Risk and Foraging in Stochastic EnvironmentsAuthor(s): Leslie Real and Thomas CaracoReviewed work(s):Source: Annual Review of Ecology and Systematics, Vol. 17 (1986), pp. 371-390Published by: Annual ReviewsStable URL: http://www.jstor.org/stable/2097001 .Accessed: 22/02/2012 22:30

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Ann. Rev. Ecol. Syst. 1986. 17:371-90 Copyright ? 1986 by Annual Reviews Inc. All rights reserved

RISK AND FORAGING IN STOCHASTIC ENVIRONMENTS

Leslie Real

Department of Zoology, North Carolina State University, Raleigh, North Carolina 27695-7617

Thomas Caraco

Department of Biological Sciences, State University of New York, Albany, New York 12222

INTRODUCTION

Foraging animals confront problems conceptually similar to those facing an economically minded consumer (46, 47), and foraging theory shares a meth- odology in common with economics. Indeed, the last 20 years have seen wide application of economic models in biology. A growing consensus suggests that ecological and economic theories are ultimately indistinguishable (6, 30).

Economic analyses begin with an economic agent that chooses from alternative objects or activities. The analyst extracts a set of measurable attributes from this collection of objects (e.g. energy content, costs, etc) that are combined into a single index. This index has the property that objects whose attributes generate higher values are preferred by the consumer over objects generating lower values. In economics, this index (the 'utility func- tion') is the basic organizing principle (36). Behavioral ecologists use this technique to learn what attributes should be included in an organism's utility function to determine how various attributes should be combined and weighted, and then to predict choices by maximizing the utility function under applicable constraints. The organism's utility function is, therefore, both an object of empirical study and a predictive tool. Economic analysis usually stops here, but evolutionary biologists go farther. They assume that the organisms' preferences are related to evolutionary fitness and that the options

371 0066-4162/86/1120-0371$02.00

372 REAL & CARACO

more preferred must lead to greater survival and reproduction. Hence, we hypothesize that the utility function is isomorphic with fitness. This, of course, is an assertion, and one that needs more empirical investigation.

The underlying assumptions and limitations of economic optimization have been clarified (53), criticized (26), and defended (48). We feel that these techniques, although controversial, are the best available for deducing empir- ically falsifiable hypotheses based on first principles for behavior.

Our theories necessarily examine one (or just a few) phenotypic characters in any particular model. Many traits interactively govern survival and reproduction, but we are forced to limit the phenotypic dimensionality of a theory if it is to be tested rigorously. However, in our interpretation of theory we recognize that single-attribute strategic models concern phenomena con- ditioned by other traits and competing selection pressures (11, 42, 53, 59). Interpreting a model for a single trait's adaptive significance requires that we remain wary of (but not inhibited by) genetic, environmental, and organismal sources of variation in the character of interest.

We might treat a theory and an organism's behavior like a map and a territory. More formally, we replace a theory's algebraic interpretation with a probabilistic interpretation. Many models in behavioral ecology, and nearly all models of foraging strategy, can be written as questions of preference and choice. In these cases a probabilistic interpretation of a theory means that we replace a strong utility hypothesis with a weak utility hypothesis (12, 18, 45). The following example contrasts these alternatives.

Suppose an organism must choose between two options. Let the associated probability densities of benefits be fi and f2. The ecologist constructs a theoretical fitness/utility function W and deduces predictions by comparing the expected fitnesses E(Wlfi). Letfi Pf2 designate preference forfi overf2. The ecologist seeks to predict preference (or indifference) with the model.

If E(W If,) : E(W If2), then an algebraic interpretation of the model yields a strong utility hypothesis of the form:

E(W|fi) > E(Wjf2) -* Pr(fi P f2) = 1. 1.

The strong utility interpretation predicts exclusive choice [or equiprobable choice when E(WIfi) = E(W1f2)]. Ecologically significant choice behavior seldom exhibits this sort of deterministic regularity (11, 18, 45).

Now suppose that the stochastic nature of selection processes, interactions with other traits, and other factors render the organism's behavior less than perfect. The model W still distinguishes any difference betweenf, and f2, but the predicted response to the difference (the trait of interest) is better treated as probabilistic. This interpretation of the model yields a weak utility hypothesis of the form:

RISK-SENSITIVE FORAGING 373

E(WIf,) : E(WIf2) -* Pr(fI Pf2) : 1/2. 2.

Iff1 advances fitness more than does f2, there should be a tendency to prefer fl. As the difference between the E(W) values increases, preference for the more profitable option presumably increases. The weak utility interpretation maintains a distinction between the model and the organism and more realisti- cally describes the consequences of selection than does the determinism of the strong utility interpretation.

In this paper we review some models for foraging decisions in stochastic environments. To do so, we must examine both the nature of the stochasticity and the way in which the forager might efficiently respond.

CLARIFYING RISK

Economists often distinguish between two types of decision-making (2). In the first type of decision the economic agent knows the objective probability distributions characterizing outcomes for feasible actions correspond to con- ditions of 'risk.' The decision-maker presumably knows (via experience or selection on a genetic program) the likelihood (probability) of an event's occurrence. In the second type, organisms with no such knowledge, (or an organism possessing only subjective estimates of the distributions) face con- ditions of 'uncertainty.' To predict behavior we must assume some form of underlying probabilistic structure. Thus, economists analyze uncertainty in the same manner as conditions of risk. However, the conditions are quite distinct; some aspects of economic behavior arguably result from uncertainty but not from risk (38). From a biological perspective, we emphasize that natural selection operates on objective probabilities, but the organism may still be deciding subjectively (at least during an initial sampling period).

The ecological literature mentions different kinds of risk, and some confu- sion has resulted. For example, the 'risk of predation,' though of obvious importance, is not related to risk as we use the term. In fact, most models of predation risk are 'riskless' in that a discrete, random outcome (the individual survives or is killed by a predator) is replaced by the population's average mortality hazard due to predation. The model organism responds not to probability distributions but to fixed attributes of patches, habitats, etc. In any case, our discussion of risk concerns an organism's preferences over probabil- ity distributions.

