation of a hypothetical r i s k - s e n s i t i v e (z-score model) forager with those of a hypothetical r i s k -insensitive (mean-maximizing) forager. I also examine the s e n s i t i v i t y of the predictions of the z-score model to changes in some of i t s assumptions. The z-score model seldom s i g n i f i c a n t l y reduce the risk of starvation despite the often considerable differences in foraging preferences. Foraging preferences predicted by the z-score model are very sensitive to plausible changes to some of i t s assumptions. Expected p r o b a b i l i t i e s of starvation are r e l a t i v e l y insensitive to major changes in preferences. I consider the apparent paradox that rufous hummingbirds are r i s k - s e n s i t i v e even though th i s foraging strategy probably does not reduce the ris k of starvation. My experimental observations of r i s k - s e n s i t i v e foraging probably do not represent an experimental a r t i f a c t . I am unable to analyze the v a l i d i t y of my computer simulations because the assumptions used in the simulations prevent dir e c t comparison to natural situations. The most productive avenue of research in the future may be a comparative study of a range of species which examines possible h i s t o r i c a l and ecological influences on the observed patterns of r i s k - s e n s i t i v e foraging behaviour. > iv TABLE OF CONTENTS ABSTRACT i i LIST OF TABLES v LIST OF FIGURES v i i ACKNOWLEDGEMENTS v i i i Chapter 1 GENERAL INTRODUCTION 1 Why Study Ri s k - S e n s i t i v i t y in Hummingbirds? 5 Chapter Summaries 6 Chapter 2 HOW CONSTANT IS THE CONSTANT OF RISK-AVERSION? 9 Introduction 9 The models 10 Variance Discounting 10 Z-score Model 14 Comparison of the Models 15 How Constant i s k? 19 Materials and Methods 21 Results 30 Discussion 37 Chapter 3 ASSUMPTIONS IN OPTIMAL FORAGING MODELS: A RE-EXAMINATION OF RISK-SENSITIVITY 41 Introduction 41 Models 44 Simulation Results 48 Preferences 48 Probabil i t y of starvation 52 Discussion 53 Chapter 4 The paradox o f r i s k - s e n s i t i v e foraging 59 Resolving the paradox 60 Future research 66 Literature c i t e d 68 LIST OF TABLES Table 2.1. Feeder preferences Table 4.1. Mean and standard deviation of standing nectar crops v i i LIST OF FIGURES Figure 1.1. Frequency histogram of r i s k - s e n s i t i v i t y papers 3 Figure 2.1. The indifference relationships predicted by the variance discounting model 12 Figure 2.2. The indifference relationships predicted by the z-score model 17 Figure 2.3. Internal view of experimental boxes 22 Figure 2.4. Cross-section view of feeder assembly 25 Figure 2.5. The relationship between mean preference scores and Rc - b 35 Figure 3.1. Simulation results 50 v i i i ACKNOWLEDGEMENTS I would l i k e to thank Lee Gass, whose supervision, tolerance and support made thi s thesis much more than just an academic exercise. I would l i k e to thank Doug Armstrong and Stefan Tamm (even though they l e f t too e a r l y ) , Gayle Brown and Glenn Sutherland for the i r friendship, emotional and i n t e l l e c t u a l support, and off-beat sense of humour. Without these people, things l i f e would have been much less enjoyable. Thanks to D.W. Stephens for his help and advice with Chapter 1. The following people read and provided valuable c r i t i c i s m s on one or more chapters of th i s thesis: Lee G., Jamie Smith, Don Ludwig, Don Wilkie, Steve Lima, Doug A., Stefan T., and Gayle B. Special thanks to Dinsdale and Herman Glukes for their help. Financial support was provided by an NSERC Postgraduate Scholarship, and NSERC grant 58-9876 to C.L. Gass. CHAPTER 1 GENERAL INTRODUCTION . . . Yet we a l l know that in trying to determine something on the boundary l i n e of the unknown, i t i s necessary to make assumptions. The assumptions can be made over a grey area of uncertainty and one can shade them in one dire c t i o n or the other with perfect honesty, but in accord with - uh - the emotions of the moment (Asimov 1972). Optimal foraging theory (OFT) is a diverse framework for the investigation of animal foraging behaviour. One of the major underlying assumptions in a l l OFT models i s that natural selection has resulted in 'optimal' behaviour (Maynard Smith 1978). McFarland (1977) expressed t h i s assumption c l e a r l y when he wrote, . . . we can expect natural selection to shape the decision making process of animals in such a way that the resultant behavior sequences are optimally adapted to the current environmental circumstances. In other words, we can regard natural selection as a designing agent capable of producing the optimal design for any given set of conditions. The history of OFT spans approximately twenty years and has been reviewed a number of times (Pyke et a_l. 1977; Maynard Smith 1978; Krebs et a l . 1981; Krebs and McCleery 1984; Pyke 1984; Gray 1986). OFT has been applied to the problems of diet choice, a l l o c a t i o n of e f f o r t , exploitation of patches and learning (Krebs and McCleery 1984), t e r r i t o r i a l i t y , and central place foraging. 2 The study of how animals do/should forage under conditions of uncertainty in their environment i s a r e l a t i v e l y recent development in OFT. Its origins can be traced to the work of economists such as Luce and Raiffa (1957), Farrar (1962), Degroot (1970), Gould (1974), and Keeney and Raiffa (1976) who developed theories predicting human behaviour in the face of economic uncertainty. Since the early 1950's, psychologists have studied how animals respond to variation in mean reinforcement, and to variable reinforcement delays. A number of these works were c i t e d frequently in some of the f i r s t ecological papers on r i s k - s e n s i t i v e foraging behaviour. The most commonly c i t e d of these papers are: Leventhal et a l . 1959, Pubols 1962, Herrnstein 1964, Logan 1965 a,b, and Fantino 1967. The development of r i s k - s e n s i t i v e foraging models in ecology began when i t was recognized that many aspects of foraging are stochastic in nature and are most accurately described by "statements of pr o b a b i l i t y " (Stephens and Charnov 1982). U n t i l recently, however, most OFT models have been deterministic, substituting expected values wherever p r o b a b i l i t i e s e x i s t . The number of papers published by ecologists which describe or test theories describing foraging behaviour under conditions of uncertainty has steadily increased since Oaten (1977) (see F i g . 1.1). In 1980-1981 three d i f f e r e n t authors (Real 1980 a,b; Caraco 1980; Stephens 1981) developed models which predicted how foragers may combine information about the mean and variance of net energetic values of prey items. These models a l l assume 3 Figure 1.1. Frequency of r i s k - s e n s i t i v i t y papers published since 1957. Publications f a l l into four categories: (1) Theoretical ecology papers, papers dealing with foraging under conditions of uncertainty, (2) Ecology, describing tests of r i s k - s e n s i t i v e foraging models, (3) Psychology, dealing with variable reinforcement and/or variable reward delay experiments, and (4) Economics, dealing with investment strategies under conditions of uncertainty. Groups (3) and (4) are not exhaustive surveys of the l i t e r a t u r e , but represent the most commonly c i t e d papers from groups (1) and (2). Ecology (theoretical) o c c r 12 11 10 9 8 7 6 5 4 3 2 1 0 :IXL I Ecology (empirical) • Psychology Economics 1 D f 57 59 61 63 '65 '67' '69' 71' '73 W W i J ' W W Year of Publication 5 that animals forage in a manner that maximizes their f i t n e s s . Furthermore, these models assume that fitness i s a monotonically increasing function of some simple combination of mean energetic reward and reward variance. This type of behaviour has been termed ' r i s k - s e n s i t i v e foraging' (Stephens 1981). When I began working on t h i s thesis in the Spring of 1984, r i s k - s e n s i t i v i t y had been documented in a number of species. One of the most important problems at the time was to determine which of the current r i s k - s e n s i t i v i t y models best described foraging behaviour. This thesis was begun with the intention of testing and r e f i n i n g the accuracy of these models. Why Study R i s k - S e n s i t i v i t y in Hummingbirds? There are many a p r i o r i reasons to assume that hummingbirds are an ideal group of animals for studying r i s k - s e n s i t i v i t y , and most of them are consequences of the small size of most species (< 7-8 g). Species range in size from the Cuban Bee Hummingbird (Mellisuga helenae), less than 2.0 g, to the Giant Hummingbird (Patagonia qigas), 20.2 g (Brown and Bowers 1985). The species that I used, the rufous hummingbird (Selaphorus rufus), averages 3.5 g, which i s less than the median weight of 5.0 g for the 191 species l i s t e d by Brown and Bowers. Because of their small siz e , hummingbirds have higher resting metabolic rates than a l l other vertebrates with the exception of some shrews (Johnsgard 1983). Their extremely high metabolic rates - and expensive mode of locomotion require 6 frequent feeding, in some cases over 5000 flower v i s i t s per day. Rufous hummingbirds spend as l i t t l e as 10% of t h e i r time foraging in good quality habitats (Sutherland e_t a l . 1 982) to over 70% in marginal habitats (Gass 1978). In the laboratory rufous hummingbirds commonly consume more than their own weight of 30% (w/w) sucrose solution each day (personal observation). High energetic demands imply that any behaviours a f f e c t i n g energy intake rates w i l l have very rapid physiological e f f e c t s . Under s t r e s s f u l laboratory conditions, Rufous hummingbirds can lose over 20% of their body weight in one day: starvation can occur in birds not fed for as l i t t l e as two days. Several studies have shown that several aspects of hummingbird behaviour are sensitive to changes in the available energy in their environment (Gass et a l . 1976; Gass 1979; Montgomerie and Gass 1981; Hixon et a l . 