David W. Stephens

The logic of risk-sensitive foraging preferences

The Logic of Risk-Sensitive Foraging Preferences It has recently been shown that the foraging preferences of animals are sensitive to variance in the probability distribution of a food reward (Caraco et al. 1980). This is an important result for optimal foraging models, because they often assume that the expected value of the food reward distribution is the quantity that should be maximized and that variance may be safely ignored (see Pyke et al. 1977 for a review). Caraco (1980) has suggested that some aspects of risk sensitivity may be explained by minimization of the probability of starvation, where starvation is defined as falling short of some threshold food requirement at the end of the day. In this note I formalize this alternative optimality hypothesis and show analytically that it accounts for the existence and some aspects of the nature of the risk sensitivity reported by Caraco et al. (1980). Caraco et al. (1980) offered yellow-eyed juncos (Junco phaenotus) a choice between two feeding stations. One station always provided a constant number of seeds, while the alternative station provided a variable number determined by a probability distribution. The number of seeds at the constant station was equal to the expected number provided at the variable station. Caraco et al. (1980) demonstrated that juncos preferred the constant station (or were risk-averse) when their expected daily energy intake exceeded their expected daily energy expenditure. More surprisingly, they also demonstrated that juncos preferred the variable station (or were riskprone) when their expected daily energy expenditure exceeded their expected daily intake. Juncos show preferences which are consistent with a simple rule: be riskaverse if your expected 24-h energy budget is positive, be risk-prone if your expected 24-h energy budget is negative. This rule has been suggested, on intuitive grounds, by Caraco (1980). In this note I show that the expected energy budget rule follows directly from minimization of the probability of starvation. Consider a small bird foraging in winter. Imagine that at least R units of food reserve are required to survive the night, Let So be the random variable that describes the birds food reserve when it becomes too dark to forage. The probability of starvation is equivalent to P(So < R). (This assumes that starvation only occurs at night, which may not be true; however the results are not affected as long as food reserves are unlikely to become dangerously low during the day.) Assume that the days foraging is divided into n intervals. At the beginning of the ith interval the forager makes some decision about how to forage and receives X~ units of food. The X~s are random variables and are assumed to be independently distributed with E(XO = la~ and Var(X0 = ~ . Assume that exactly r units of food are expended during each interval (this assumes that r is independent of the foraging decision, which cannot be strictly true). Let Sn be the birds internal food supply when there are n decisions left before nightfall. We can express the reserves at the end of the day as

Optimal foraging: Some simple stochastic models.

SummarySome simple stochastic models of optimal foraging are considered. Firstly, mathematical renewal theory is used to make a general model of the combined processes of search, encounter, capture and handling. In the case where patches or prey items are encountered according to a Poisson process the limiting probability distribution of energy gain is found. This distribution is found to be normal and its mean and variance are specified. This result supports the use of Hollings disc equation to specify the rate of energy intake in foraging models. Secondly, a model based on minimization of the probability of death due to an energetic shortfall is presented. The model gives a graphical solution to the problem of optimal choices when mean and variance are related. Thirdly, a worked example using these results is presented. This example suggests that there may be natural relationships between mean and variance which make solutions to the problems of ‘energy maximization’ and ‘minimization of the probability of starvation’ similar. Finally, current trends in stochastic modeling of foraging behavior are critically discussed.

On economically tracking a variable environment

Abstract This paper presents a simple model of how an animal should best use experience to track a changing environment. The model supposes that the environment switches between good and bad states according to a first-order Markov chain. The optimal sampling behavior is characterized in terms of the stability of runs (the probability that the environment will stay in the same state from one time to the next) and the relative costs of two kinds of errors: sampling and overrun errors. This model suggests further experimental and theoretical problems.

Testing models of non-kin cooperation: mutualism and the Prisoner's Dilemma

Abstract Since 1981, the iterated Prisoners Dilemma has dominated studies of non-kin cooperation. Alternative models have received relatively little attention. The simplest alternative is mutualism, in which mutual cooperation always pays best. The behaviour of three pairs of blue jays, Cyanocitta cristata , was tested in precisely controlled iterated mutualism and Prisoners Dilemma games. Although the jays readily cooperated in the mutualism game, cooperation neither developed nor persisted in a Prisoners Dilemma. No empirical justification was found for the status of the iterated Prisoners Dilemma as the basic paradigm of non-kin cooperation.

On the evolution of host selection in solitary parasitoids

Host selection by solitary parasitoids is modeled as a problem in life-historical evolution. Our basic approach is to assume fitness maximization in the face of trade-offs. We consider two kinds of trade-offs: (1) patchy versus fine-grained host distribution, and (2) an egg-production rate that is either negatively related to adult survival or simply a fixed constant. These lead to four general cases, and we derive the range of hosts attacked under each. The resulting decision rules are similar to several decision rules from classical foraging theory.