As indicated above, a forager in a stochastic environment may face two distinct but complementary problems (23, 69). First, the organism must learn the reward probability distributions associated with different behaviors (a problem of information), and then it must select a strategy for exploiting those distributions (a problem of risk).

374 REAL & CARACO

Information problems are usually analyzed with Bayesian models in which the organism modifies its estimates of probabilities based on its experience while foraging. For example, Green (27, 28) assumes that a bird searches a patch until it exhausts the patch or decides to leave. Each patch contains n places, or bits, where a prey item may occur. Given a level of patch quality (the probability that a bit contains a prey item), total prey per patch is a binomial variate. Patch quality itself is a Beta random variable. The optimal Bayesian bird leaves a patch when it has found only k prey after sampling tk different bits. The Bayesian dimension is explicit since the decision to leave a patch depends on experience there and the overall pattern of patch quality (27, 28, 49, 51).

Analyses of problems of information have generated a variety of new hypotheses differing qualitatively from the results of corresponding de- terministic models. For example, consider Clark & Mangel's (21, 22) theory for the evolution of group foraging, or Real's (62) model for habitat selection in fish. Houston et al (31) and Krebs et al (40) have reviewed the theory for problems of information. Experimental research on questions of information has begun only recently in behavioral ecology (39, 43). Therefore, the remainder of this review restricts attention to problems of risk, where the theory has been subjected to a reasonable number of tests.

RISK-SENSITIVITY AND FORAGING BEHAVIOR

Most theories for risk-sensitive foraging propose decision criteria that ex- plicitly or approximately involve an interaction between the mean and vari- ance of food consumed or the time spent acquiring required energy (8, 9, 32, 50, 57, 58, 59, 69). These criteria arise naturally when the relevant probabil- ity distributions are functions of two independent parameters. However, maximizing the expectation of a random variable (without explicit considera- tion of the variance) will be appropriate in certain simple problems (13). Suppose the random outcome of a foraging decision follows a one-parameter distribution, so that all cumulative probabilities are specified by the parame- ter's value. If fitness is maximal when the probability of starvation (or other penalty) is minimal, ranking the means can be sufficient to solve the problem.

Let the i-th option imply that identical food items will be consumed at constant probabilistic rate Ai. If T is the time available, and X is the total food consumption (a discrete, nonnegative random variable), then choice of the i-th option means that X will be a Poisson variate with expectation AiT.

Suppose the forager starves or fails to reproduce if X ? R. We hypothesize that selection favors minimizing Pr(X ? R), so that fitness is a step function of total food consumption. The best strategy chooses Ai according to:

RISK-SENSITIVE FORAGING 375

R

min Pr(X ? R1A) = min e -A (AiT)xlX! 3.

= minr(R + 1, AiT)/R!, 4.

where

F(R + 1, AiT) = fAT e-ttRdt. 5.

Equation (5) indicates the equivalence between minimizing Pr (X? R) and maximizing AiT = E(X) in this simple example. A useful generalization of this sort of problem invokes the order relationship 'stochastically larger' (13, 65).

Let XI and X2 be discrete nonnegative random variables where XI is stochastically larger than X2 (denoted XI ' ST X2). Then, by definition (65):

Pr(XI > R) ? Pr(X2 > R) for all R. 6.

Taking complements in (6), we have

Pr(XI ? R) ? Pr(X2 ?R) for all R. 7.

Using (6) again and summing over all possible values of R, we obtain:

E(X1) = E Pr(XI > R) ' E Pr(X2 > R) = E(X2) 8. R=O R=O

As above, let X be total prey consumption and let R be the requirement for survival. Expression (7) indicates that choosing XI cannot lead to a greater probability of starvation when XI : ST X2- If outcomes Xi can be ranked by the order relationship stochastically larger, the chance of starvation cannot increase (and usually will decrease) as E(Xi) increases (13). However, such a simple equivalence between fitness and mean rewards will hold only in special cases. For example, if total energy intake follows a normal distribu- tion, the probability of starvation depends on combined effects of both mean and variance. Consequently, a useful theory for risk-sensitive foraging re- quires more than ranking strategic options by their associated averages.

To examine mean-variance effects, let W(X) map energy intake into fitness. In theory, some W specifies present and expected future survival and reproduction for any X, but we require more tangible currencies for empirical work. For a nonbreeding forager, W(X) can be survivorship over a given time interval. McNamara & Houston (50) show that, under certain conditions, lifetime fitness will be proportional to short-term survival during the

376 REAL & CARACO

nonbreeding season. Therefore, we need not always project very far into the future when hypothesizing a form for W(X).

For some organisms, W can be taken as survival over a single day (18, 19, 57, 67, 69). Let X now represent total daily energy intake, a continuous random variable. X is a sum acquired over n foraging opportunities during the time available for feeding. For sufficiently large n, X approaches normality by the central-limit theorem (8, 67). We let E(X) = , and let V(X) = (J2, and take Pr(X - R) as the probability of starvation.

Stephens & Charnov (69) assume selection acts to minimize the probability of starvation, and they identify the consequences in a useful manner. Since X approaches normality, the distribution of z [where z = (X - p)Io-] approaches the standard normal. Then minimizing Pr(X 2 R) is equivalent to minimizing:

Pr[z ' (R - O)IMi = 4(ZR) 9.

Hence the z-score model. Equation 9 reveals that dJ(zR)hd9p < 0. For given or, an increase in mean

reward will obviously be favored. The effect of varying the standard deviation depends on the difference between the required and expected intake, R - ,. If , > R, the forager can expect to surpass its requirement, so that its energy budget is positive (10, 12). In this case do(zR)ld/o > 0; for a given , an increase in reward variance increases the chance of starvation. Consequently, we anticipate risk-aversion when ,u > R (10, 18, 19, 32, 57, 69). If , < R, the forager's expected energy budget is negative and d8 (zR)/doJ < 0. For a given , an increase in benefit variance decreases the chance of starvation. Therefore, we anticipate a risk-prone response to benefit variance when ,t <