1983). It i s reasonable to expect that an animal that i s highly sensitive to energy intake rate, w i l l also be sensitive to variance in t h i s rate. They should, therfore, display clear signs of r i s k - s e n s i t i v e foraging behaviour. Chapter Summaries Two major classes of models describe r i s k - s e n s i t i v e foraging behaviour. The f i r s t class of models, now refered to as variance discounting models, were developed by Real (1980 a,b), Caraco 1980, and others. The second class, represented by the z-score model, was developed by Stephens (1981) and Stephens and Charnov (1982). In Chapter 2 I present an experiment designed to determine whether the variance discounting model or 7 the z-score model best describes the foraging behaviour of rufous hummingbirds and, by extrapolation, other r i s k - s e n s i t i v e foragers. The results of t h i s experiment c l e a r l y show that rufous hummingbirds are r i s k - s e n s i t i v e and that their behaviour vi o l a t e s the variance discounting model's prediction of constant risk-aversion. The results agree q u a l i t a t i v e l y with the predictions of the z-score model; however, the model f a i l e d to explain the large amount of individual v a r i a b i l i t y in preferences. Chapter 2 i s in press as Stephens and Paton. A l l models are based on assumptions, but modelers commonly f a i l to examine the dependence of their predictions on these assumptions. Risk-sensitive foraging models are based on the assumption that in a stochastic environment a forager's survivorship i s enhanced by s e n s i t i v i t y to uncertainty in energetic gains. The models employ a number of additional assumptions, both e x p l i c i t and i m p l i c i t , which have never been adequately examined. Although i t is r e l a t i v e l y easy to test the predictions of these models, i t i s much more d i f f i c u l t to s i m i l a r l y test the v a l i d i t y of individual assumptions. In Chapter 3 I use computer simulations to explore the s e n s i t i v i t y of the z-score model to changes in some of i t s underlying assumptions. The assumptions examined include the foragers' a b i l i t y to discriminate between alternative reward d i s t r i b u t i o n s , the number of prey items they take per day, and the nature of how foragers make decisions about how, when and where to feed. I compare a simple deterministic foraging model with the z-score model to gauge the advantage p o t e n t i a l l y 8 realized through r i s k - s e n s i t i v e foraging. Using graphical comparisons, I show that simulated foraging preferences are very sensitive to changes in the assumptions examined, but that predicted p r o b a b i l i t i e s of starvation are r e l a t i v e l y i n s e n s i t i v e . Under most of the conditions examined, I found that the deterministic model and the z-score model predicts nearly i d e n t i c a l survival rates even though the predicted foraging preferences are often very d i f f e r e n t . On the basis of these findings, I c r i t i c a l l y examine some current trends in optimal foraging theory and experimentation. In Chapter 4 I explore the apparently contradictory conclusions of Chapters 2 and 3 by examining the ecological relevance of my experiment and my simulations. An analysis of data on nectar standing crops suggests that my conclusion in Chapter 3 about the value of r i s k - s e n s i t i v e foraging may have been too severe for some natural nectar d i s t r i b u t i o n s . It remains unknown, however, how many of the factors involved in foraging that have not been examined may affe c t the a b i l i t y of r i s k - s e n s i t i v e foraging to reduce the pro b a b i l i t y of starvation. I conclude that, while rufous hummingbirds do demonstrate r i s k - s e n s i t i v e foraging behaviour in the laboratory, this behaviour i s inadaquately described by current theories. It remains unclear what role r i s k - s e n s i t i v i t y plays in the behaviour of wild hummingbirds. I believe that a comparative study of r i s k - s e n s i t i v e foraging behaviour that focuses on species differences in r i s k - s e n s i t i v i t y may be helpful in resolving t h i s issue. 9 CHAPTER 2 HOW CONSTANT IS THE CONSTANT OF RISK-AVERSION? Introduction A forager may prefer a constant gain of 10 c a l o r i e s to a gamble that y i e l d s 0 or 20 c a l o r i e s with equal p r o b a b i l i t y , or i t may prefer the gamble. If i t prefers the constant gain, then i t s preference i s c a l l e d risk-averse, but i f i t prefers to gamble, then i t s preference i s c a l l e d risk-prone. Many studies have shown that such prefences e x i s t , (Leventhal et a l . 1959; Caraco et a l . 1980; Real 1981; Waddington et a l . 1981, Caraco 1983; Barnard and Brown 1985; Ba t t a l i o et a_l. in press; Wunderle and O'Brien in press). Two studies have examined the more complicated case in which the alternatives vary in both mean and variance (Real e_t a_l. 1982; Caraco and Lima 1985). When food gains are normally d i s t r i b u t e d , risk (of energetic s h o r t f a l l ) i s equivalent to variance. Foraging behaviours which take into account both mean and variance have been termed ' r i s k -s e n s i t i v e ' . Since i t s introduction into the l i t e r a t u r e , r i s k -s e n s i t i v i t y has forced a re-evaluation of foraging theory because o r i g i n a l l y foraging theory had assumed that knowing the mean gain was s u f f i c i e n t to predict forager preferences. The existence of r i s k - s e n s i t i v e preferences has been well established. The exact nature of the relationship between the mean and variance, however, remains unclear. For example, how much mean w i l l a risk-averse forager s a c r i f i c e to gain a unit 10 reduction of variance; do foragers trade off means and variances? Behavioural ecologists have proposed two diff e r e n t relationships between mean and variance: the variance discounting and z-score models. The models make di f f e r e n t assumptions about how mean and variance interact to influence preference. I w i l l b r i e f l y describe each model, and describe the results of an experiment designed to distinguish between the two models. The models Variance Discounting Many authors (Oster and Wilson 1978; Caraco 1980; Real I980a,b) have argued that foragers should maximize a linear combination of the mean and variance. Real (1980a,b) has c a l l e d t h i s the variance discounting model. This model proposes that foragers should maximize: M _ k c r 2 where M is the mean food gain, o2 i s the variance in food gain, and 'k' i s assumed to be 'a constant associated with the degree to which variance (uncertainty) i s ev o l u t i o n a r i l y or behaviorally undesirable' (Real 1981) and is c a l l e d the 'constant of risk-aversion.' This model assumes that the 'fitnes s ' value of food increases continuously and smoothly with increasing amounts of food. S p e c i f i c a l l y , the u t i l i t y function 11 is assumed to be exponential. According to this model a forager should be w i l l i n g to ' s a c r i f i c e ' up to 'k' units of mean for a unit decrease in variance. A unit decrease in variance i s worth 'k' units of mean regardless of the mean's si z e . This property i s c a l l e d constant risk-aversion: mean and variance are perfect substitutes for one another, and 'k' i s the constant marginal rate of sub s t i t u t i o n . An analysis of indifference curves shows the implications of constant risk-aversion. Variance discounting predicts that the forager should be in d i f f e r e n t between a l l mean/variance pairs on the l i n e : F = u - ko 2 or ju = F + ko2 where F i s a constant. F i g . 2.1a shows that these indifference curves are p a r a l l e l straight l i n e s with slope equal to 'k' and M-intercepts at diff e r e n t F values. The model predicts that foragers w i l l prefer any variance/mean pair (CT2,M) on a higher indifference l i n e to any other point on a lower indifference l i n e . Variance discounting predicts that the forager should maximize the ^'intercept (F) given a fixed slope of 'k'. Consider, for example, an experiment which offers a forager two a l t e r n a t i v e s : the f i r s t a l ternative provides food with a high variance ( a 2 2 ) and a high mean (M 2 ) and the second alte r n a t i v e provides food with a lower variance (a 2,) and mean (MI) (Fig. 2.1b). According to the variance discounting model, 12 Figure 2.2. (A) The indifference relationships that the z-score model predicts. Lines a l l have the same mean-intercept (Rnet). A standard deviation-mean pair on a higher li n e i s preferred to a pair on any lower l i n e . (B) represents the hypothetical preference tests in which a forager i s offered a choice between the end points of one of the broken l i n e s . According to the z-score model preferences should change when there i s a general s h i f t in means that does not affect the variances. * marks the preferred a l t e r n a t i v e . 13 ( A ) — ' i a? al Variance (a2) 1 4 the best choice depends on the value of 'k'. If 'k' i s less than the slope of the li n e segment connecting (a2^,n^) and ( a 2 2 » f 2 ) " t n e broken l i n e s in F i g . 2.1b - then the high mean, high variance alternative should be preferred. If 'k' i s greater than this slope, then the low mean, low variance alternative should be preferred. The slope of the connection l i n e would not change i f the means of both alternatives were increased by the same amount. In other words, preferences are independent of the absolute magnitude of the means. This observation r e f l e c t s the assumption of constant risk-aversion: i f a r e l a t i v e l y high mean is worth accepting a higher variance when means are low then a r e l a t i v e l y high mean (with a high variance) must s t i l l be preferred when means are generally high. Z-score Model Several authors (Caraco 1980; Stephens 1981; Pulliam and Mi l l i k a n 1982; Stephens and Charnov 1982; McNamara and Houston 1982) have suggested that r i s k - s e n s i t i v i t y might be explained on the basis of minimizing the pr o b a b i l i t y of f a l l i n g short of food. For example, a small non-breeding bird may require a fixed number of ca l o r i e s to survive a winter night. Stephens (1981) and Stephens and Charnov (1982) have shown that a ' s h o r t f a l l minimizer' should care about both the mean and the variance in food gains. According to thi s model, when food gains are normally d i s t r i b u t e d , minimizing the prob a b i l i t y of a s h o r t f a l l i s equivalent to minimizing: 15 Z = (Rnet - u) / a where M i s the mean food gain, a i s the standard deviation of food gains, and Rnet i s the minimum amount of food that the forager requires (usually for s u r v i v a l ) . Unlike the variance discounting model, the z-score model does not predict constant risk-aversion. Instead, i t predicts that the amount of mean a forager w i l l 'pay' to gain a unit reduction in variance depends on the mean ' i t s e l f . Caraco and Lima (1985) pointed out that the z-score model predicts decreasing risk-aversion: as the mean increases a unit increase in variance requires a smaller increase in mean to maintain indifference. Under decreasing risk-aversion, mean and variance are imperfect substitutes, and mean i s substituted for variance at a decreasing marginal rate. Comparison of the Models In order to compare the z-score and variance discounting models, consider the indifference relationships predicted by the z-score model. A forager, according to the z-score view, should be i n d i f f e r e n t between a l l (a,n) points that l i e on the line described by: Z = (Rnet - n)/o or M = Rnet - Za where Z i s a constant. A l l indifference l i n e s have an intercept at the requirement (Rnet) on the mean axis (Fig. 2.2a). A 1 6 forager should prefer any (o,u) point on a li n e of higher slope to any other point on a li n e of lower slope: a high slope means a low Z, because the slope is -Z. These indifference l i n e s d i f f e r from those predicted by the variance discounting model in two respects. F i r s t , the z-score model predicts a linear indifference relationship between mean and standard deviation whereas variance discounting predicts a linear relationship between mean and variance. Second, the z-score model predicts slope maximizing, in contrast to the intercept maximizing of the variance discounting model. Consider the two alternative experiment discussed e a r l i e r . According to the z-score model, a forager's choice should depend on the values of the high-variance/high-mean ( a 2 , M 2 ) » a n d lower-variance/lower-mean ( a , , M i ) a l t e r n a t i v e s r e l a t i v e to Rnet. The li n e between these points defines an intercept, c a l l e d (b), on the mean axis. If the requirement (Rnet) i s greater than b then the z-score model predicts preference for the high-variance/high-mean alter n a t i v e . Conversely, i f Rnet i s less than b the z-score model predicts preference for the low-variance/low-mean al t e r n a t i v e . F i g . 