Pigeons Game Theory

20.12.2020by admin

Interest by ecologists in foraging grew rapidly after the mid-1960s. Scientists in areas such as agricultural and range research already had long-standing interests in the subject (see chap. 6 in this volume). Entomologists, wildlife biologists, naturalists, and others had long been describing animal diets. So what was new? What generated the excitement and interest among ecologists?

We believe that the answer to this question is symbolized by a paper published by the economist Gordon Tullock in 1971, entitled 'The coal tit as a careful shopper.' Tullock had read the studies of Gibb (1966) on foraging by small woodland birds on insects, and he suggested in his paper that one could apply microeconomic principles to understand what they were doing. (We do not mean to suggest that Tullock originated this approach, merely that his paper clearly expressed what many ecologists were thinking.) The idea of using an established concept set to investigate the foraging process from first principles animated many ecologists. This motivation fused with developing notions about natural selection (Williams 1966) and the importance of energy in ecological systems to give birth to 'optimal foraging theory' (OFT). The new idea of optimal foraging theory was that feeding strategies evolved by natural selection, and it was a natural next step to use the techniques of optimization models.

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Although the terminology differs somewhat among authors, the elements of a foraging model have remained the same since the publication of Stephens and Krebs's book. At their core, models based on optimal foraging theory possess (1) an objective function or goal (e.g., energy maximization or starvation minimization), (2) a set of choice variables or options under the control of the organism, and (3) constraints on the set of choices available to the organism (set by limitations based on genetics, physiology, neurology, morphology, and the laws of chemistry and physics). In short, foraging models generally take the form, 'Choose the option that maximizes the objective, subject to constraints.' A specific case may be matched with a detailed model (e.g., Beauchamp et al. 1992), or a model may conceptualize general principles to investigate the logic underlying foraging decisions, such as whether an encountered item should be eaten or passed over in favor of searching for a better item.

We now regard the rubric 'optimal foraging theory,' used until the mid-1980s, as unfortunate. Although optimality models were important, they were not the only component of foraging theory, and the term emphasized the wrong aspects of the problem. 'Optimality' became a major focus and entangled those interested in the science offoraging in debates on philosophical perspectives and even political stances, which, needless to say, did more to obscure than to illuminate the scientific questions. A few key publications will enable the reader to appreciate this history and the intensity of debate. Stephens and Krebs (1986) reviewed the issues up to 1986 (seePykeet al. 1977; Kamil and Sargent 1981; and Krebs et al. 1983 for earlier reviews). Perry and Pianka (1997) provided a more recent review, and showed that while the titles of published papers dropped the words 'optimal' and 'theory' after the mid-1980s, foraging remained an active area of research. Sensing opprobrium from their colleagues, scientists evidently began to shy away from identifying with optimal foraging theory. If the reader doubts that this was a real factor, he or she should read the article by Pierce and Ollason (1987) entitled 'Eight reasons why optimal foraging theory is a complete waste of time.' In a more classic (and subtle) vein, Gould and Lewontin (1979) criticized the general idea of optimality in their famous paper entitled 'The spandrels of San Marco and the Panglossian paradigm: A critique of the adaptationist programme' (later lampooned by Queller [1995] in a piece entitled 'The spaniels of St. Marx'). Many other publications have addressed these and related themes.

A persistent source of confusion has been just what 'optimality' refers to. Critics assert that it is unreasonable to view organisms as 'optimal,' using biological arguments such as the claim that natural selection is a coarse mechanism that rarely has enough time to perfect traits, or that important features of organisms may originate as by-products of selection for other traits. These arguments graded into ideological stances, such as claims that use of 'optimality' promotes a worldview that justifies profound socioeconomic inequalities. It is difficult to disentangle useful views in this literature from overheated rhetoric, a problem exacerbated by careless terminology and glib applications on both sides. Our view is that most ofthis debate misses the point that 'optimality' should not be taken to describe the organisms or systems investigated. 'Optimality' is properly viewed as an investigative technique that makes use of an established set of mathematical procedures. Foraging research uses this and many other experimental, observational, and modeling techniques.

Nor does optimality reasoning require that animals perform advanced mathematics. As an analogue, a physicist can use optimality models to analyze the trajectories that athletes use to catch a pass or throw to a target. However, no one supposes that any athlete is performing calculus as he runs down a well-hit ball (see section 1.10 below).