R (12, 19, 57, 69). Suppose two alternative foraging options yield respective total benefits X1

and X2. The Xi are random variables with V(X1) < V(X2). A risk-average forager will prefer X1, because its variance is lower, unless E(X2) is suf- ficiently larger than E(X1). A risk-prone forager will prefer the more variable X2, unless E[X1] is sufficiently larger than E[X2]. The preceding analysis predicts that a forager's expected energy budget governs its response to risk when the animal must satisfy a short-term physiological requirement, and that reward mean and variance both will influence choice over two-parameter distributions of benefits or costs (13, 50, 57, 59, 69). Furthermore, the z-score model suggests an explicit mean-variance interaction. By analyzing iso-fitness contours, Caraco & Lima (17) show that the z-score model predicts decreasing risk-sensitivity (see below), so that the way in which mean and variance collectively govern preference depends not only on the sign of (, -

R), but also on the value of ,u. When W(X) is survival over a single day, all values of X such that X > R

RISK-SENSITIVE FORAGING 377

impart the same fitness. That is, the z-score model hypothesizes a step function for W(X), but other models for risk-sensitive foraging treat con- tinuous fitness functions. Real (58) hypothesizes a concave W(X). Caraco (8) and McNamara & Houston (50) conclude that W(X) will be convex-concave when some sort of short-term requirement is biologically significant.

For a nonbreeding forager, survivorship should always increase with ener- gy intake, so we assume W' > 0. This assumption should hold unless excess weight imposes a mortality hazard (44). Surviving winter could imply that the forager monitors an average daily requirement; a reasonable possibility is that energy intake balances expected 24-hour expenditures (19). We let R* represent the average daily requirement; a total energy intake of at least yR* is required over y days (-y > 1). For X well below R*, survival is low (but immediate starvation need not be certain since y > 1). As X approaches R*, survivorship increases with increasing rapidity. Therefore, W is convex (W" > 0) for X < R*. Above R* each additional amount of energy quite likely decreases in value to an adequately fed animal (58), but energy consumed beyond R* can be carried over to the next day (68). Therefore, W is concave (W" < 0) for X > R*. As discussed below, this form for the fitness function predicts risk-prone preferences for X < R*, and risk-averse preferences for X > R*.

When fitness cannot be approximated by the individual's short-term sur- vival or reproductive success, the convex-concave form for W may not apply. For example, colony-level fitness in social insects might suggest that W(X) is strictly concave for the individual forager, since individual physiological requirements are presumably less important selectively than is the energy budget of the more permanent colony (53). If this is true, risk-aversion at all reward levels follows. As a second example, we note that body size in birds and mammals should correlate positively with the length of time an individual can survive without food (7, 25). Therefore, the time horizon (-y days) over which the average requirement R* is defined could easily increase with body size. As a consequence, the convexity of W for X < R* may approach linearity as body size increases. Houston & McNamara (33) consider a forager with an infinite time horizon (-y -* oo); their theoretical objective becomes maximization of the expected time until starvation occurs. Given a set of assumptions concerning prey profitabilities and encounters with prey, they find that a forager should always take a prey item if it yields an increase in total stored energy. Under certain conditions, this strategy might imply strict risk-aversion (Houston, personal communication). Therefore, larger animals might adhere to risk-aversion, provided that larger body size tends to free the forager from short-term energetic requirements. If the shape of W(X) does indeed depend predictably on body size, then the regulatory models proposed by Staddon (66) may prove valuable to both ecologists and operant psychologists.

378 REAL & CARACO

Returning to foragers with short-term requirements, suppose that W(X) is convex-concave with an inflection point at X = R*. If we hypothesize an analytical form for W, the expected fitness associated with the i-th strategic option is:

Ei[W(X)] = f W(X) f1(X) dx, 10.

wherefi(X) is the probability density of energy intake resulting from choice of the i-th option. The best decision maximizes expected fitness. However, we often will want a more general procedure that does not require specifying W(X) analytically. A simple alternative is expanding W(X) about A with a Taylor series approximation (1, 34, 58). Expanding through the third de- rivative and then taking expected values, we have:

E[W(X)] W(g) + (1/2) W" (i) c.2 + (1/6) W'.'(..) 3, 11.

where 3 is the skew of X, E[(X - 3]. If X is distributed symmetrically about its mean, p3 = 0 and Expression 11 then suggests variance discounting (58, 59). When X has a skewed distribution (i.e. when 3 * 0), g and o-2

generally are dependent (1). However, we can separately examine the con- tribution to expected fitness of mean, variance, and skew, both theoretically and experimentally (15, 41).

The first term on the righthand side of Expression 11 is the mean's contribution to expected fitness. Assuming W' > 0, an increased mean reward ordinarily will enhance expected fitness. The second term in Expres- sion 11 is proportional to W" (I) o-2. If ,u < R*, then W" > 0 and expected fitness increases as reward variance increases [by Jensen's inequality: (8, 24)]. Hence, the forager should be risk-prone when expecting an energetic deficit. If ,u > R*, then WI" < 0 and expected fitness decreases as o-2

increases. Therefore, risk-aversion follows from an expected energetic sur- plus. Note that the simpler step function models generate essentially the same qualitative predictions concerning response to reward variance as do the more realistic convex-concave models.

The effect of skew on expected fitness will depend on WI'I (which is assumed to exist). To analyze WI'I, it is convenient to categorize risk- sensitivity as constant or variable. For simplicity, we restrict our discussion to risk-aversion; for a broader treatment see Caraco & Chasin (15).

Constant risk-aversion follows when a given o-2 depresses expected fitness by the same amount at any level of ,u, meaning that W" does not depend on X. Decreasing risk-aversion follows when the negative effect of a given - 2

decreases as ,u increases, implying that WI" (,) declines as , increases. For increasing risk-aversion dW" I (g)/ld > 0, and a given o 2 exerts the opposite

RISK-SENSITIVE FORAGING 379

effect on expected fitness. Decreasing risk-aversion has the greatest intuitive appeal.

We can portray constant and variable risk-aversion via Pratt's (55) index of 'local risk-aversion': p(X) = - WI /W'I. Since W' > 0, p(X)>0 for all forms of risk-aversion. Under constant risk-aversion, dp(X)/dX = 0. For decreasing risk-aversion, dp(X)/dX < 0; for increasing risk-aversion, dp(X)/dX > 0.