2.2b shows that i f the means are low, then the forager should prefer the high-variance/high-mean alter n a t i v e . However, i f some amount (c) i s added to both alternatives and i f thi s addition i s large enough, then the forager should switch i t s preference to the low-variance/low-mean al t e r n a t i v e . This prediction i s an example of the model's general prediction of decreasing risk-aversion: a r e l a t i v e l y high mean may be worth suffering the higher variance 17 Figure 2.2. (A) The indifference relationships that the z-score model predicts. Lines a l l have the same mean-intercept (Rnet). A standard deviation-mean pair on a higher l i n e i s preferred to a pair on any lower l i n e . (B) represents the hypothetical preference tests in which a forager i s offered a choice between the end points of one of the broken l i n e s . According to the z-score model preferences should change when there i s a general s h i f t in means that does not affe c t the variances. * marks the preferred a l t e r n a t i v e . 18 1 1 Standard Deviation (o) 19 when means are generally low, but when means are generally high, a lower variance becomes more important than a higher mean. How Constant is k? The two models d i f f e r when faced with the general s h i f t in means shown in Figs. 2.1b and 2.2b. When the slope between the alternatives i s maintained, the z-score model predicts that general changes in mean can affe c t preferences, but the variance-discounting model predicts that general changes in mean cannot a f f e c t preference (compare F i g . 2.1b with F i g . 2.2b). I used t h i s difference to distinguish between variance-discounting' s, prediction of constant risk-aversion. Put another way, I ask: how constant i s the constant of risk-aversion? This question i s important for two reasons. F i r s t , i t is important to know whether the value of variance i s r e a l l y independent of the mean as variance discounting predicts, or whether i t i s a function of the mean as the z-score predicts. Second, the value of 'k' in the variance discounting model is not defined independently of observed behaviour, and must be determined experimentally by observing foraging preferences. This means that variance discounting can only make quantitative predictions i f a 'k' measured in one situation predicts preferences in another (Staddon 1983). In contrast, the z-score model defines i t s parameter (Rnet), the food requirement, independently of foraging behaviour observed in the experiment. The predictive value of variance discounting hinges on the nature of 'k'. If 'k' i s a constant which describes a species 20 s p e c i f i c or individual s p e c i f i c response to r i s k , then variance discounting i s a powerful predictive model. The following experiment was designed to ask the question, "Is the constant of risk-aversion r e a l l y constant?" In t h i s experiment foragers were presented with two alternatives from which to feed. One al t e r n a t i v e provided food with a high mean and a high variance, and the other provided food with a low mean and low variance. By observing the proportion of v i s i t s to each alt e r n a t i v e , we measured foraging preferences. After measuring preferences, the means of both alternatives were increased without changing variances by adding the same amount to both a l t e r n a t i v e s . If the constant of risk-aversion i s not a constant (with respect to means), the s h i f t in means should change preferences. This would- contradict the variance discounting prediction of constant risk-aversion. The predictions of the z-score model depend on the relationship of the /i-intercept (b) to Rnet. If b i s on di f f e r e n t sides of Rnet for the two treatments then the z-score model predicts a change in preferences from the high-variance/high-mean alternative to the low-variance/low-mean al t e r n a t i v e after a constant has been added to the means. Fa i l u r e to observe th i s switch would contradict the z-score model prediction of decreasing r i s k -aversion. I f , however, the intercepts l i e on the same side of Rnet, the z-score model predicts the same preference in both treatments (the preference w i l l depend on whether the intercepts are above or below Rnet). If preferences change under these conditions, the z-score model i s again contradicted. 21 Materials and Methods Six wild-caught adult rufous hummingbirds (Selaphorus rufus: three males and three females) were used for this experiment. A l l subjects were kept in experimental boxes (see Fi g . 2.3) throughout the experiment. Each box was kept on a 10:14 light:dark cycle and at a temperature of 10±1°C. A l l birds had ad libitum access to a feeder (dispensing 25% sucrose solution) when not taking part in an experiment. The ad libitum feeder was removed on the night before an experiment. Experiments were begun at f i r s t l i g h t (08:00) and usually ended before 12:30. The ad libitum feeder was not returned u n t i l 12:30 on experimental days regardless of whether birds finished e a r l i e r . When an experiment did not conclude by 12:30, I returned the bird's feeder immediately after the experiment ended. In order to insure that birds had enough dietary protein and to reduce any cumulated energy d e f i c i t s I conducted experiments only six days a week. On the days that experiments were not performed, the ad libitum feeder contained a special high protein food (see Tooze and Gass 1985) and adult Drosophilia were provided. The six experimental boxes that I used in t h i s experiment were i d e n t i c a l and measured 60 x 95 x 40 cm (width x length x height), (see Fig. 2.3). The home (perch) chamber was illuminated throughout the day. At the far end of the two other (feeder) chambers I placed red c i r c l e s ('flowers'). The birds were trained to extract a solution of water and sucrose through 22 Figure 2.3. Perspective view of experimental boxes. Boxes were divided into a home chamber (with perch) and two reward chambers (with 'flowers'). The top and two sides of the box have not been drawn in order that the chambers can be seen. 24 a small hole in the centre of these red c i r c l e s . Access into the feeder chambers was through a hole in the p a r t i t i o n separating the home chamber from the two feeder chambers. I controlled the birds' access to each flower by deciding which of the two chambers to illuminate (birds were rarely observed to enter an u n l i t chamber). A spring loaded device holding a 250 ul microcentrifuge tube (cut to 21 mm length) was located behind a metal p a r t i t i o n (Fig. 2.4). A Brinkman adjustable microdispenser was used to add 25% sucrose solution to the tubes. The flowers were arranged so that in order to obtain the sucrose solution the bird's beak had to pass through a computer monitored photodarlington (see Gass 1985). The computer flashed t e l l - t a l e l i g h t s , v i s i b l e only to the experimenter, to show when a v i s i t had occurred, and when food tubes should be changed. The computer controlled a l l feeding opportunities by co n t r o l l i n g the l i g h t s in the feeder chambers; hummingbirds always f l y immediately from a dark chamber to a lighted chamber. When a bird v i s i t e d a flower the computer turned off the chambers' l i g h t s 1 s after the bird withdrew i t s beak. The computer offered another feeding opportunity 1 min after the previous flower v i s i t had ended (beak withdrawal). I did not completely control the int e r v a l between flower v i s i t s because birds could feed any time after the l i g h t ( s ) had been turned on. The d a i l y experimental protocol consisted of 80 feeding opportunities which were divided into 'forced choice' and free choice t r i a l s . In forced choice t r i a l s the computer illuminated 25 Figure 2.4. Cross-sectional view of feeder assembly. Microcentrifuge tubes are held in place by a p l a s t i c c l i p (not shown) . 27 only one feeder chamber. The forced choice t r i a l s were intended to insure that the birds were equally familiar with both al t e r n a t i v e s . The f i r s t 40 feeding opportunities each day were forced choices: 20 forced choice t r i a l s on each side. The computer determined the order of presentation using six randomized schedules. The la s t 40 t r i a l s were free choices in which both feeder chambers were illuminated. I tested hummingbird preferences in two treatments as suggested by Figs. 2.1b and 2.2b. The low-line treatment offered a 'large' alternative that provided volumes of 45 ul or 5 M1 of food with equal p r o b a b i l i t i e s of 0.5: a mean of 25 nl and a standard deviation of 20. The low-line treatment also offered a 'small' a l t e r n a t i v e that provided volumes of 10 ixl and 15 M1 with equal p r o b a b i l i t i e s : a mean of 12.5 ixl and a standard deviation of 2.5. Notice that the 'small' a l t e r n a t i v e has a small mean and smaller standard deviation than the 'large' a l t e r n a t i v e . The high-line treatment also offered both a large and a small a l t e r n a t i v e , but both alternatives had larger means than the low-line treatment alternatives because I added 20 ixl to each possible outcome. Adding a constant value increases the mean of each alternative without changing the variance. S p e c i f i c a l l y , in the high-line treatment the 'large' alternative provided volumes of 65 nl or 25 ixl: mean = 45 jul, SD = 20. The 'small' alternative provided volumes of 30 ixl or 35 jul: mean = 32.5, SD = 2.5. According to the variance discounting model preferences 28 should be the same in each treatment. This means that i f the 'large' a l t e r n a t i v e i s preferred in the low-line treatment, then i t must also be preferred in the high-line treatment. However, the z-score model predicts that the preference for the 'large' alternative should be stronger in the low-line treatment. I assigned birds 1, 3 and 4 to the high-line treatment and birds 2, 5 and 6 to the low-line treatment. Birds 1 and 3 experienced the low-line treatment after f i n i s h i n g the high-line treatment, and birds 2 and 5 experienced the high-line treatment after f i n i s h i n g the low-line treatment. Birds 4 and 6 experienced only one treatment. Measuring preferences presents two problems. F i r s t , both models assume that the forager 'knows' what the alternatives are. In order to allow birds to 'learn about' the nature of the alternatives being offered, I did not score a bird's choices u n t i l i t s behaviour had s t a b i l i z e d . I considered the behaviour to be stable when the proportion of choices taken from each side did not change by more than 10%. Furthermore, I required that treatments l a s t for at least three days (the average treatment lasted 4.9 days). I scored preference using only the l a s t two 'stable' days in a treatment. Second, many animals have strong side preferences in dichotomous preference t e s t s . To control for side preferences, I repeated each test after switching the sides of the 'small' and 'large' alternatives. The 80 choices (40 from each of the la s t two 'stable' days of a treatment) were divided into eight successive blocks of ten choices, and I measured the number of 29 times a bird chose the 'large' a l t e r n a t i v e . Symbolically, t h i s preference test yielded eight scores: B 1 f B 2, . . B 8 (B stands for 'before side switch'). This procedure was repeated after the sides were switched, y i e l d i n g eight new scores: A,, A 2, . . A 8 (A stands for 'after side switch'). The scores used to test whether the 'large' or 'small' alternative was preferred are the eight sums: B..