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Puffibc ECOLOGY BACK IBTO OPTIMAL FOBAGIBC THEORY By PUffIBC ECOLOGY BACK IBTO OPTIMAL FOBAGIBC THEORY by Richard F. Green Technical Report 89-11 Department of Mathematics and Statistics University of Minnesota Duluth, Minnesota 55812 October 1989 1 PUTTING ECOLOGY BACK INTO OPTIMAL FORAGING THEORY Abstract: Optimal foraging theory is one of the most popular areas in modem evolutionary biology. In this paper, I outline some of the main ideas of optimal foraging theory and describe two basic foraging models in detail. I argue that one of these mod- els is less interesting than it should be because it does not include realistic assumptions about the environment or about the behavior of the forager. Then I describe a model that does make explicit behavioral and ecological assumptions. I argue for the use of particular, quantitative models and describe an experimental test of one such model. I discuss some criticisms of optimal foraging theory and will try to predict what the future holds. In particular, I show how optimal foraging theory might help to bridge the gap between individual behavior and population biology. INTRODUCTION For many years ecologists have been interested in the food habits of animals. Their question is, How do animals look for and select food? Optimal foraging asks the ques- tion: How should animals look for and select food? The first papers to ask the question 2 in this way were published in 1966.1• These and other early papers were mainly about optimal diet. An early review of the subject was provided by Schoener,3 and a well- known account was given by MacArthur,4 who pointed out that as the overall density of prey decreases, predators should be less selective in their choice of prey. A consequence of this is that predators which compete with each other for prey should show greater overlap in their use of prey and should compete more strongly as prey density decreases. A great impetus to the study of foraging theory was provided by Chamov,5 - 8 who provided an explicitly mathematical theory and used the striking phrase "optimal forag- ing" to refer to what Schoener3 referred to, perhaps more accurately, as "feeding strate- gies." A broad view of foraging theory was laid out in an influential review by Pyke, Pul- liam and Chamov,9 and most experimental work on foraging behavior has been based on the ideas contained in this review. Recently, three volumes have been published on foraging. Two of these are the proceedings of symposia, one10 held at Harvard Univer- sity in 1983 and the other11 held at Brown University in 1984. The third volume is a monograph by Stephens and Krebs.12 I think that a reader interested in learning about optimal foraging theory would do best to begin with the book by Stephens and Krebs, which gives an excellent description of foraging models and the strategy of modeling and 2 its criticisms, and with Schoener's13 informative, personal history of the early years of optimal foraging theory. Early work on foraging theory was basically ecological, and Schoener's historical sketch reflects this point of view. More recently, foraging theory has been treated as al- most exclusively behavioral, and ecological aspects have been neglected. This modem view is seen in the monograph by Stephens and Krebs, 12 who seek to understand for- aging as a behavioral adaptation, with little attention being paid to its ecological con- text and none to its ecological consequences. In the next section, I will begin by follow- ing Stephens and Krebs in their treatment of the two basic foraging models, but I will present an alternative to one of these models and try to show that foraging theory can be ma.de more interesting by paying attention to the ecological context of the behavior. The model that I present permits a quantitative treatment of foraging behavior, which is more useful than the older, qualitative treatment because it permits the investigation of some ecological consequences of foraging behavior. OPTIMAL FORAGING MODELS In their review of optimal foraging theory, Pyke, Pulliam and Charnov9 described four problems that a forager must face: (1) What prey to take, and if prey are distributed in patches, (2) What patch types to search; (3) When to leave a patch, and ( 4) How to move between patches. Foraging theory may be divided into parts according to which of these questions it addresses. Most work on foraging theory has been done on problems (1) and (3). The least amount of work has been done on (4), perhaps because it is diffi- cult to give a realistic, yet tractable, formulation of the problem. Stephens and Krebs12 have lumped the problems involving choice [(1) and (2)] into one, which they refer to as the "prey" problem. They contrast this with the problem involving residence time, which they refer to as the "patch" problem. I will describe the basic prey and patch models as Stephens and Krebs describe them, and then I will present an alternative to the basic patch model. For all the models, an optimal forager is one that maximizes its long-term average rate of energy uptake or its long-term average rate of finding prey while foraging. The prey problem: What types of prey should a forager accept? The prey problem is to decide which of several types of prey (indexed by the sub- script i) the forager should take. The basic prey model assumes: ( 1) Foragers cannot search for prey and handle prey at the same time. (2) Individuals of each prey type are encountered one at a time, at random, indepen- dently of other types, according to a Poisson process. 3 (3) The net energy gain, ei, the handling time, hi, and the encounter rate, Ai, are fixed for each type i, and do not depend on the choices that the forager makes. ( 4) The forager is able to discriminate among prey types perfectly and instantaneously, without cost. ( 5) The forager has complete information about the parameters of the model, but does not use information acquired while foraging. The basic prey model predicts that: (1) A forager will choose the most profitable prey types, where profitability is measured by the ratio e, /h,. (2) Foragers will be more selective when encounter rates are higher (that is, prey are more abundant). (3) Whether a particular type of prey will be ta.ken depends only on the abundance of more profitable prey; it does not depend on the abundance of the prey in question, or the abundance of less profitable prey. (4) Prey of a given type should be accepted always or never. There should be no "par- tial preferences," with the forager taking some, but not all, items encountered of a particular prey type. 12 14 Stephens and Krebs (following Krebs, Stephens and Sutherland ) cite many stud- ies that test some or all of these predictions of the prey model. Most of these studies are classified as giving at least qualitative support to the theory. The most frequently tested prediction is that foragers prefer the most profitable prey. It is not surprising that this prediction is usually confirmed. The patch model: How long should a forager spend in a patch? In the patch problem prey are distributed in patches and a forager must decide when to leave one patch and go on to another. The basic patch model assumes: (1) Searching for patches and searching for prey within patches are mutually exclusive activities. (2) Patches are encountered at random according to a Poisson process with fixed rate A, for each patch type i. (3) The time chosen to leave a patch is independent of the forager's experience within that patch. 4 (4) For each patch type, i, the net expected energy gain is a fixed, continuous func- tion, 9i(t), of the time that a forager remains in the patch. For each patch type, this function takes value O at time 0, increases initially, and is eventually negatively accelerating. (5) A forager is assumed to have complete information about the encounter rates Ai and the gain functions 9i(t), but no additional information about patch quality is obtained while the patch is being searched. The optimal foraging strategy for the basic patch model is given by Charnov's8 "marginal value theorem," which says that the optimal time to remain in patch type i is the time ti when the derivative (the "marginal value") of the gain function gHti) = C, where C is the highest possible long-term average rate of energy gain. Charnov's theo- rem is illustrated, for one patch type, in Fig. la. (Put Fig. 1 about here.) The basic patch model predicts: (1) Foragers should stay longer in a patch of a given type when the environment is poorer overall, or if travel time between patches is longer. (2) A forager should search all patches encountered until the net rate of energy gain in the patches is reduced to the same level. (3) A forager should take more resources from better patches. Again, a large number of studies12,H tend to show qualitative agreement with these predictions. By far the most frequently tested prediction is the first-that foragers should remain longer in patches when conditions are worse. There is a difficulty with Charnov's original marginal value theorem: it assumes that the gain function for a particular patch type is a continuous, deterministic func- tion of time in a patch. For a real predator capturing individual prey, the gain function would be discontinuous and stochastic. In one patch, the gain function might look like that shown in Fig. lb, where the slope is almost always zero, and the marginal value theorem does not apply.
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