Since p(X) = -W"/W',

d p(X)/dX = [(W'v)2 - W ' W'] I (W')2. 12.

Knowing the sign of d p(X)/dX may allow us, from Equation 12, to determine the sign of WI'I and, hence, the effect of skew on expected fitness. Caraco & Chasin (15) show that positive skew tends to advance expected fitness when risk-aversion is either constant or decreasing. Skew in X may arise from rare events or from uniformly skew rewards at individual foraging bouts (15). An efficient forager might respond to reward skew, but responses to mean and variance presumably dominate in nature.

If we truncate Expression 11 at the second moment, we have a variance discounting formula: E(W) W(,u) + (1/2) W" (,) o-2. Without specifying W(X), we might take WI '/2 as a 'constant of risk-aversion' (60, 64). That is, we might predict constant risk-aversion by assuming W" is independent of X. However, as an approximation to a postulated analytical form for the fitness function, variance discounting can accommodate constant or variable risk- sensitivity.

For instance, if we assume that W(X) = log X and evaluate this function according to the Pratt measure of local risk-aversion, we see that the coeffi- cient of risk diminishes:

p(X) = - W"/W' = X > 0 p'(X) = - 1/X2 < 0.

If we assume that W(X) is quadratic, i.e. W(X) = aX - bX2, then the organism will show increasing risk-aversion. Alternatively, for W(X) = a -

e-bX [a form used by Caraco (8)] then risk-aversion will be constant. Pratt (55) catalogs a variety of different utility/fitness functions that will generate different types of risk aversion.

Analysis of the indifference curves for the different fitness functions will reveal the implicit response to increasing expectation (Figure 1). Curves bending upward correspond to fitness showing diminishing risk-aversion. Straight lines correspond to constant risk-aversion. Strictly concave in- difference curves indicate increasing risk-aversion.

The Taylor Series (Equation 11) can be used to approximate any of these

380 REAL & CARACO

a (N

b

Mean (,u) Figure 1 Generalized indifference curves for three different specified fitness functions W(X): (a) W(X) = aX - bX2, showing increasing risk-aversion, (b) W(X) = a - e-'x, showing constant risk-aversion, and (c) W(X) = log X, showing diminishing risk-aversion. The curvature of the indifference relation reveals the organism's sensitivity to risk with variation in the expectation of the distribution.

functions. However, termination of the approximation at the second moment requires either (a) the variables are normally distributed, or (b) W(X) is quadratic. The approximation will only be exact then if we admit the seeming- ly unreasonable assumption of increasing risk-aversion. Nonetheless, as an approximation the variance discount model will hold for any function W(X) and need not reflect a given type of variability in risk aversion. For example, for W(X) = log X the variance discount approximation would be:

RISK-SENSITIVE FORAGING 381

E[W(X)] = log , - 1/2 o-24lr2,

which clearly has a diminishing coefficient of risk. Our review of risk-sensitivity has mentioned only static models. If a forager

can switch among available options during the course of a day, a feedback control policy would decrease (or at least never increase) the probability of starvation. Houston & McNamara (32) discuss the optimal control policy when there is no cost to switching between two options with identical mean reward rates but with different levels of reward variability. The forager should use the option with the lesser variability as long as the animal expects to surpass its daily requirement. However, if the forager's energetic condition ever indicates that a deficit is more probable than an excess, the animal then should immediately switch to the more variable option. The best dynamic policy shows a strong similarity to the corresponding static model, but the optimal control also suggests predictable changes in sensitivity to risk.

In this section we have treated risk-sensitivity in a normative manner. The next step is, of course, to evaluate the theoretical predictions in light of empirical results.

EMPIRICAL RESULTS

To investigate risk-sensitive foraging experimentally, the investigator con- trols the reward distributions available to an animal. One class of experiments examines preference over reward variance when mean rewards are fixed. A second class tests predicted trade-offs between mean and variance by simulta- neously changing both parameters.

Responses to Variance with Fixed Means

The most direct test for risk-sensitivity lets a forager choose between two resources or feeding stations. One resource is held constant. The other resource varies randomly over space or time, but its expected value equals the constant reward. The organism selects the amount of time or effort allocated to each resource. The null hypothesis is indifference; any significant deviation from equal use of the two resources demonstrates risk-sensitivity.

Real (60) constrained bumblebees (Bombus sandersoni) to an artificial patch where an equal number of yellow and blue flowers were arrayed randomly on a large grid. A known quantity of artificial nectar was dispensed into each flower. Flowers of one color always provided 2 ,ul of nectar. Flowers of the other color provided a variable reward: Two thirds were empty, and the rest contained 6 ,il of nectar. Approximately 85% of all visits were to the constant reward, a response indicating risk-aversion. The bees still preferred the constant reward when all flowers contained at least 0.5 ,ul.

382 REAL & CARACO

Hence the observed risk-aversion is clearly more than avoidance of a floral type that sometimes provides no reward.

Real (60) also found that wasps (Vespula maculifrons) foraged risk- aversively in the same patch of artificial flowers. But the wasps' preference for the constant reward appeared weaker than that observed in the bees. Unlike bumblebees, the wasps are carnivorous and do not require nectar for reproduction. The difference in choice probabilities could mean that organ- isms are more sensitive to variance in those resources more critically linked to reproduction.

Waddington et al (71) demonstrated risk-aversion in a second species of bumblebee, Bombus edwardsii. In these experiments reward variation was imposed temporally; bumblebees apparently respond to both spatial and temporal patterns of reward variance in a risk-averse manner.

Wunderle & O'Brien (74) studied hand-reared Bananaquits (Coereba flaveola) foraging on artificial flowers of two different colors. In their first experiment, one floral type provided a constant volume of nectar. The other type varied, but its mean volume equalled the constant reward. Foragers generally avoided variance; less experienced birds chose the constant reward more than other subjects did.

In a second experiment, Wunderle & O'Brien (74) showed that the Banana- quits responded similarly when resource quality, rather than quantity, varied randomly. Wunderle & O'Brien filled every flower of both types with the same volume of nectar and made sugar concentration the random variable. The birds again preferred lesser variance, and measures of their risk-aversion did not differ significantly from those observed in the manipulation of nectar quantity.