+A,, B2+A2, . . . B B+A 8. This gave eight preference scores, ranging between 0 and 20, for each b i r d in each treatment. A score of 0 means that the bird always took the 'small' alternative, a score of 10 indicates indifference, and a score of 20 means that the bird always took the 'large' a l t e r n a t i v e . Most authors have assumed that the z-score model's Rnet (requirement) i s the forager's d a i l y food requirement, although i t could be the requirement for a longer or shorter period. Following Caraco et a l . (1980) I estimated t h i s by observing how much food they ate per day under experimental temperature and .photoperiod conditions (see Table 2.1). I took these measurements on days when no preference tests were made. Analysis of variance shows that requirements d i f f e r s i g n i f i c a n t l y (at the 5% level) between birds. This gave me an estimate of the amount a forager had to eat in the 10 h of l i g h t under experimental conditions in order to maintain constant body weight. To calculate the average amount that an individual needed in each of the forty free choice t r i a l s (Rc), I used the formula Rc = (0.45Rnet - Gfor)/40, where Rnet i s the measured 10 h requirement (Table 2.1 shows Rnet values for each i n d i v i d u a l ) , 30 which is multi p l i e d by 0.45, because the birds had ad libitum food for 5.5 h after the experiment each day. Gfor i s the gain during the forced choice t r i a l s (750 /ul in the low-line treatment, and 1550 M1 in the high-line treatment). Both models predict only three types of preference (but see Chapter 3). According to both of these models birds should choose either always v i s i t the large or the small a l t e r n a t i v e , or they should choose equally between alternatives ( i n d i f f e r e n c e ) . Equal preferences occur when mean preference scores are equal and not when preferences are simply in the same d i r e c t i o n . Therefore, I analyzed my data for quantitative changes in preference. Results Table 2.1 shows the data for each bird in each treatment. I analyzed these data in three steps. F i r s t , I asked whether the same constant of risk-aversion could explain a l l the observed hummingbird foraging behaviour. In other words: i s 'k' a constant property of the species? Second, I asked whether 'k' is constant between treatments for the same i n d i v i d u a l : i s 'k' a constant of individuals? Last, I asked whether the z-score model accurately predicts the observed preferences. Is 'k' a constant for species? To answer t h i s question I considered only the f i r s t treatment of each of the six birds. Ideally I would have l i k e d to do a nested analysis of variance of these data, with individuals nested within treatments (high-31 Table 2.1. Summary of minimum food requirement estimates (Rnet) and preference scores for a l l birds. A preference score of 0 means that the bird always took the 'small' a l t e r n a t i v e , and a score of 20 means the bi r d always took the 'large' a l t e r n a t i v e . Bird Rio+S .K.(N) Preference Scores  (sex) ml 1st 2nd 3rd 4th 5th 6th 7th 8th MeajHS.E. 1 (F) 2 (F) 3 (M) 4 (F) 5 High-line 10 12 10 10 12 10 10 10 10.5+0.33 9.25+0.77(4) Low-line 15 14 17 16 17 17 16 17 16.1+0.72 High-line 11 15 16 17 15 13 17 15 14.9+0.72 6.86+0.47(3) Low-line 19 18 20 18 19 18 20 19 18.9+0.29 High-line 10 10 11 10 11 10 12 11 10.6+0.26 7.2+0.48(6) Low-line 14 18 16 16 17 17 15 16 16.1+0.44 High-line 10 11 10 10 11 12 11 11 10.7+0.18 7.67+0.66(3) Low-line High-line 3 8 5 3 3 5 7 6 5.0+0.68 , M. 6.29<0.55(5) (M) Low-line 17 17 14 13 19 15 10 16 15.1+0.99 R High-line 4.71+0.32(7) Low-line 10 11 12 8 15 10 11 10 10.9+0.72 32 l i n e and low-line). However, these data were markedly non-normal with unequal variances even after the appropriate transformations (Sokal and Rohlf 1981), so analysis of variance was not appropriate. Instead, I performed a Friedman test viewing each of the six individuals as a treatment, and each of the eight preference scores as a block (see Conover 1980). This test showed s i g n i f i c a n t differences between the individuals (the 5% l e v e l of significance i s used throughout). However, I wanted to know whether the treatment or i n t r i n s i c v a r i a b i l i t y of individuals caused these differences. I tested the difference between preferences of individuals in the high-line and low-line treatments using a test for unplanned comparisons in the Friedman test (Conover 1980). This test showed s i g n i f i c a n t differences between birds in the high-line group and the birds in the low-line group. I conclude that the constant of r i s k -aversion is unlikely to be a constant of the species. The 'k' value estimated from the low-line group could not successfully predict the behaviour of the high-line group. Is 'k' a constant for individuals? To answer th i s question I considered only the four birds that experienced both the high-l i n e and the low-line treatments. For a given b i r d the data consisted of eight differences obtained by pairing each high-l i n e preference score with the corresponding low-line score and then subtracting the high-line scores from the low-line scores. These differences met the assumptions of analysis of variance. I allowed for the effects of individual birds and for the e f f e c t s of repeated measurement by performing a repeated 33 analysis of variance on these differences (score sequence, 1 through 8, was the 'treatment' in t h i s a n a l y s i s ) . This analysis tested the hypothesis that the mean difference equaled zero (using BMDP s t a t i s t i c a l software, Dixon 1983) as the variance discounting model predicts. This test showed that the mean difference was s i g n i f i c a n t l y d i f f e r e n t from zero. I conclude that the constant of risk-aversion is not a constant property of individual hummingbirds, because d i f f e r e n t 'k' values would be required to predict an individual's behaviour in the high and low-line treatments. How well does the z-score model explain the observed behaviour? My data supported the z-score model over the variance discounting model, because individuals and groups of individuals behaved d i f f e r e n t l y in the high and low-line treatments. In other words, the variance discounting prediction of constant risk-aversion is not supported by the data. The z-score model prediction of decreasing risk-aversion i s supported since preferences for the high-mean/high-variance alternative was stronger in the low-line treatment than in the high-line treatment. The z-score model makes stronger predictions i f an \v KV estimate of the forager's requirement (Rnet) can be made. Since Rnet varies between individuals, the z-score model might explain some of the differences between individuals that I observed. Each treatment defines an intercept (b) on the M~axis of F i g . 2.1b and 2.2b (bfhigh] = 30.69 ixl and b[low] = 10.69 jul). When the requirement i s greater than t h i s intercept the z-score model 34 predicts exclusive preference for the high-mean al t e r n a t i v e , but when the requirement is less than t h i s intercept, the z-score model predicts exclusive preference for the low-mean al t e r n a t i v e . The birds are expected to be in d i f f e r e n t between alternatives only when intercepts are very close to requirements. I calculated Rc - b for each b i r d in each treatment, and I expected (under a s t r i c t interpretation of the z-score model) that preference scores should change abruptly from near 20, when Rc - b i s large, to 10, when Rc - b i s near zero, and then to 0, when Rc - b i s small (much less than zero). A plot of preference scores against Rc - b (Fig. 2.5) shows no clear evidence of a step change change in preferences as predicted by the z-score model. Furthermore, preferences never approach the exclusive preference score of 20 predicted by the model. I have assumed in thi s experiment that preferences are measured only after the birds have obtained complete knowledge of the s t a t i s t i c a l nature of both al t e r n a t i v e s . If I relax t h i s assumption, the graded preference view of Stephens (1985) predicts an increase in preference for the high-mean/high-variance alternative with Rc -• b. Non-parametric measures of co r r e l a t i o n (Kendall's (0.61), and Spearman's (0.77), rank c o r r e l a t i o n c o e f f i c i e n t s ) showed s i g n i f i c a n t p o s i t i v e c orrelations between the preference for the high-mean alter n a t i v e and Rc - b. Moreover, the z-score predicts that indifference, i . e . preference scores of 10, should occur when Rc - b = 0. The intercept of the regression l i n e through these 3 5 Figure 2.5. Mean preference scores from a l l birds in a l l treatments are plotted against Rc - b, the difference between the requirement per choice (Rc) and the mean-intercept defined by the alter n a t i v e s offered (b). The regression l i n e shown has the equation: Mean Score = 0.13(RC - b) + 9.07. The parametric co r r e l a t i o n c o e f f i c i e n t is 0.74, and i t i s s i g n i f i c a n t l y d i f f e r e n t from zero. 3 6 37 points l i e s between 5.32 and 12.84 with 95% confidence, and the estimated intercept is 9.07, (Fig. 2.5). These data suggest that neither variance discounting or the z-score model adequately account for a l l aspects of r i s k - s e n s i t i v e foraging behaviour in rufous hummingbirds. Future r i s k - s e n s i t i v e foraging models should, however, incorporate the z-score prediction of decreasing risk-aversion with increasing mean prey p r o f i t a b i l i t y . Discussion My results contradict the variance discounting model's prediction that preferences are unaffected by a general increase in the mean value of food (constant risk-aversion). The re l a t i v e l e v e l of mean food gain affected the trade off between mean and variance as predicted by the z-score model. The constant of risk-aversion, 'k' in the variance discounting model, i s not constant for rufous hummingbirds. A 'k' value that i s consistent with one observed preference is inconsistent with preferences after a general increase in means of 20 ul. My results do not support the use of the variance discounting model, instead of the z-score model, to predict hummingbird foraging behaviour. The z-score model, however, is only capable of accurately predicting q u a l i t a t i v e trends in foraging preferences. U n t i l now the strongest evidence supporting the z-score model was the changing r i s k - s e n s i t i v i t y observed in granivorous 38 birds and shrews (Caraco et a l . 