Decision ecology: Foraging and the ecology of animal decision making

In this article, I review the approach taken by behavioral ecologists to the study of animal foraging behavior and explore connections with general analyses of decision making. I use the example of patch exploitation decisions in this article in order to develop several key points about the properties of naturally occurring foraging decisions. First, I argue that experimental preparations based on binary, mutually exclusive choice are not good models of foraging decisions. Instead, foraging choices have a sequential foreground-background structure, in which one option is in the background of all other options. Second, behavioral ecologists view foraging as a hierarchy of decisions that range from habitat selection to food choice. Finally, data suggest that foraging animals are sensitive to several important trade-offs. These trade-offs include the effects of competitors and group mates, as well as the problem of predator avoidance.

Components of change in the evolution of learning and unlearned preference

Several phenomena in animal learning seem to call for evolutionary explanations, such as patterns of what animals learn and do not learn. While several models consider how evolution should influence learning, we have very little data testing these models. Theorists agree that environmental change is a central factor in the evolution of learning. We describe a mathematical model and an experiment, testing two components of change: reliability of experience and predictability of the best action. Using replicate populations of Drosophila we varied statistical patterns of change across 30 generations. Our results provide the first experimental demonstration that some types of environmental change favour learning while others select against it, giving the first experimental support for a more nuanced interpretation of the selective factors influencing the evolution of learning.

Impulsiveness without discounting: the ecological rationality hypothesis

Observed animal impulsiveness challenges ideas from foraging theory about the fitness value of food rewards, and may play a role in important behavioural phenomena such as cooperation and addiction. Behavioural ecologists usually invoke temporal discounting to explain the evolution of animal impulsiveness. According to the discounting hypothesis, delay reduces the fitness value of the delayed food. We develop an alternative model for the evolution of impulsiveness that does not require discounting. We show that impulsive or short–sighted rules can maximize long–term rates of food intake. The advantages of impulsive rules come from two sources. First, naturally occurring choices have a foreground–background structure that reduces the long–term cost of impulsiveness. Second, impulsive rules have a discrimination advantage because they tend to compare smaller quantities. Discounting contributes little to this result. Although we find that impulsive rules are optimal in a simple foreground–background choice situation in the absence of discounting, in contrast we do not find comparable impulsiveness in binary choice situations even when there is strong discounting.

How important are partial preferences

each of five different f (N) . The f ( N ) were designed such that while Nopt remained constant at five the dW/dN at N<Nopt was changed in relation to dW/dN at N> Nopt. TWO extreme examples of the f ( N ) used are presented in Fig. 1. The simulations confirm that N~ can never be smaller than Nopt, a result derived analytically by Pulliam & Caraco (1984). Our simulations indicate, however, that Siblys conclusion that Nopt < N~ is incorrect since in some cases N~= Nopt (Fig. 1). This occurs when the fitness of joining a group of optimal size is less than that of remaining alone (see Fig. 1). The generality of both Clark & Mangels (1984) and Pulliam & Caracos (1984) conclusion that individuals will more often be in groups larger than Nop t because Nop t is unstable resides in the generality o f f (N) for which Nopt < Ns. At present, there is no reason to believe that this type of f (N) is more common than any other. In fact, when competition is important, we would expect f ( N ) to be of the shape that stabilizes Nopt. This is because competition could add costs to joining supra-optimal group sizes. These costs could be the result of harassment, aggression or even exclusion from the group. Spotted hyaenas for instance, establish dominance hierarchies that exclude lower-ranking individuals from having access to smaller carcasses (Tilson & Hamilton 1984). The costs of joining groups of supra-optimal size for lower-ranking individuals may be greater than the costs of joining groups of sub-optimal size. When lions hunt Thompsons gazelle in the Serengeti, Nopt (two lions) provides the maximum attainable harvesting rate (Caraco & Wolf 1975). Groups larger than the optimum could not provide the minimum daily food requirement of adult lions (Caraco & Wolf 1975). It is clear that lions joining a group in excess of Nopt would reduce their fitness considerably such that Nopt would be stable. Contrary to Siblys conclusions therefore, one should not assume that Ns and Nobs are greater than Nop t before investigating the shape of the fitness functions. We wish to acknowledge the members of the McGill Graduate Seminar on Spatial Dispersion. Graham Bell, Don Kramer, Louis Lefebvre, Rob Peters, Marc Hauser and an anonymous referee provided constructive criticisms of an earlier version of the manuscript. We thank Sylvie Lanct6t and Roger Giraldeau for providing the computer. Financial support came from F,C,A.C. and N.S.E.R.C. scholarships to L.-A.G. and D.G. respectively. Luc-ALAIN GIRALDEAU* DARREN GILLIS

How constant is the constant of risk-aversion?

We describe an experiment designed to distinguish between two models of risk-sensitive feeding behaviour: the variance discounting model and the z-score model. The variance discounting model assumes that mean reward levels do not affect preferences over reward variability, but the z-score model assumes that mean reward levels do affect preferences over variability. We presented two choices to feeding rufous hummingibrds, Selaphoruous rufus. One alternative had a higher mean and a higher variance than the other. After measuring preference, we increased the mean of both alternatives by adding the same amount to all possible outcomes. The variance discounting model predicts that such a general shift should not change preferences, but the z-score model predicts that preferences will change. Our results support the z-score model. The variance discounting models assumption of constant risk-aversion fails.