Foraging for a Fixed Time

Bluehead chubs alter their foraging in response to changes in energetic returns and danger from green sunfish. The best explanation for their behavior combines food and danger in a life history context (Skalski and Gilliam 2002). To build models of foraging under danger of predation, we start from the first principle of foraging theory—that food is good. We assume that higher foraging success leads to greater reproductive success in the future. To include danger, we need a second principle—that death is bad for fitness. Early research was uncertain on how to incorporate danger into foraging models (see box 1.1), perhaps because it is not obvious how to combine the benefits of foraging and the costs of predation. Because the costs and benefits are in different units, we need to translate both foraging gain and danger into some measure of fitness. A life history perspective is essential, and it leads to a simple solution that exists precisely because the costs and benefits of foraging under danger of predation are linked.

Decisions made under danger of predation are life history problems because, if predation occurs, the forager's life is history. In a life history, the basic currency to maximize is expected reproductive value, b + SV, where b is current reproduction, S is survival to the following breeding season, and V is the expected reproductive value for an animal that does survive to the next breeding season (see Stearns 1992). I concentrate here on foraging and fitness during a period without current reproduction (b = 0), so the measure of fitness is SV, the future benefits multiplied by the odds of surviving to realize them. I expect increased foraging to decrease survival to the time of reproduction, but to increase future reproduction if the animal does survive.

Death lowers expected future fitness to zero. Therefore, the cost ofbeing killed is the reproductive success a forager could have had if it had survived. This linkage means that when we ask how much risk a forager should accept to produce one additional offspring, we need to know how many offspring it would produce otherwise. For example, a forager that would otherwise expect to produce one offspring might risk a lot to produce a second, while a forager that would otherwise expect to produce three offspring should risk less to produce a fourth, and a forager that would otherwise expect to produce a dozen should risk little to produce a thirteenth. This linkage of costs and benefits sets up an automatic state dependence: the potential losses from being killed increase with previous foraging success, so the relative value of further foraging is likely to be lower (see Clark 1994). Even if the fitness gains of foraging are constant, the costs should increase, since the expected reproductive value increases, and that entire value would be lost in death. In line with this logic, juvenile coho salmon are more cautious when they are larger, because larger individuals expect greater reproduction if they survive to breed (Reinhardt and Healey 1999).

Now I will repeat these arguments mathematically. For a nonreproducing animal, fitness equals the future value offoraging discounted by the probability of surviving from now until then, W(u) = S(u)V(u), where u is a measure of foraging effort, Wis fitness, S is survival, and Vis future reproductive value.

Fitness, survival, and future reproductive value are all functions of foraging effort u. In general, we expect survival to decrease and future reproductive value to increase with foraging effort. More specifically, we expect survival to decrease exponentially with mortality, S(u) = exp[-M(u)], where M(u) is mortality.

Mortality rate, M(u), and future reproductive value, V(u), could take various mathematical forms. For simplicity, I define foraging effort as a fraction of the maximum possible effort, so that u varies from zero to one and does not have units. This allows mortality, M(u), and future reproduction, V(u), to be given as simple functions of foraging effort.

Mortality is a function of the amount of time spent exposed to attack, the attack rate per unit time, and the probability of dying when attacked (see Lima and Dill 1990). Greater overall foraging effort could affect any of these components. For now I use a descriptive equation for mortality, M(u) = kuz, where k is a constant and the exponent z gives the overall shape of the trade-off. Later we will examine two specific cases to see what k and z might mean biologically, but for now I will simply label k as the mortality constant and z as the mortality exponent. The general principle is that mortality should increase with foraging effort at a linear or accelerating rate; that is, M(u) = kuz with z > 1. If foragers exercise their safest options first, we expect an accelerating function because additional food comes from increasingly dangerous options. A mathematically convenient value for the exponent, z = 2, matches observed changes in behavior well enough (Werner and Anholt 1993), but other values are not ruled out, so I also examine a linear relationship (z = 1) as well as more sharply accelerating ones (z = 3 and z = 4). For all values, survival declines as foraging effort increases, but the contours of the decline depend on the exponent of the mortality function, z (fig. 9.3). As we

Figure 9.3. Survival declines with foraging effort. The swiftness of the decline varies with z, the exponent of the curve relating foraging effort to mortality.

shall see near the end of this chapter, the value of this exponent determines whether foragers should over- or underestimate danger.

For the relationship between foraging effort and future reproductive value, wewilluse V(u) = Km.In this equation, the constant K translates foraging effort into future offspring, and u is foraging effort. We expect future reproductive value to increase with total foraging effort. Studies have shown that greater foraging success leads to greater fitness in adult crab spiders (Morse and Stephens 1996), water striders (Blanckenhorn 1991), and water pipits (Frey-Roos et al. 1995). Particularly for any organisms that are able to grow, reduced foraging in the presence ofpredators can lead to considerable long-term losses of potential reproduction (Martin and Lopez 1999; see also Lima 1998, table III).

A linear relationship between foraging and future reproductive value is useful for its simplicity. Other relationships may occur in nature, and the relationship may differ between the sexes even within a species (Merilaita and Jor-malainen 2000). I use a linear relationship here because it allows simple models with clear conclusions, even though these models may somewhat understate the effects of danger. The results of more complex models, in which future fitness is a decelerating function of foraging gain, strongly support the conclusions I reach using this simpler linear relationship.

To complete the modeling framework, assume that foraging effort must be greater than some required effort, R. This requirement, R, is the required rate offeeding divided by the maximum rate offeeding and so is a proportion without units. A forager starves if its foraging effort is less than the requirement, and avoids starvation as long as its foraging effort is greater than the requirement. A forager gains some amount of fitness, V(R), by just meeting the requirement, but increases its future reproductive value by foraging at a rate higher than the requirement.

Assembling the pieces described above, we get the overall equation for fitness: W(u) = S(u) V(u) = [exp(-kuz)][Ku]. We can find the optimal foraging effort, u*, if we differentiate W(u), set the derivative to zero, and solve for u. We find that

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