1980; Caraco 1981, 1983; Barnard and Brown 1985). These studies showed that foragers preferred low-variance alternatives ( i . e . they were r i s k -averse) when expected gains were greater than d a i l y requirements (M > Rnet), and preferred high variance alternatives ( i . e . they were risk-prone) when expected gains were less than the da i l y requirement (Rnet > M). This behaviour supports the z-score model, and contradicts the variance discounting model because the constant of risk-aversion in the f i r s t case (k > 0) could not predict the observed behaviour in the second case (k < 0). The proponents of variance discounting could argue that, although risk-prone behaviour can be demonstrated in the laboratory, such behaviour i s not common in nature, and both models explain day-to-day risk-averse foraging behaviour equally well. Caraco and Lima's (1985) results support t h i s argument. Caraco and Lima t r i e d to determine whether indifference curves (l i n e s along which a l l alternatives are equally preferred) were line a r (as predicted by the z-score model) or quadratic (as predicted by the variance discounting model) when plotted using mean-standard deviation coordinates. They found that both models worked equally well. However, estimating even a single indifference point takes many hours of observation, so Caraco and Lima were handicapped by trying to dis t i n g u i s h a linear from a c u r v i l i n e a r function with only a handful of data. My results show that the two models make di f f e r e n t predictions about the interactions of mean and variance. These differences might be important in day-to-day foraging decisions, 39 even though both models predict that animals w i l l usually be risk-averse. For example, my experiment shows that a hummingbird's preferences might be diff e r e n t in r i c h and poor habitats, even i f means are always correlated with variances. The variance discounting model might be saved by one of two types of \KX\P K arguments. F i r s t , a variance discounting proponent might make a s p e c i f i c claim about the conditions under which 'k' i s expected to be constant. For example, 'k' w i l l be constant when the change in means i s less than X% of the energy requirement. Second, the proponent might argue that the model i s purely descriptive and that 'k' simply measures observed risk - t a k i n g . This would, however, relegate variance discounting to a s t a t i s t i c a l book keeping technique, rather than a the o r e t i c a l insight. Variance discounting has one unassailable use. It i s a convenient formulation that modelers might use to study how r i s k - s e n s i t i v i t y a f f e c t s mean-maximizing models. Variance discounting, broadly defined, can include models of the form: max(M - k(n)a2} where k(u) i s now some function of the mean (L. Real, personal communication). My results and comments deal only with the version of variance discounting where M M ) i s a constant. However, a constant 'k' value, and i t s consequence of constant risk-aversion, i s e x p l i c i t l y or i m p l i c i t l y assumed in most uses of variance discounting (e.g. Real 1980a,.b, 1981; Real et a l . 1982; Lacey et a l . 1983; Caraco and Lima 1985; Wunderle and 40 O'Brien in press). Moreover, i f the ' c o e f f i c i e n t of r i s k -s e n s i t i v i t y ' (k(y)) i s l e f t as an unspecified function of the mean, then variance discounting becomes untestably vague: k(n) can be changed to accommodate any observation. A c r i t i c might argue that both of the models considered here are so simple that any attempt to d i s t i n g u i s h between them would be a battle of straw men. I agree that neither model explains everything about r i s k - s e n s i t i v e foraging behaviour. The models are important because they help us to f i x ideas: viewing them as alternatives suggested my experiment. However, my experiment distinguishes between two contrasting properties of the simple models considered here, and my results suggest that the successors of these models should not assume constant risk-aversion as variance discounting has. 41 CHAPTER 3 ASSUMPTIONS IN OPTIMAL FORAGING MODELS: A RE-EXAMINATION OF RISK-SENSITIVITY Introduction A l l models contain assumptions, either i m p l i c i t or e x p l i c i t , about the systems that they describe. In b i o l o g i c a l models these assumptions are made for many reasons, but primarily they are made to simplify complex natural processes because facets of the system in question are unknown or poorly understood. Few non-mathematical b i o l o g i s t s appreciate how c r i t i c a l l y the results of th e o r e t i c a l work depend on their b i o l o g i c a l assumptions (Gilbert et a l . , 1979). Few modelers examine the s e n s i t i v i t y of their models to changes in their assumptions, despite the fact that many of these assumptions are untested, implausable, or have been shown to be f a l s e . Despite the use of such assumptions, modelers often make fine tuned predictions and then devise post hoc explanations to ra t i o n a l i z e the f a i l u r e of their models to make accurate quantitative predictions. This approach to modelling i s t y p i f i e d by the recent work on the foraging behaviour of animals in stochastic environments. Caraco (1980), Real (I980a,b), Stephens (1981) and Stephens and Charnov (1982) argued that, in a stochastic environment, animals should take into account the variance in net energetic gain as well as i t s mean when choosing how to forage. This behaviour 42 has been termed " r i s k - s e n s i t i v e foraging" after economic theory from which much of the terminology has been borrowed. A number of experiments have demonstrated that animals in a number of taxonomic groups are sensitive to variance in net energetic gain (Birds: Caraco et a l . 1980; Caraco 1981, 1982, 1983; Lima 1984; Caraco and Lima 1985; Wunderle and O'Brien (in press); also see Chapter 2. Bees: Real 1981; Waddington et a_l. 1981; Real et a l . 1982. Shrews: Barnard and Brown 1985; Branard et a l . 1985). In many of these studies, the observed foraging behaviour agreed q u a l i t a t i v e l y , but not quantitatively with theoretical predictions. Several authors (e.g. Krebs and McCleery 1984) have offered a number of possible factors, such as inherently stochastic foraging behaviour, i n s u f f i c i e n t learning t r i a l s , , etc., as explanations for the recurrent observation of non-quantitative agreement with theory. Others have suggested that i t i s highly unlikely that simple models which consider only a few parameters can ever make quantitatively accurate predictions of the behaviour of complex systems (Maynard Smith, 1978). Few modelers have considered, however, how the b i o l o g i c a l assumptions of their models influence their predictions. Pyke (1984) pointed out that i t is often d i f f i c u l t , i f not impossible, to obtain independent tests of the underlying assumptions of optimal foraging theory. In l i g h t of these d i f f i c u l t i e s , he suggests that the most l o g i c a l approach for addressing discrepancies between predictions and observations i s to explore the consequences of variation in the model's 43 assumptions, and from th i s to devise c r i t i c a l experiments from which the most r e a l i s t i c model can be determined. I w i l l apply t h i s technique to the z-score r i s k - s e n s i t i v e foraging model of Stephens (1981), and Stephens and Charnov (1982). One of the most fundamental assumptions of the z-score model, and other r i s k - s e n s i t i v e foraging models, is that r i s k -sensitive foraging confers a selective advantage over r i s k -insensitive foraging because i t reduces their p r o b a b i l i t y of starvation. Specific situations can be imagined where i t would undoubtedly be advantageous to be sensitive to the variance of reward d i s t r i b u t i o n s (see Stephens and Charnov 1982). However, no one has as yet t h e o r e t i c a l l y evaluated t h i s advantage over a wide range of conditions l i k e l y to be encountered by animals in their natural environments. What is the extent of the advantage, in terms of decreased risk of starvation, of a z-score model forager over i t s deterministic analog? The z-score model i m p l i c i t l y assumes that foragers can always discriminate between alternative reward d i s t r i b u t i o n s on the basis of the z - s t a t i s t i c . In addition, the model assumes that foraging preferences are 'decided' at the beginning of a foraging period based on the projected energetic reserves at the end o f that period. Once a decision has been made i t i s used throughout that period regardless of the food actually obtained. Neither of these assumptions has been tested. How do changes in these assumptions aff e c t the predicted foraging preferences and the expected p r o b a b i l i t i e s of starvation? Using computer simulations, I show that changing some of 44 the untested assumptions of the z-score model to equally plausible alternatives can dramatically a l t e r i t s predictions. I also show that the advantage, in terms of decreased r i s k of starvation, r ealized by a forager following the z-score model over the analogous deterministic forager i s only s i g n i f i c a n t over a very small range of the parameter space examined. I use predictions from these simulations to suggest ways to test the realism of several assumptions. F i n a l l y , I c r i t i c i z e the complacent use of "untested g e n e r a l i t i e s that i n h i b i t rather than enhance understanding" (Holling 1964) when discussing some recent t h e o r e t i c a l and empirical r i s k - s e n s i t i v e foraging papers. Models I simulated a simple two-alternative foraging paradigm similar to that used in a number of recent r i s k - s e n s i t i v e experiments (see Caraco et a_l. 1980; Caraco and Lima 1 985; Banard and Brown 1985). The situation that I have chosen aids in simplifying the discussion of results and should allow a comparison of my predictions with previously published data. My conclusions apply to many foraging situations and models. Each simulation consists of a fixed number (N) of feeding opportunities presented at regular i n t e r v a l s over some time period. The time period i s considered to be one day, but the results are independent of the time scale chosen. Foragers must accumulate a minimum energy reserve (Rnet) by the end of the foraging period (dusk) in order to survive u n t i l the beginning 45 of the next foraging period (dawn). Two alternative feeders are presented simultaneously and d i f f e r only in the temporal d i s t r i b u t i o n of the food that they provide. One always provides the same energetic gains (Gc). The second provides net energetic gains (Gv) that are symmetrically d i s t r i b u t e d around a mean (E(Gv)). In a l l cases, the model forager knows the exact nature of both d i s t r i b u t i o n s at a l l times. Foragers begin the foraging day with a zero net energetic reserve, and are assumed to die from starvation during the night unless they obtain energetic reserves of at least Rnet units of food by the end of the day. Starvation before the end of the day (the 'death in the afternoon' problem: Stephens 1982), the eff e c t of fat storage, or factors influencing nocturnal energy expenditure (e.g. torpor, see Tooze and Gass 1985) are not considered in th i s study. I simulated foraging behaviour over a range of constant and variable mean energetic gains (from 0 to 20 units) for two di f