Risk-sensitivity in small granivorous birds has been explored extensively by Caraco (10, 11, 12, 14, 19). A bird is confined in a large aviary where two feeding stations are separated by a partition. At each experimental trial the bird leaves a perch on the aviary's centerline and visits one of the feeding stations where the animal obtains a known number of millet seeds (Panicum miliaceum). The subject's expected daily energy budget can be manipulated by adjusting the length of preexperimental deprivation and the average feed- ing rate which the bird experiences during an experiment. The daily physi- ological requirement (the food intake balancing all energetic costs) is es- timated from analyses of oxygen consumption or food intake under ad lib availability.

Caraco et al (19) examined risk-sensitivity in yellow-eyed juncos (Junco phaeonotus) at both positive and negative expected energy budgets. The birds faced choices between a constant number of seeds and a random number of seeds; the mean of the variable reward equalled the constant reward. The juncos preferred the constant reward when their energy budget was positive

RISK-SENSITIVE FORAGING 383

but switched to preference for the variable reward when their energy budget was negative. The shift between risk-averse and risk-prone preferences was independent of the order of treatments an individual experienced.

Similar experiments with dark-eyed juncos (Junco hyemalis) yielded sim- ilar results; energy budgets predicted preference over reward variance (10). In addition, Caraco found that indifference to reward variance was the most common result for 'balanced' energy budgets. This was an intermediate condition where the feeding rate during an experiment was just sufficient to meet costs for a bird spending all available time foraging.

Caraco (11) next subjected white-crowned sparrows (Zonotrichia leucophrys) to the same experiments that revealed risk-aversion in juncos. White-crowned sparrows weigh 50% more than dark-eyed juncos, but their preference behavior did not reveal a significant interspecific difference in risk-aversion. Most small birds are constrained by daily energy requirements during the winter, so that similar responses to risk could be expected. As an independent variable for predicting risk-sensitivity, body size (in birds and mammals) may be interesting only when its variation is sufficient to induce variation in the time horizon over which foragers attempt to regulate energy intake relative to expenditures.

Experiments reviewed to this point all involve choice between a constant and a variable reward. But animals must discriminate among, and strategical- ly use, different variable rewards if risk-sensitivity is to influence behavior in a significant way. Caraco (12) reports that white-crowned sparrows chose between two variable rewards (random numbers of millet seeds) with the same expected value. The birds preferred the smaller or larger reward stan- dard deviation according to their expected energy budget.

The transition from risk-averse to risk-prone preferences has been observed in two other experimental systems. Barnard & Brown (3) enclosed individual common shrews (Sorex araneus) in plastic tanks with two feeding stations where the animals could obtain mealworm segments. One station held a constant reward while the other provided a variable reward; expected rewards were identical. When the shrews were fed at a rate below their physiological requirement, they preferred variability. However, they were risk-averse at positive energy budgets and chose the constant reward on 74% of the ex- perimental trials.

In a clever extension of their work, Barnard & Brown (4) presented the same choices to shrews under conditions of apparent resource competition. When a competitor was introduced (actually an apparent competitor, since the introduced animal was not allowed to feed), subject shrews were indifferent to reward variance. Competition eliminated evidence of risk preference in the shrews, but the adaptive significance (if any) remains a mystery. Kagel et al (35) suggest an economic interpretation.

384 REAL & CARACO

Moore & Simm (52) linked a switch between risk-averse and risk-prone foraging preferences to the annual cycle of the migratory yellow-rumped warbler (Dendroica coronata). Energy requirements peak during the pre- migratory period. The birds must deposit fat sufficient to satisfy the demands of migratory travel. When birds in this premigratory status were given a choice between a constant and a variable amount of food (with a mean equal to the constant reward), they were risk-prone. Control birds (i.e. birds not in premigratory disposition) were risk-averse when presented the very same choices. Furthermore, the migratory birds' preferences changed from risk- prone to risk-averse after they reached maximal body size. Note that the warblers' behavior showed both preference for and aversion to reward vari- ance, but their daily energy budget was never negative. That is, the transition in risk-preference was based on more than a comparison of daily intake and daily expenditure. When the birds' weight was less than the migratory level, they preferred the variable reward. But they became risk-averse when they attained the weight appropriate to the physiological requirements of migra- tion. In terms of our hypothetical W(X), this suggests that the average daily requirement (R* in the discussion above) varies seasonally in yellow-rumped warblers in accordance with the demands of their annual cycle.

Daily energy budgets predict the nature of risk-sensitivity in organisms whose fitness should be proportional to short-term survivorship: small, nonbreeding birds and mammals. But some other results indicate that risk- sensitivity is a more complex phenomenon, and we summarize those observa- tions here.

Recall that Barnard & Brown (4) found that common shrews' sensitivity to risk depended on their expected energy budget, but their risk-sensitivity disappeared when an apparent competitor was introduced. Shrews usually forage solitarily, but the result may suggest that direct competitive in- terference or dominance interactions could influence foraging preferences in nature.

K. D. Waddington, B. Heinrich, & L. Real independently have attempted to induce risk-prone behavior by depriving bumblebees of food, but the bees remained risk-averse toward variance in reward size (personal communication and personal observation). As we indicated above, an individual bee's food intake may always map into colony fitness in a concave manner (promoting risk-aversion). A second adaptive explanation proposes that bumblebees re- duce their energetic expenditures by lowering their metabolic rate when food is scarce, so that risk-proneness offers no strategic advantage. Of course, the mechanical rule of thumb (40) used by a forager need not involve scaling its expected intake to any requirement. A bumblebee might ordinarily forage quite efficiently by simply avoiding resources with large reward variances. For still another view, see Wells & Wells (73).

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Battalio et al (5) found that laboratory rats exhibited risk-averse preferences over randomly sized rewards whether their energy budgets were positive or negative. As previously discussed, a large animal's survival is seldom con- strained by immediate energy requirements, and risk-aversion becomes more likely as the time horizon for regulating food consumption lengthens with increases in body size. Large animals can depend on energy stores to mod- erate the influence of temporal fluctuations in food availability. Therefore, their risk-sensitivity need not be attuned directly to daily benefits and costs.