The word 'theory' was also a stumbling block for many ecologists, who regarded it as a sterile pursuit with little relevance to the rough-and-tumble reality of the field. Early foraging models were very simple, and their explanatory power in field situations may have been oversold (see, e.g., Schluter 1981). Ydenberg (chap. 8 in this volume), for example, makes clear the limitations of the basic central place foraging model put forward in 1979. But, informed by solid field studies (e.g., Brooke 1981), researchers identified the holes in the model and developed theoretical constructs to address them (e.g., Houston 1987). Errors in the formulation ofthe basic model were soon corrected (Lessells and Stephens 1983; Houston and McNamara 1985). This historical perspective shows how misrepresentative are oft-repeated claims such as, 'Empirical studies of animal foraging developed more slowly than theory' (Perry and Pianka 1997). As in most other branches of scientific inquiry, theory and empirical studies proved, in practice, to be synergistic partners. Their partnership is flourishing in foraging research, and theory and empiricism in both laboratory and field are important parts of this volume.

If the basics of foraging models have remained unchanged since the publication of Stephens and Krebs's book (1986), the range and sophistication of objective functions, choice variables, and constraint sets has expanded. Mathematics has spawned new tools for formulating and solving foraging models. And advances in computing have permitted ever more computationally intensive models. The emphasis of modeling has expanded from analytic solutions to include numerical and simulation techniques that require mind-boggling numbers of computations. The last two decades have seen a pleasing lockstep among empirical, modeling, mathematical, and computational advances.

New concepts have also emerged. Some of the biggest conceptual advances in foraging theory have come from the realization that foragers must balance food and safety (see chaps. 9,12, and 13 in this volume), an idea that ecologists had just begun to consider when Stephens and Krebs published their book in 1986. Box 1.1 outlines the history of this important idea.

BOX 1.1 Prehistory: Before Foraging Met Danger

Peter A. Bednekoff

The theory of foraging under predation danger took time to formulate. Broadly speaking, students of foraging hardly ever addressed the effects of predation during the 1970s, but they gave increasing attention to predation in the 1980s, and predation enjoyed unflagging interest through the 1990s. From the start, behavioral ecologists took the danger of predation seriously; but they treated foraging and danger separately. In the first edition of Behavioral Ecology (Krebs and Davies 1978), the chapter on foraging (Krebs 1978) is immediately followed by one dealing with predators and prey (Bertram 1978), with another chapter on antipredator defense strategies not far behind (Harvey and Greenwood 1978). The thinking seems to have been that these phenomena operated on different scales, such that danger might determine where and when animals fed, but energy maximization ruled how they fed (Charnov and Orians 1973; Charnov 1976a, 1976b). This was a useful scientific strategy: it was important to test whether energetic gain affected foraging decisions before testing whether energetic gain and danger jointly affected foraging decisions. We probably can separate foraging from some kinds of activities. For example, male manakins may spend about 80% of their time at their display courts on leks (Thery 1992). Male manakins probably need to secure food as rapidly as possible when off the lek and to display as much as possible when on the lek. Therefore, foraging and displaying are separate activities. Survival, however, is a full-time job. Animals cannot afford to switch off their antipredator behavior. Because

(Box 1.1 continued)

trade-offs between danger and foraging gain can occur at all times and on all scales, the effects of danger can enrich all types of foraging problems.

A more subtle difficulty may have delayed the integration of foraging and danger: the two models that dominated early tests of foraging theory, the diet and patch models, do not readily suggest ways to integrate danger (see Lima 1988b; Gilliam 1990; Houston and McNamara 1999 for later treatments). Several graphical models dealt with predation and other aspects of foraging (Rosenzweig 1974; Covich 1976) and one chapter juxtaposed diet choice and antipredator vigilance models, both important contributions made by Pulliam (1976). Although the pieces seem to have been available, integration did not happen quickly. Even the early experimental tests treated danger as a distraction rather than a matter of life and death (Milin-ski and Heller 1978; Sih 1980). These studies would have reached similar conclusions if they had considered competitors rather than predators.