f e r e n t numbers of feeding opportunities per day (N = 5, 80). The value for the 'large' number of feeding opportunities (N = 80) was chosen so that the binomial d i s t r i b u t i o n of the variable gains offered formed a good approximation to the normal d i s t r i b u t i o n . N = 80 is also similar to the number of foraging opportunities offered in several of the r i s k - s e n s i t i v e foraging experiments mentioned e a r l i e r . To survive the night, the forager must average Rnet/N units of energy per feeding opportunity. The standard deviation of the variable rewards was 5.0 for a l l simulations. Feeder 46 preference and survival/starvation were recorded for a l l parameter combinations. I estimated average preference and pr o b a b i l i t y of starvation (P(s)) by r e p l i c a t i n g each model 200 times for a l l combinations of N, Gc and E(Gv). Under the z-score model, i f t o t a l food gains are normally di s t r i b u t e d , then minimizing the p r o b a b i l i t y of starvation i s equivalent to minimizing: (Rnet - M) / a (1) where Rnet i s the energy required by the forager (usually for s u r v i v a l ) , M i s the mean food gain after N rewards, and a i s the standard deviation of the food reward. Preferences among alternatives i s determined by c a l c u l a t i n g which alternative minimizes equation 1. Stephens (1981) and Stephens and Charnov (1982) proposed that foraging in t h i s manner affords a lower ris k of energetic s h o r t f a l l in a stochastic environment than foraging in a r i s k - i n s e n s i t i v e (deterministic) way. Both studies assumed that foragers can discriminate between reward d i s t r i b u t i o n s regardless of the degree of difference between them. The necessity that rewards be normally d i s t r i b u t e d required the use of the central l i m i t theorem (N large) in the development of the z-score model. Houston and McNamara (1982) showed that one consequence of t h i s necessity i s that such a 47 foraging policy must be non-sequential. That i s , behaviour is fixed by a single decision at the beginning of a foraging period, based on expected outcomes over the entire foraging period. They also indicated that this decision rule (equation 1) i s the best non-sequential rule. The use of equation 1 implies that a l l alternatives are evaluated before the f i r s t reward i s taken. Once the best alternative has been found, the rule i s to use this a l t e r n a t i v e exclusively, regardless of energetic states reached during the day. Houston and McNamara argued that the optimal sequential rule, one in which decisions are based on current energy reserves, cannot be found in the same forward looking manner and i t i s necessary to work backwards from the f i n a l feeding opportunity (for an example, see Clark and Mangel 1984; Clark 1985). Therefore, foragers that could use an optimal sequential rule should suffer a smaller r i s k of starvation than foragers using the optimal non-sequential rule (z-score). The main purpose of t h i s chapter i s to examine the consequences of a l t e r i n g assumptions. I have, therefore, used a 'forward looking' sequential decision rule, rather than the- optimal 'backward looking' model of the type used by Clark (1985). The forager using a forward looking sequential decision rule simply recalculates the z-score s t a t i s t i c before each feeding opportunity and chooses whichever alternative has the smallest z-score value. This rule i s much easier to implement than the backward looking rule, and requires no additional assumptions about the forager, except that i t must recalculate the same z-48 score before each reward. I use four models to examine some of the assumptions used in the z-score model. Model 1 uses a simple deterministic decision rule based on maximizing mean net energy intake. Model 2 i s based on the o r i g i n a l z-score model as proposed by Stephens and Charnov (1982). Model 3 i s i d e n t i c a l to model 2 except that I assume the forager cannot discriminate between alternatives i f the estimated p r o b a b i l i t i e s of starvation of the alternatives d i f f e r by less than 2.5%. I used t h i s value for s t a t i s t i c a l convenience, however I w i l l show that any value greater than 0% w i l l produce similar r e s u l t s . The fourth model is also i d e n t i c a l to the second except that the forager re-evaluates equation 1 based on current energy reserves for each alternative before each feeding opportunity. In a l l models, i f the forager cannot d i s t i n g u i s h between d i s t r i b u t i o n s i t chooses a feeder at random. Simulation Results Preferences Figure 3.1 summarizes the simulation results for Models 1 to 4. A l l four models show d i f f e r e n t patterns of predicted preferences (Fig. 3.1a,b top rows). Because Model 1 foragers maximize net energy gain, they choose the alternative that provides the greatest expected net gain. This i s a pure strategy that fixes on the variable alternative when the 49 expected gain from i t (E(Gv)) is greater than that from the constant alternative (Gc), and on the constant alternative when E(Gv) < Gc. Model 1 predicts a mixed strategy only when both alternatives provide equal gains. Because Model 2 foragers minimize the prob a b i l i t y of starvation, they choose the alternative that presents the smallest z-score value before the f i r s t prey is taken. These foragers take a l l prey from the constant alt e r n a t i v e when i t provides gains greater than or equal to the minimum energy requirement (Rnet). When net gains are less than Rnet, these foragers take a l l prey from the variable a l t e r n a t i v e . Model 2 predicts indifference only under the rare condition that the z-score s t a t i s t i c s for both alternatives are equal. Preferences predicted by both Model 1 and 2 are independent of the number of foraging opportunities per day. Because a Model 3 forager cannot discriminate between alternatives whose p r o b a b i l i t i e s of starvation (P(s)) d i f f e r by less than 2.5%, they are in d i f f e r e n t much more frequently than Model 2 foragers even though they use the same foraging strategy. Model 3 predicts indifference when both alternatives provide either very low net gains (P(s) near zero), or very large net gains, (P(s) near one). The constant and variable alternative expected gains for which indifference i s predicted, increase with N, with reward variance, and inversely with discriminatory a b i l i t y . The regions of indifference respond very slowly to changes in discriminatory a b i l i t i e s . For example, when N=80, increasing the discriminatory a b i l i t y of a 5 0 Figure 3.1. Simulation results of four foraging models: Model 1 is a simple deterministic mean-maximization foraging model. Model 2 is the z-score model of Stephens (1981). Model 3 i s a version of Model 2 in which foragers have li m i t e d a b i l i t y to discriminate between alternative reward d i s t r i b u t i o n s . Model 4 i s i d e n t i c a l to Model 2 except that foragers make independent foraging choices before each feeding opportunity based on current energetic reserves, instead of only once per day. The top rows of (a) and (b) show the predicted preferences for a variable and a constant alternative for each model for N = 5 and 80, respectively. The bottom rows of (a) and (b) show the estimated p r o b a b i l i t i e s of starvation for each model. Each contour plot shows the predicted preferences (or estimated p r o b a b i l i t i e s of starvation) for a l l combinations of expected net energy gains for the constant a l t e r n a t i v e (Gc) and the variable alternative (E(Gv)). The minimum energy required for survival (Rnet) i s 10.0. A preference of 0 (see scale at bottom) means a l l rewards were taken from the constant alternative (risk-averse), and a preference of 1 means that a l l rewards were taken from the variable alternative (risk-prone). An estimated p r o b a b i l i t y of starvation (P(STARVATION)) of 0 means that the forager always survives, and an estimated pr o b a b i l i t y of starvation of 1 means that the forager w i l l always starve to death. 52 Model 3 forager by 1000 times (decreasing the minimum detectible difference from 2.5% to 0.0025%) reduces the area of indifference by less than 1%. Because Model 4 foragers re-evaluate foraging preferences before each feeding opportunity, they e x i b i t a wide range of foraging preferences. They f i x on the constant al t e r a n t i v e when Gc >= Rnet, but for a l l other cases they begin feeding from the variable a l t e r n a t i v e . If s u f f i c i e n t reserves accumulate to ensure survival before the end of the day, they then f i x on "the constant alternative for the rest of the day. Clear l y , preferences are greatly affected by single changes in some assumptions, and how these preferences are affected depends considerably on the number of prey taken per day. I w i l l show below how these preferences affect the expected p r o b a b i l i t i e s of starvation. Probability of starvation A l l four models show similar patterns of s e n s i t i v i t y of p r o b a b i l i t i e s of starvation to changes in assumptions (Fig 3.1a,b bottom rows). The lowest pr o b a b i l i t y of starvation that can be detected in my simulations i s 0.005%. This region of detectable risk (P(s) >= 0.005) i s sharply bounded along the li n e corresponding to the minimum dai l y energy requirement (Rnet = Gc = 10). In Models 1 and 3 th i s boundary deviates from Gc = 10 at E(Gv) >= 10. In Model 1 t h i s is a consequence of the forager choosing the variable alternative whenever E(Gv) > Gc, even though the risk of starvation from the constant alternative 53 is zero and the risk from the variable a l t e r n a t i v e in non-zero. This deviation decreases rapidly as N decreases. The deviation observed for Model 3 (N=80), which is a consequence of the limited a b i l i t y to detect difference in r i s k , i s small, or undetectable for a l l values of N. The zone of detectable r i s k i s also bounded by straight lines in the variable gain dimension which, with one small exception, is the same for a l l models for a given value of N. There i s a smooth change in values of P(s) from 1.0 to 0.0 in this dimension. The rate of t r a n s i t i o n between one and zero increases rapidly as N increases. The region of detectable risk decreases for a l l models as N decreases. For a l l values of N, this region i s greater for Model 1 than a l l other Models, but this difference decreases as N increases. For very large N (>1000) th i s difference w i l l be n e g l i g i b l e . The region of detectable r i s k i s v i r t u a l l y i d e n t i c a l for Models 2, 3 and 4 (with the exception of a small deviation in Model 3 for intermediate N). Discussion I have examined three d i f f e r e n t assumptions of the z-score model: (1) that r i s k - s e n s i t i v e foraging confers a s i g n i f i c a n t advantage over r i s k - i n s e n s i t i v i t y by reducing the pro b a b i l i t y of starvation, (2) that foragers always d i s t i n g u i s h between reward d i s t r i b u t i o n s based on selected sample s t a t i s t i c s regardless of the degree of difference, and (3) that foragers make a single 54 decision about where to forage based on the expected outcome of future gains. Within the boundaries of my simulations, assumption 1 i s true only in a very r e s t r i c t e d sense, and that changes to assumptions 2 and 3 lead to s i g n i f i c a n t changes in predicted foraging preferences without s i g n i f i c a n t l y changing the estimated p r o b a b i l i t y of starvation. Because these assumptions are common to a number of foraging models, my results may have a d i r e c t bearing on the a p p l i c a b i l i t y of many these. The fact that r i s k - s e n s i t i v e foraging behaviour has been observed in bumblebees, wasps, several small bird species, and shrews (see introduction) suggests that for these groups at least, the advantages of r i s k - s e n s i t i v e foraging may exceed associated costs. My results show that the advantage of r i s k -sensitive foraging over r i s k - i n s e n s i t i v e foraging (simple mean-maximization) decreases rapidly with increasing numbers of prey eaten per day. In the case of a forager such as a hummingbird, which may feed from several thousand flowers per day (Gass et a l . 