Both the theory and experiments we have reviewed to this point concern random variation in either resource quantity or resource quality. A number of studies in operant psychology examine a related but different problem. Each of two options provides the same fixed reward size. One alternative rewards the animal after a fixed delay elapses between choice and consumption; the other alternative imposes a variable delay (with expected value equal to the fixed delay). Most results indicate strong preference for the variable delay (23, 29, 37, 56). The subjects (usually pigeons, Columbia livia) were de- prived to about 80% of their normal body weight, so that risk-proneness deserves consideration as a possible explanation (40). However, foragers may treat delay times in a very different manner from reward size. If foragers discount future rewards, the economic model analyzed by Kagel et al (35) predicts that a variable delay will always be preferred over its mean. For another discussion of responses to energetic values vs responses to delays, see Staddon (66).

Mean-Variance Trade-offs

The models discussed above predict an interaction between the expectation and variance of a reward distribution. For example, a risk-averse forager prefers large means and small variances, so that its preferences will reveal a trade-off between mean and variance under appropriate conditions. The exact nature of the predicted trade-off is model specific, but most work in this area has realistically been limited to a comparison of constant and variable risk- sensitivity.

Caraco et al (19) studied certainty equivalents for reward distributions in foraging yellow-eyed juncos. That is, they located the constant reward for which a bird was indifferent between that constant number of seeds and a given variable reward. Under risk-aversion, the certainty equivalents were less than the variable distribution's mean (indicative of the mean-variance trade-off). As the mean of the variable reward increased, the difference between the mean and the certainty equivalent declined. This result suggests decreasing risk-aversion. Stephens & Paton (70) have demonstrated decreas- ing risk-aversion in rufous hummingbirds (Selasphorus rufus). Via an indirect

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statistical method, Battalio et al (5) interpret laboratory rats' preferences as examples of decreasing risk-aversion.

The way a risk-averse forager trades off mean and variance can be studied by constructing indifference curves in the mean-variance plane. However, only two such studies have been conducted to date.

Real et al (64) estimated mean-variance indifference curves from the foraging preferences of captive bumblebees (Bombus pennsylvanicus). In an artificial patch, flowers of one color provided a constant nectar volume. The variance of the other floral type was fixed, and its mean was increased or decreased until individual bees were indifferent (i.e. until the constant reward became a certainty equivalent for the variable reward). Next, the variance of the variable reward was increased and its mean then was adjusted until the bees' foraging choices showed indifference again. For a single constant reward, the procedure identifies three sets of responses to variable rewards: a set with each element preferred over the constant reward, a set where the constant reward is preferred over each element, and an indifference set with elements preferred equally to the constant reward. Regression analysis of the indifference set reveals the slope and curvature of the mean-variance trade-off (17, 64).

Real et al (64) completed four experiments of this sort. In three ex- periments, they detected a significant trade-off between mean reward and reward variability. The quantitative character of the trade-off was sensitive to a variety of ecological attributes, including the patchiness and color of the floral types. Two of the three significant trade-offs indicated constant risk- aversion; the third significant result suggested either constant or decreasing risk-aversion.

Caraco & Lima (17) performed similar experiments with dark-eyed juncos. In each of two series of experiments, they found a significant trade-off between mean reward and reward variability. Both constant and decreasing risk-aversion accounted equally well for the estimated indifference curves.

The experiments discussed in this section collectively reveal that risk- averse foragers clearly do trade off reward variance against mean reward. However, neither constant- nor variable-risk-sensitivity emerges as the domi- nant response. Recalling our earlier comments on algebraic vs probabilistic interpretation of a theory, we feel that a qualitative demonstration of a trade-off is the biologically significant result.

Both theoretical and empirical work on risk-sensitive foraging began only a few years ago. But there already has been a healthy exchange between model and experiment, resulting in a variety of new ideas (63). Our models need to be refined; we also need field tests and a greater understanding of the mechanisms involved. What is encouraging is that when we look at resource variance, it nearly always influences foraging behavior. The details of risk-

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sensitive foraging require a great deal of elucidation, but its existence is unquestionable.

IMPLICATIONS

The concepts of risk-sensitive decision-making currently are being applied to a variety of ecological problems. Our sampling will be brief and hardly exhaustive.

Houston & McNamara (33) present several models for optimal dietary choice when minimizing the probability of starvation is the strategic objec- tive. In an environment where two prey types are encountered sequentially and independently, the preference ranking of the prey types, in contrast to that in the classical model, is not absolute (40). When net energy per item is a random variable, the ranking of the prey types and the decision to specialize or generalize depend on the forager's energy reserve, means and variances of food items' energetic values, and the amount of time left in the day.

Caraco & Gillespie (16) analyze choice of foraging mode as a problem with risk. The model attempts to explain variation in mobility among certain female orb-weaving spiders during the reproductive season. When foraging sites vary both temporally and spatially, risk-aversion corresponds to a mobile strategy (changing web site each night), and risk-proneness corresponds to the sit-and-wait strategy (staying at the same site). The option yielding the greater probability of successful reproduction depends on the difference between the expected prey consumption and the intake required for production of an egg sac.

A number of authors have noted that risk-sensitivity may help explain aspects of social foraging (9, 13, 20, 21, 22, 57). When food is distributed in patches, the variance in the time an individual spends searching for food can vary inversely with group size. Similarly, larger groups may reduce the variance in daily food intake when food is found in patches and foraging time is constrained (20).

Caraco & Brown (14) investigated the evolution of food-sharing in stochas- tic environments. In certain communal species, nonrelated reproductives foster both their own and each other's dependent offspring. Sharing food can reduce the variance in the time spent providing the young their required food. But sharing will be an evolutionarily stable strategy only when food density is high, foraging time constraints are sufficiently relaxed, and mortality hazard due to predation is positively correlated among the young.

The implications of risk-sensitive foraging for the dynamics and evolution of plant-pollinator interactions are obvious. The distribution of nectar rewards can influence both the frequency of floral visitation and the pattern of pollen dispersal (61, 72).

388 REAL & CARACO

L. Real (personal observation) finds that pollinators' risk-sensitivity can depend on the scaling of reward sizes. The significance of risk-sensitivity in insect pollinators rests critically on the correlation of mean reward and its variance across floral species.

