The second chapter of Stephens and Krebs (1986) develops the foundational models offoraging, the so-called "diet" and "patch" models. The treatment is clear and rigorous, and the beginning student is encouraged to use their chapter as an excellent starting point. In addition to the classic review articles listed above, one can find recent reviews ofthe published tests ofthese models in Sih and Christensen (2001; 134 published studies of the diet model) and Nonacs (2001; 26 studies of the patch model).

The significance ofthese two models lies in the types ofdecisions analyzed. The terms "diet" and "patch" are misnomers in the sense that the decisions are more general than choices about food items or patch residence time. Stephens and Krebs (1986) termed these models the "attack" and "exploitation" models to underscore this point, but these terms have never caught on.

The diet model analyzes the decision to attack or not to attack. The items attacked are types ofprey items, and the forager decides whether to spend the necessary time "handling" and eating an item or to pass it over to search for something else. The model identifies the rules for attack that maximize the long-term rate of energy gain. Specifically, the model predicts that foragers should ignore low-profitability prey types when more profitable items are sufficiently common, because using the time that would be spent handling low-profitability items to search for more profitable items gives a higher rate of energy gain. The diet model introduced the principle of lost opportunity to ecologists, who have since used the concept in many other settings (e.g., "optimal escape"; Ydenberg and Dill 1986). The diet model considers energy gain, but the same rules apply in non-foraging situations of choice among items that vary in value and involvement time.

The patch model asks how much time a forager should invest in exploiting a resource that offers diminishing returns before moving on to find and exploit the next such resource. The "patches" are localized concentrations of prey between which the predator must travel, and the rule that maximizes the overall rate ofenergy gain is to depart when more can be obtained by moving on. In this sense, the patch model also considers lost opportunity, but its real value was to introduce the notion of diminishing returns. If the capture rate in a patch falls as the predator exploits it—a general property of patches— then the maximum "long-term" rate of gain (i.e., over many patch visits) is that patch residence time at which the "marginal value" (i.e., the intake rate expected over the next instant) is equal to the long-term rate of gain using that patch residence rule. Because diminishing returns are ubiquitous, this so-called "marginal value theorem" (Charnov 1976b) can be used in many situations. For example, we can think of eiders as "loading" oxygen into their tissues prior to a dive. The rate at which they can do so depends on the difference in partial pressure between the tissues and the atmosphere, and hence the process must involve diminishing returns. How much oxygen they should load depends on the situation, and the "patch" model gives us a way to analyze the problem (Box 1.2).

BOX 1.2 Diving and Foraging by the Common Eider

Common eiders and other diving birds capture prey underwater during "breath-hold" diving. During pauses on the surface between dives, they "dump" the carbon dioxide that has accumulated in their tissues and "load" oxygen in preparation for the next dive. (Heat loss may also be a significant factor in some systems, but is not considered here.) Figure 1.2.1 schematically portrays a complete dive cycle. This graph shows a slightly offbeat version of the marginal value theorem.

Figure 1.2.1. The relationship between dive time (composed of round-trip travel time to the bottom plus feeding time on the bottom) and the total amount of time required for a dive plus subsequent full recovery (pause time). The relationship accelerates because increasingly lengthy pauses are required to recover after longer dives. Small prey are consumed at rate c during the feeding portion of the dive. The problem is to adjust feeding time (ft] - ft) to maximize the rate of intake over the dive as a whole. The tangent construction in the figure shows the solution. The reader can check the central prediction of this model by redrawing the graph to portray dives in deeperwater (i.e., make travel time longer). The repositioned tangentwill show that dives should increase in length if energy intake is to be maximized.

Figure 1.2.1. The relationship between dive time (composed of round-trip travel time to the bottom plus feeding time on the bottom) and the total amount of time required for a dive plus subsequent full recovery (pause time). The relationship accelerates because increasingly lengthy pauses are required to recover after longer dives. Small prey are consumed at rate c during the feeding portion of the dive. The problem is to adjust feeding time (ft] - ft) to maximize the rate of intake over the dive as a whole. The tangent construction in the figure shows the solution. The reader can check the central prediction of this model by redrawing the graph to portray dives in deeperwater (i.e., make travel time longer). The repositioned tangentwill show that dives should increase in length if energy intake is to be maximized.

A dive consists of round-trip travel time to the bottom (it) and time on the bottom spent finding and consuming small mussels (feeding time). Travel time is a constraint, and it is longer in deeper water or, as in the eider example in the prologue, faster currents. Dive time (id) consists of travel time plus feeding time. Dive-cycle time consists of dive time plus the pause time on the surface between dives (is). How should an eider organize its dives to maximize the feeding rate?

(Box 1.2 continued)

Fs(ts) = O2 intake from a pause of length ts, Fd(td) = O2 depletion from a dive of length td, Y(td) = energy intake (number of mussels times energy per mussel) from a dive oflength td.

The average rate of food intake is thus

which is maximized subject to the condition that oxygen intake must equal oxygen usage, so

To solve this problem graphically, first solve equation (1.2.2) for ts as a function of td:

Here $ (td) represents the pause time required to recover oxygen reserves after a dive of length td. One would expect that $'(td) would increase with td. This is the source of the diminishing returns in this model—increasingly longer times are required to recover after longer dives. An attractive feature of this model is that it requires an estimate of $(td), which can be obtained from observational data, rather than the separate functions Fs and Fd.

Suppose that Y(td) = 0 if td < tt (no food can be consumed if the dive is not long enough to travel to the bottom and back), and that if td > tt, then

meaning that energy is ingested at the rate c during the portion ofthe dive spent feeding on the bottom. The optimization problem is to adjust the length of the dive (td; td > tt) to maximize the rate of energy gain, which is c •(td - tt) (1 2 5)

Write (td) = td + $(td) = total dive time plus pause time. Then maximizing equation (1.2.5) is equivalent to adjusting tdtominimize (td)/(td — tt). This is shown in the graph, and the optimal dive time is easily found.

(Box 1.2 continued)

The model predicts that dive and surface time both increase with travel time (dive depth), that the level of oxygen loading increases with depth, and that the optimal dive length is independent of resource quality (c).

While these simple models do not apply universally like Newton's laws, they are foundational, and it is hard to overstate their importance in the logical development offoraging theory. The patch model may in fact be the most successful empirical model in behavioral ecology; its basic predictions have been widely confirmed, at least qualitatively, although it is not always clear that the logic of the patch model correctly describes the situation being modeled. Stephens and Krebs (1986) considered mainly long-term average rate maximizing, but investigators have since shown that animals sometimes behave as "efficiency" maximizers (Ydenberg 1998). The links between efficiency-maximizing and rate-maximizing currencies have interesting implications for energy metabolism and workloads (chap. 8 in this volume explores this topic further).

The simplicity of both the diet and patch models is deceptive, and the beginning student will have to work hard to master their subtleties. They show that the modeler's real art is not mathematics per se (after all, the math is elementary), but rather in distilling the essentials from so many and such varied biological situations.

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