1976), the difference in the expected risk from these two foraging strategies i s probably n e g l i g i b l e . Conversely, the advantage of r i s k - s e n s i t i v e foraging for a forager which takes r e l a t i v e l y few, highly variable prey per day, may be quite high. While most of the species used to study r i s k - s e n s i t i v e foraging have shown some degree of r i s k - s e n s i t i v i t y , they have a l l foraged suboptimally compared to the predictions of the models being tested. It i s possible that some of the explanations of Krebs and McCleery (1984), such as inherently 55 stochastic behaviour, i n s u f f i c i e n t learning t r i a l s , etc., may account for some of these observations. I believe, however, that much of t h i s apparently suboptimal behaviour occurs because the advantage of r i s k - s e n s i t i v i t y i s often very close to zero and the cost of suboptimal behaviour i s low. Further, in nature, where prey types may not be e a s i l y recognizable and the parameters of prey d i s t r i b u t i o n s may be changing continually and in unpredictable ways, t h i s advantage is probably neg l i g i b l e v i r t u a l l y a l l the time. I have shown that plausible major changes in foraging preferences often result in neglible changes in the expected pr o b a b i l i t y of starvation. I have also shown that r i s k -sensitive foraging seldom reduces the p r o b a b i l i t y of starvation. I do not f i n d i t surprising, then, that there have been so many observations of suboptimal behaviour when the cost of such behaviour, in terms of increased r i s k , i s n e g l i g i b l e . Several recent studies (Regelmann 1984; Caraco and Lima 1985; and see Chapter 2) have attempted to determine which of two a l t e r n a t i v e models, z-score and variance discounting, best describe foraging behaviour. Each study depended, at least in part, on accurate quantitative predictions to discriminate between models; models whose predictions were inconsistent with observations were rejected. This approach i s only v a l i d for rejecting s p e c i f i c sets of assumptions (Maynard Smith 1978). Unfortunately, none of these studies considered how c r i t i c a l l y the power of their tests depended on the assumptions on the models. I have shown that some of these assumptions have 56 s i g n i f i c a n t e f f e c t s on predicted preferences. I w i l l show below why taking these facts into account would have been b e n e f i c i a l to a l l of these studies. Caraco and Lima (1985) attempted to discriminate between the two models on the basis of indifference curves. Foragers are predicted to be indi f f e r e n t ( i . e . take equal numbers of prey from both alternatives) between a l l points that l i e on a given indifference curve. The z-score model predicts that indifference l i n e s should be linear when plotted using mean-standard deviation coordinates. Whereas the variance discounting model predicts that indifference l i n e s should be linear when plotted using mean-variance coordinates. Because of the scatter of the i r data Caraco and Lima were unable to decide in favour of one model over the other. In Chapter 2 I used the fact that the variance discounting model predicts i n s e n s i t i v i t y to changes in reward means which do not change the variance (constant risk-aversion). The z-score model predicts that preferences are sensitive to changes in reward means which do not change the variance (decreasing r i s k -aversion). I rejected the variance discounting model, but found that the scatter in my data prevented me from concluding that the z-score accurately represented foraging behaviour. Regelmann's (1984) proposed method of discriminating between these models depended on the fact that, under some conditions, they predict greatly d i f f e r e n t foraging strategies: exclusive (pure) preference versus mixed preference. By structuring experiments around these conditions he suggested 57 that support for one model over the other may be obtained. The method r e l i e d on the fact that the z-score model v i r t u a l l y always predicts exclusive preferences, but the variance discounting model often predicts mixed preferences. A l l of these studies may have been naive in ignoring the dependence of the z-score and variance discounting models on their assumptions. None of these studies considered how dependent predicted preferences are on these assumptions, nor did they consider how l i t t l e the risk of starvation may actually depend on the foraging strategy used. Caraco and Lima's (1985) method was dependent on the existence of a strong rel a t i o n s h i p between preferences and risk ( i . e . for any given combination of mean and variance there should exist a very characterstic preference). Since that rel a t i o n s h i p probably does not ex i s t , their method was unlikely to succeed. In Chapter 2 I may have had more success in supporting the z-score model had I used a species that was l i k e l y to be much more sensitive to variance in prey p r o f i t a b i l i t y (e.g. a species that requires a few, high variance prey per day). The method proposed by Regelmann (1984) is seriously flawed since i t depends heavily on the unique predictions of the o r i g i n a l models. By manipulating only a few untested assumptions from these models i t is possible to generate a wide range of overlapping predictions. Without knowing the true nature of these assumptions, i t i s almost impossible to choose a set of reward parameters which w i l l unambiguously support one model in favour of the other. Many authors have already c r i t i c i z e d the use of untested 58 and untestable assumptions in optimal foraging theory (see Pulliam 1981; Orians 1981; Zach and Smith 1981; Rapport 1981; Staddon and Gendron 1983; Pyke 1984). Despite the wealth of c r i t i c i s m s , models continue to incorporate these assumptions without regard to the uncertainty that is inherent in them. Because of these u n r e a l i s t i c assumptions, many future experiments w i l l probably f a i l . I have shown that computer simulations allow the examination of the assumptions of models in a way not possible by any other method. S e n s i t i v i t y analysis reveals how assumptions, i n d i v i d u a l l y or in combination, can af f e c t a model's predictions. Simulations can generate predictions which can be used to id e n t i f y u n r e a l i s t i c assumptions. Ideally t h i s tool should be applied during the developmental stage of a model before any empirical work has begun. 59 CHAPTER 4 THE PARADOX OF RISK-SENSITIVE FORAGING One of the fundamental assumptions of r i s k - s e n s i t i v e foraging models i s that "natural selection has acted on the preference behaviour of those animals faced with environmental v a r i a t i o n " (Caraco e_t a l . 1980). Further, natural selection is assumed to have favoured those animals whose behaviour minimizes the p r o b a b i l i t y of energetic s h o r t f a l l . Both of these assumptions are e x p l i c i t l y included in the z-score model (Stephens 1981; Stephens and Charnov 1982). In Chapter 2 I concluded that rufous hummingbirds foraged in a r i s k - s e n s i t i v e manner, as predicted by that model. In Chapter 3 I concluded that r i s k - s e n s i t i v e foraging rarely s i g n i f i c a n t l y reduces the probability of a forager starving compared to mean-maximizing. Despite this conclusion, rufous hummingbirds (and at least 10 other species in several major taxa) are r i s k - s e n s i t i v e (Chapter 2), at least under some experimental conditions. The apparently contradictory conclusions of Chapters 2 and 3 represent a paradox: even though many species are r i s k -sensitive, the selective forces postulated for the existence of this s e n s i t i v i t y do not appear to e x i s t . The paradox would be resolved i f the conclusions of either the laboratory experiment or my simulation analyses of the z-score model were not relevant to nature. The laboratory results would be i n v a l i d indications of natural c a p a b i l i t i e s i f they could be shown to be a r t i f a c t s of my experimental design. The conclusions from my simulations about the value of r i s k - s e n s i t i v e foraging would be i n v a l i d i f I 60 underestimated natural variation in d a i l y food intake. Resolving the paradox Was the behaviour that I observed in the laboratory purely an a r t i f a c t of my experiment? This i s a problem in any type of behavioural research; e s p e c i a l l y experimental research, and more esp e c i a l l y in the laboratory. It is often a p r a c t i c a l necessity to take animals out of their natural environments and study them in highly a r t i f i c i a l environments. Moreover, the questions asked are often extreme abstractions of real l i f e problems; for example I represented the choice between d i f f e r e n t prey types in nature by preferences between two coloured c i r c l e s at the ends of small chambers (see Chapter 2). It i s never certain how much the a r t i f i c i a l i t y of the laboratory a f f e c t s , or i s even the cause of the observed behaviours. Zach and Smith (1981) c r i t i c i z e d laboratory tests of optimal foraging theory because they "have l i t t l e relevance to conditions in the wild." Johnston (1985) suggested that, at best, laboratory experiments are a r t i f a c t u a l , and as such they have l i t t l e bearing on behaviour in the wild. Because my experimental environment was very a r t i f i c i a l , i t is possible that the behaviour I observed has no analogy in nature. My response to thi s is that, even i f my own results were a r t i f i c i a l , r i s k - s e n s i t i v e foraging has been documented in too many species using a range of experimental approaches (see Chapter 2) for me to believe that i t does not represent a real phenomenon of some kind. The ecological relevance of thi s 61 phenomenon i s , however, s t i l l uncertain. Did I misjudge the value of r i s k - s e n s i t i v e foraging in my simulations? By d e f i n i t i o n , energetic s h o r t f a l l occurs when t o t a l energy gain does not meet requirements. I concluded from my simulation analyses that r i s k - s e n s i t i v e foraging i s advantageous only when mean da i l y intake i s near requirements and the variance of da i l y intake i s large enough to generate s i g n i f i c a n t l i k e l i h o o d of energetic s h o r t f a l l . In a l l of my simulations, I assumed that the number of food items eaten per day (N) i s fixed for the day. Daily N is not fixed for many mobile foragers, however, who simply eat more items when their requirements increase or when average item qua l i t y decreases (Krebs and Davies 1984; Kamil and Sargent 1981). Therefore, individuals tend to be buffered against energetic s h o r t f a l l by th i s f l e x i b i l i t y . Physiological mechanisms such as torpor also buffer them, making them less dependent on achieving "requirements" every day (Hainsworth et a_l. 1977; Tooze and Gass 1985). A l l animals must maintain positive or neutral energy budgets on some time scale, but as I argued in Chapter 1, hummingbirds should be more dependent on meeting requirements every day than most animals. A good example of behavioural buffering against energetic s h o r t f a l l is temperate species of hummingbirds, including rufous, which defend feeding t e r r i t o r i e s in dense meadows of flowers. These hummingbirds are very f l e x i b l e in many ways (Gass, . 