Ott et al (54) show that risk-sensitivity influences pollinator-mediated gene flow. Bumblebees (Bombus pennsylvanicus) increased the distance moved between floral visits as the variance in nectar volume increased about a fixed mean volume. Additionally, the variance in flight distance increased as reward variance increased. By equating the variance in flight distance with variance in gene dispersal, Ott et al (54) demonstrated that neighborhood size increases linearly as reward variance increases. Hence, risk-sensitivity in pollinators may be an important factor regulating gene flow among plants.

We hope that we have provided a simple introduction to both the concepts underlying theories of risk-sensitive foraging and the experimental work designed to test those theories. Of course, the most important evaluations of the theory will come from the results of field studies, many of which are just beginning.

ACKNOWLEDGMENTS

This research was supported by NSF Grants BNS 8418714 to Thomas Caraco and BSR 8214599 and BSR 8500203 to Leslie Real.

Literature Cited

1. Arditti, F. D. 1967. Risk and the re- quired return on equity. J. Finance 22:19-36

2. Arrow, K. 1974. Essays in the Theory of Risk-Bearing. Amsterdam: North Hol- land

3. Barnard, C. J., Brown, C. A. J. 1985. Risk-sensitive foraging in common shrews (Sorex araneus L.). Behav. Ecol. Sociobiol. 16:161-64

4. Barnard, C. J., Brown, C. A. J. 1985. Competition affects risk-sensitivity in foraging shrews. Behav. Ecol. Socio- biol. 16:379-82

5. Battalio, R. C., Kagel, J. H., MacDon- ald, D. N. 1985. Animals' choices over uncertain outcomes: Some initial ex- perimental results. Am. Econ. Review 75:597-613

6. Boulding, K. 1978. Ecodynamics. Bev- erly Hills: Sage

7. Calder, W. A. 1974. Consequences of body size for avian energetics. In Avian Energetics, ed. R. A. Paynter, pp. 86- 144. Nuttal Ornithol. Club Publ. 15. Cambridge, Mass.

8. Caraco, T. 1980. On foraging time allocation in a stochastic environment. Ecology 61:119-28

9. Caraco, T. 1981. Risk-sensitivity and foraging groups. Ecology 62:527-31

10. Caraco, T. 1981. Energy budgets, risk and foraging preferences in dark-eyed juncos. Behav. Ecol. Sociobiol. 8:213- 17

11. Caraco, T. 1982. Aspects of risk- aversion in foraging white-crowned sparrows. Anim. Behav. 30:719-27

12. Caraco, T. 1983. White-crowned spar- rows: Foraging preferences in a risky environment. Behav. Ecol. Sociobiol. 12:63-69

13. Caraco, T. 1986. Foraging games in a random environment. See Ref. 35a. In press

14. Caraco, T., Brown, J. L. 1986. A game between communal breeders: When is food sharing stable? J. Theor. Biol. 118:379-93

15. Caraco, T., Chasin, J. 1984. Foraging preferences: Response to reward skew. Anim. Behav. 32:76-85

16. Caraco, T., Gillespie, R. G. 1986. Risk- sensitivity and foraging mode. Ecology. In press

17. Caraco, T., Lima, S. L. 1985. Foraging juncos: Interaction of reward mean and variability. Anim. Behav. 33:216-24

RISK-SENSITIVE FORAGING 389

18. Caraco, T., Lima, S. L. 1986. Survival, energy budgets, and foraging risk. In Quantitative Analyses of Behavior, Vol. VI:1-21 Foraging, ed. M. L. Com- mons, A. C. Kacelnik, S. J. Shettle- worth. New Jersey; Erlbaum. In press

19. Caraco, T., Martindale, S., Whittam, T. S. 1980. An empirical demonstration of risk-sensitive foraging preferences. Anim. Behav. 28:820-30

20. Caraco, T., Pulliam, H. R. 1984. Sociality and survivorship in animals ex- posed to predation. In A New Ecology, Novel Approaches to Interactive Sys- tems, ed. P. W. Price, C. N. Slobodchi- koff, W. S. Gaud, pp. 279-309. New York: Wiley

21. Clark, W. W., Mangel, M. 1984. Foraging and flocking strategies: In- formation in an uncertain environment. Am. Nat. 123:626-41

22. Clark, C. W., Mangel, M. 1986. The evolutionary advantages of group forag- ing. Theor. Popul. Biol. In press

23. Davison, M. 1969. Preference for mixed-interval vs. fixed-interval sched- ules. J. Exp. Anal. Behav. 12:247-25

24. DeGroot, M. H. 1970. Optimal Statis- tical Decisions. New York: McGraw- Hill

25. Downhower, J. F. 1976. Darwin's finches and the evolution of sexual di- morphism in body size. Nature 263:558-63

26. Gould, S. J., Lewontin, R. C. 1978. The spandrels of San Marco and the Pan- glossian paradigm: A critique of the adaptationist programme. Proc. R. Soc. London 205:581-59

27. Green, R. F. 1980. Bayesian birds: A simple example of Oaten's stochastic model of optimal foraging. Theor. Pop- ul. Biol. 18:244-56

28. Green, R. F. 1984. Stopping rules for optimal foragers. Am. Nat. 122:30-43

29. Herrnstein, R. S. 1964. Aperiodicity as a factor in choice. J. Exp. Anal. Behav. 7:179-82

30. Hirschliefer, J. 1977. Economics from a biological perspective. J. Law Econ. 20:1-53

31. Houston, A., Kacelnik, A., McNamara, J. 1982. Some learning rules for acquir- ing information. In Functional Ontogeny ed. by D. McFarland, pp. 140-91. Lon- don: Pitman

32. Houston, A., McNamara, J. 1982. A sequential approach to risk-taking. Anim. Behav. 30:1260-61

33. Houston, A., McNamara, J. 1986. The choice of two prey types that minimizes the probability of starvation. Behav. Ecol. Sociobiol. 17:135-41

34. Jean, W. H. 1971. The extension of

portfolio analysis to three or more pa- rameters. J. Finance Quant. Anal. 6:505-15