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The first mature theory of foraging and predation concentrated on habitat choice and did not consider the details of foraging within habitats (Gilliam 1982). This theory assumed that animals grew toward a set size with no time limit. It showed that animals should always choose the habitat that offers the highest ratio of growth rate, g, to mortality rate, M. In order to avoid potentially dividing by zero, Gilliam expressed his solution in terms of minimizing the mortality per unit of growth, so we call this important result the mu-over-g rule. Departures from the basic assumptions lead to modifications of the M/g rule. This rule is a special case of a more general minimization of

v where r is the intrinsic rate of growth for the population, b is current reproduction, and Vis expected future reproduction (Gilliam 1982; Werner and Gilliam 1984). The familiar special case applies to juveniles in a stable population: juveniles are not yet reproducing, so b is zero, and the population is stable, so its growth rate, r, is also zero (Gilliam 1982; Werner and Gilliam 1984). Gilliam never published this work from his dissertation, but Stephens and Krebs (1986) cogently summarized the special case. Although the M/g rule is incomplete for various situations (Ludwig and Rowe 1990; Houston et al. 1993), it is surprisingly robust (see Werner and Anholt 1993). Modified versions may be solutions for problems that do not superficially

(Box 1.1 continued)

resemble the one analyzed by Gilliam (Houston et al. 1993), and Gilliam's M/g criterion may reappear from analysis of specific problems (e.g., Clark andDukas 1994; see also Lima 1998, 221—222, and chap. 9 in this volume).

In hindsight, we can see that various studies in the early 1980s pointed to the pervasive effects of danger on foraging (e.g., Mittelbach 1981; Dill and Fraser 1984; Kotler 1984), but these effects were not immediately integrated into the body of literature on foraging. Besides Gilliam's studies, Stephens and Krebs mentioned only one other study of foraging under predation danger, which found that black-capped chickadees sacrifice their rate of energetic gain in order to reduce the amount of time spent exposed at a feeder (Lima 1985a). This influential book seems to have just preceded a flood of results. In the mid-1980s, students of foraging found that danger influences many details of foraging and other decisions made by animals (Lima and Dill 1990). The general framework has continued to be productive and currently shows no sign of slowing its expansion (see Lima 1998).

A second profoundly important concept is 'state dependence,' the idea that the tactical choices of a forager might depend on state variables, such as hunger or fat reserves. This concept developed in ecology in the late 1970s and 1980s and is described in sections 1.8 and 1.9 below. Stephens and Krebs (1986) used the idea of state dependence in two chapters and anticipated the still-growing impact of this concept.

A third important conceptual advance not considered at all in Stephens and Krebs (1986) lies in social foraging games and the consequences of foraging as a group. Foraging games between predator and prey represent an extension of both game theory and foraging theory. Here the objective function of the prey takes into account its own behavior as well as that of the predator, and the predator's objective function considers the consequences of its behavior and that of its prey. We anticipate that these models will find application in a variety of basic and applied settings.

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The word 'theory' was also a stumbling block for many ecologists, who regarded it as a sterile pursuit with little relevance to the rough-and-tumble reality of the field. Early foraging models were very simple, and their explanatory power in field situations may have been oversold (see, e.g., Schluter 1981). Ydenberg (chap. 8 in this volume), for example, makes clear the limitations of the basic central place foraging model put forward in 1979. But, informed by solid field studies (e.g., Brooke 1981), researchers identified the holes in the model and developed theoretical constructs to address them (e.g., Houston 1987). Errors in the formulation ofthe basic model were soon corrected (Lessells and Stephens 1983; Houston and McNamara 1985). /game-pigeon-battleship-cheats.html. This historical perspective shows how misrepresentative are oft-repeated claims such as, 'Empirical studies of animal foraging developed more slowly than theory' (Perry and Pianka 1997). As in most other branches of scientific inquiry, theory and empirical studies proved, in practice, to be synergistic partners. Their partnership is flourishing in foraging research, and theory and empiricism in both laboratory and field are important parts of this volume.

BOX 1.1 Prehistory: Before Foraging Met Danger

Peter A. Bednekoff

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(Box 1.1 continued)

trade-offs between danger and foraging gain can occur at all times and on all scales, the effects of danger can enrich all types of foraging problems.

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(Box 1.1 continued)

A second profoundly important concept is 'state dependence,' the idea that the tactical choices of a forager might depend on state variables, such as hunger or fat reserves. /trapped-escape-game-pigeon-forge-reviews.html. This concept developed in ecology in the late 1970s and 1980s and is described in sections 1.8 and 1.9 below. Stephens and Krebs (1986) used the idea of state dependence in two chapters and anticipated the still-growing impact of this concept.

Continue reading here: Attack and Exploitation Models

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