1986), one of which i s that individuals regulate the numbers of flowers in their t e r r i t o r i e s as conditions change 62 (Gass et a l . 1976; Kodric-Brown and Brown 1978; Gass 1979; Hixon et a l . 1983; Gass and Sutherland 1985). This maintains d a i l y nectar production per t e r r i t o r y very near Rnet in most years (Montgomerie and Gass 1981), although in some years there i s so l i t t l e competition for t e r r i t o r i e s that t e r r i t o r y quality i s much higher than Rnet (Carpenter, 1986). This f l e x i b i l i t y v i o l a t e s my assumption of fixed N, so t e r r i t o r i a l hummingbirds should not gain from r i s k - s e n s i t i v e foraging unless N were small or variation among flowers in nectar standing crop were very large. N i s not small in these systems. Very r i c h t e r r i t o r i e s have several hundred flowers (Gass e_t a l . 1976), and less densely-flowered t e r r i t o r i e s can s t i l l a f f ord positive energy budgets from low e f f o r t i f they contain several thousand flowers (Gass et a l . 1976; Carpenter, 1986). Several related arguments suggest that v a r i a t i o n in nectar standing crop is very low in temperate t e r r i t o r i a l systems. F i r s t , optimal foraging theory predicts that individuals forage non-randomly with respect to nectar a v a i l a b i l i t y , s e l e c t i v e l y harvesting the best options from the d i s t r i b u t i o n s available to them. The theory thus predicts that foragers experience lower variance in p r o f i t a b i l i t y (and a higher mean) than exists in their environments. Second, t e r r i t o r i a l hummingbirds do forage non-randomly with respect to several types of variation including v a r i a t i o n in nectar standing crop (Pyke 1978; Gass and Montgomerie 1981; Wolf and Hainsworth 1983; Gass and Sutherland 1985). Third, most flowers defended by hummingbirds produce nectar slowly (hence the large number of flowers in 63 t e r r i t o r i e s ) , and accumulate l i t t l e . Variation in standing crop is strongly related to mean standing crop (Table 4.1; r 2 = 0.93; F = 303.3, p << 0.001). This regression included 22 plant species separated by over 40 degrees of la t i t u d e . Note that the sample represents three orders of magnitude in both mean and standard deviation, and includes species from t r o p i c a l and temperate zones. The three lowest variances in Table 4.1 (Aquileqia formosa, C a s t i l l e j a miniata, and Ribes cereum) are from flowers that are known to be defended by hummingbirds (Armstrong 1986; Gass et a l . 1976). If these species exhibit low variance in standing crop, hummingbirds foraging on them should experience even lower variance, for the reasons outlined above. The combination of low variance in nectar harvest per flower and large number of flowers v i s i t e d per day suggests that the standard error of mean da i l y harvest is low for these hummingbird species, and that they are strongly buffered against energetic s h o r t f a l l . N, and therefore da i l y intake, may be r e l a t i v e l y fixed in at least one hummingbird system, at least some of the time. Some t r o p i c a l hummingbirds (termed "high reward t r a p l i n e r s " ; Feinsinger and Colwell 1978) f l y long distances to forage from small numbers of individual flowers that produce large quantities of nectar. Traplines can be stable for extended periods of time because the plants of the species on which these traplines are b u i l t open only a few flowers at a time for weeks or months. Risk-sensitive foraging could be a s i g n i f i c a n t advantage to trapline hummingbirds under these conditions (small 6 4 Table 4.1. Mean and standard deviation of nectar standing crops for a range of temperate North American and t r o p i c a l nectar producing plant species. Spec ies Source Standing Mean (st, Crop dev. ) Agastache c f . p r i n g l e i C a s t i l l e j a t e n u i f o l i a C a s t i l l e j a i n t e g r i f o l i a Cuphea llavea ( p i s t i l a t e phase) Cuphea llavea (staminate phase) Hedeoma c i l i o l a t a Lamourauxia r h i n a t h i f o l i a Lobelia c a r d i n a l i s Operculina a l a t i p e s Penstemon kunthi i Penstemon barbatus Psitticanthus caylculatus Salvia c a r d i n a l i s Salvia elegans Salvia gregi i Salvia pubescens C a s t i l l e j a miniata Ribes cereum Aquilegia  Erythrina  Erythrina Justica formosa f usca poeppingiana secunda Justica secunda Madevilla hirsuta Heliconia  Heliconia  Heliconia Heliconia  H e l i c o n i a - 2 H l e i c o n i a - 3 Heliconia - 1 6 Heliconia- 1 7 Heliconia- 1 8 wagnerlana  imbr icata  latispatha mar iae (Trinidad (Trinidad (Trinidad (Tabago (Trinidad (Costa Rica (Costa Rica (Costa Rica (Costa Rica (Costa Rica (Costa Rica (Costa Rica (Costa Rica (Costa Rica 2 2 3 4 4 5 5 5 6 6 6 6 6 6 6 6 6 10.10 21 .67 30.24 36.80 71 , 5 , 21 40 8 8 17 30.07 67.37 15.29 15.79 88.37 79. 13 24.86 19.32 85.68 2.27 1 .34 5.88 345.40 105.39 99.66 35.69 435.70 482.71 322.01 380.56 200.74 589.67 296.93 317.84 464.21 376.39 (5 (9 (15 (45 (42 (4 (8 (34 (38 (5 (4 (57 (29 (16 (6 (43 (2 (1 (8 (202 (33 (27 (13 (186 (229 (153 (181 (95 (280 (141 (151 (220 (179 .04! .58: .46: . 1 9 : .84: . 0 3 : .06: .61 .81 . 2 1 . 5 4 : . 1 2 : . 2 3 : . 4 6 : . 2 2 : . 6 8 : .69: . 0 1 .58: . 7 4 : . 2 9 : . 9 4 : . 3 7 : . 0 4 : . 7 2 : . 2 4 : . 11 . 5 3 : .61 .31 . 2 6 : . 9 2 : . 1 2 : Sources: (1) Cruden et a l . (1983) (2) Armstrong TT986) (3) Gass (1974) (4) Feinsinger et a l . (1979) (5) Linhart and Feinsinger (1980) ( 6 ) S t i l e s (1975), * standard deviations estimated from a linear regression of the f i r s t 24 ent r i e s . 65 N, high mean per flower) i f variance among flowers were large. If so, the chance of several small energy gains in succession (e.g. because another hummingbird v i s i t e d some flowers in the trapline) i s a real p o s s i b i l i t y ; reduced d a i l y intake and possible energetic s h o r t f a l l could r e s u l t . There are few data in the l i t e r a t u r e of means and standard deviations of nectar standing crops for plants v i s i t e d by trapline foragers. However, standing crops for nine species of Heliconia ( S t i l e s 1975), a l l of them v i s i t e d by t r a p l i n i n g hermit hummingbirds, a l l have very high means (Table 4.1). As I showed e a r l i e r , high variances tend to be associated with high means (one of the two very high-mean species included in that c o r r e l a t i o n analysis i s also v i s i t e d by hermits), so these species of hermit hummingbirds probably experience high variances in nectar a v a i l a b i l i t y . Whether hermit hummingbirds foraging on these species are actually subjected to s i g n i f i c a n t r i s k s of starvation depends on how close d a i l y net gains are to requirements. If net gains are well above Rnet, the variances w i l l have to be very large in order for there to be a s i g n i f i c a n t r i s k of starvation. Unfortunately, these data do not ex i s t , so i t is not possible to assess the ris k of starvation, and therefore the advantage of r i s k - s e n s i t i v e foraging, for these hummingbirds. Why are rufous hummingbirds r i s k - s e n s i t i v e ? Very simply, I do not know. The behaviour ex i s t s , even though the assumption that i t reduces the pr o b a b i l i t y of starvation i s probably f a l s e . It i s possible that t h i s behaviour i s an h i s t o r i c a l e f f e c t , the 66 result of evolutionary forces no longer in existence. Hummingbirds probably evolved during the Upper Pleistocene (approximately 1 m i l l i o n years ago) (Fishes and Peterson 1964) in the tropics of South America (Mayr 1964). It i s possible that the ancestors of rufous hummingbirds evolved under conditions similar to those experienced by modern t r o p i c a l t r a p l i n e foraging species. Under these conditions r i s k -sensitive foraging behaviour may have been s e l e c t i v e l y advantageous. Today, th i s selective advantage may no longer exist for rufous. Instead, t h i s t r a i t may be s e l e c t i v e l y neutral, representing "unexplored p o t e n t i a l " (D. Brooks, personal communication), and may show up only under laboratory conditions. Future research There are many other assumptions in both the z-score model and my simulations, that I have not considered. For example, how do the advantages of r i s k - s e n s i t i v e foraging d i f f e r between t e r r i t o r i a l , t e r r i t o r y - p a r a s i t i c , high and low-reward t r a p l i n e r s , and generalists? How do they change between migratory and non-migratory species, between breeding and non-breeding seasons, between males and females? A l l of these questions await further research. I think a broad, comparative study of r i s k - s e n s i t i v i t y in a wide range of species would be very helpful in understanding more about the nature of r i s k - s e n s i t i v e foraging. This research should focus on differences in r i s k - s e n s i t i v i t y between 67 di f f e r e n t classes of hummingbirds, p a r t i c u l a r l y between high-reward t r a p l i n e r s and temperate zone t e r r i t o r i a l i s t s . An analysis of the differences in r i s k - s e n s i t i v i t y between these classes may reveal some of the important ecological factors related to the existence of r i s k - s e n s i t i v e foraging behaviour. For example, i f the size of N and the amount of variance are important in determining the advantage of r i s k - s e n s i t i v e foraging then I would expect to see greater r i s k - s e n s i t i v i t y in the t r a p l i n i n g hummingbird species than in the temperate t e r r i t o r i a l species. Without understanding more about what ecological factors influence r i s k - s e n s i t i v e foraging behaviour, i t w i l l be d i f f i c u l t , or even impossible, to develop models capable of making accurate general predictions about patterns of r i s k - s e n s i t i v e foraging behaviour. 68 LITERATURE CITED Armstrong, D.P. 1986. Some aspects of the economics of t e r r i t o r i a l i t y in North American hummingbirds. M.Sc. Thesis. 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