35. Kagel, J. H., Green, L., Caraco, T. 1986. When foragers discount the fu- ture: Constraint or adaptation? Anim. Behav. 34:271-83

35a. Kamil, A. C., Krebs, J. K. 1986. Second Conference on Foraging Be- havior. New York: Garland

36. Keeney, R. L., Raiffa, H. 1976. De- cisions with Multiple Objectives: Prefer- ences and Value Tradeoffs. New York: Wiley

37. Killeen, P. 1968. On the measurement of reinforcement frequency in the study of preferences. J. Exp. Anal. Behav. 11:263-69

38. Knight, F. H. 1921. Risk, Uncertainty, and Profit. New York: Harper and Row

39. Krebs, J. R., Kacelnik, A., Taylor, P. 1978. Test of optimal sampling by foraging great tits. Nature 275:27-31

40. Krebs, J. R., Stephens, D. W., Suther- land, W. J. 1983. Perspectives in op- timal foraging. In Perspectives in Ornithology, ed. A. H. Brush, G. A. Clark, Jr., pp. 165-221. Cambridge: Cambridge Univ. Press

41. Lacey, E. P., Real, L., Antonovics, J., Heckel, D. G. 1983. Variance models in the study of life histories. Am. Nat. 122:114-31

42. Lande, R. 1982. A quantitative genetic theory of life history evolution. Ecology 63:607-15

43. Lima, S. L. 1985. Sampling behavior of starlings in simple patchy environments. Behav. Ecol. Sociobiol. 16:135-42

44. Lima, S. L. 1986. Predation risk and unpredictable feeding conditions: Deter- minants of body mass in birds. Ecology. 67:377-85

45. Luce, R. D., Suppes, P. 1965. Prefer- ence, utility and subjective probability. In Handbook of Mathematical Psycholo- gy, Vol. 3, ed. R. D. Luce, R. R. Bush, E. Galanter, pp. 249-410. New York: Wiley

46. MacArthur, R. 1972. Geographical Ecology. Princeton, NJ: Princeton Univ. Press

47. MacArthur, R. H., Pianka, E. R. 1966. On the optimal use of a patchy environ- ment. Am. Nat. 100:603-09

48. Maynard Smith, J. 1978. Optimization theory in evolution. Ann. Rev. Ecol. Syst. 9:31-56

49. McNamara, J. 1982. Optimal patch use in a stochastic environment. Theor. Pop- ul. Biol. 21:269-88

50. McNamara, J. M., Houston, A. I. 1982. Short-term behaviour and life-time fit- ness. In Functional Ontogeny, ed. D.

390 REAL & CARACO

McFarland, pp. 60-87. London: Pitman 51. McNamara, J., Houston, A. 1986. A

simple model of information use in the exploitation of patchily distributed food. J. Theor. Biol. In press

52. Moore, F. R., Simm, P. A. 1985. Risk- sensitive foraging by a migratory war- bler (Dendroica coronata). Ms.

53. Oster, G. F., Wilson, E. 0. 1978. Caste and Ecology in the Social Insects. Princeton, NJ: Princeton Univ. Press

54. Ott, J. R., Real, L., Silverfine, E. 1985. The effect of nectar variance on bumble- bee patterns of movement and potential gene flow. Oikos. 45:333-40

55. Pratt, J. W. 1964. Risk aversion in the small and in the large. Econometrica 32: 122-36

56. Pubols, B. H. 1962. Constant versus variable delay of reinforcement. J. Comp. Physiol. Psychol. 55:52-56

57. Pulliam, H. R., Millikan, G. C. 1982. Social organization in the non- reproductive season. In Avian Biology, Vol. VI, ed. D. S. Farmer, J. R. King, pp. 169-97. New York: Academic

58. Real, L. A. 1980. Fitness, uncertainty, and the role of diversification in evolu- tion and behavior. Am. Nat. 115:623-38

59. Real, L. A. 1980. On certainty and the law of diminishing returns in evolution and behavior. In Limits to Action: The Allocation of Individual Behavior, ed. J. E. R. Staddon, pp. 37-64. New York: Academic

60. Real, L. A. 1981. Uncertainty and polli- nator-plant interactions: The foraging behavior of bees and wasps on artificial flowers. Ecology 62:20-26

61. Real, L. 1983. Microbehavior and mac- rostructure in pollinator-plant in- teractions. In Pollination Biology, ed. L. Real, pp. 287-304. New York: Aca- demic

62. Real, L. 1985. Habitat selection and adaptive foraging theory. Ms.

63. Real, L. 1985. The logic of risk- sensitive foraging revisited. Ms.

64. Real, L., Ott, J., Silverfine, E. 1982. On the trade-off between the mean and the variance in foraging: An ex- perimental analysis with bumblebees. Ecology 63:1617-23

65. Ross, S. M. 1983. Stochastic Processes. New York: Wiley

66. Staddon, J. E. R. 1986. Optimal forag- ing in field and Skinner box. See Ref. 35a. In press

67. Stephens, D. W. 1981. The logic of risk-sensitive foraging preferences. Anim. Behav. 29:628-29

68. Stephens, D. W. 1982. Stochasticity in foraging theory: Risk and information. PhD thesis. Oxford Univ., Oxford, En- gland

69. Stephens, D. W., Charnov, E. L. 1982. Optimal foraging: Some simple stochas- tic models. Behav. Ecol. Sociobiol. 10:251-63

70. Stephens, D. W., Paton, S. R. 1985. How constant is the constant of risk- aversion? Ms.

71. Waddington, K. D., Allen, T., Hein- rich, B. 1981. Floral preferences of bumblebees (Bombus edwardsii) in rela- tion to intermittent versus continuous re- wards. Anim. Behav. 29:779-84

72. Waser, N. M. 1983. The adaptive nature of floral traits: Ideas and evidence. In Pollination Biology, ed. L. Real, pp. 242-85. New York: Academic

73. Wells, H., Wells, P. H. 1983. Honey bee foraging ecology: Optimal diet, minimal uncertainty or individual con- stancy? J. Anim. Ecol. 52:829-36

74. Wunderle, J. M., O'Brien, T. G. 1986. Risk aversion in hand-reared Banana- quits. Behav. Ecol. Sociobiol. In press