The expectation operator, E, in equation (7.16) may look complicated, but makes it more readable than ifwe explicitly expressed the probabilities for the two random variables f (d ), cn(d ) that it encompasses. For simplicity reasons, modelers frequently use two complementary probabilities rather than some continuous distribution in this type of model. For example, the cn(d) variable could be a small energy loss during mild nights, cn(oooD), with probability Pgood, and a larger loss during cold nights, cn(BAD), with probability 1 — PGOOD. The max0<£<1 expression tells us that only the optimal foraging effort needs to be considered. The daily survival probability S(x, £) from equation (7.13) is a mass-dependent variant of equation (7.2). Fat reserves can never have negative values. Thus x' in equation (7.16) is replaced by 0 if the right-hand expression is negative. In addition, we have

reflecting the assumption that the bird has died if x = X(d) = 0.

Equation (7.16) shows that the trade-off between starvation and predation risk that we discussed previously has two linked components. First, greater foraging effort £ increases reserves at dusk, X1(d), and this increases the probability of overnight survival. But greater foraging effort also increases predation risk, |l(x)£. Second, increased foraging activity also increases the expected level of future reserves, X0(d'), where d' is any day between d and D. Increased reserves lower the risk of future starvation (for example, during a prolonged period ofunusually cold nights), but they increase mass-dependent metabolic costs and predation risks.

If we imagine how natural selection optimizes foraging behavior with respect to this trade-off, intuition might suggest an optimum at which equally many birds die from starvation and predation. Field observations of northern passerines do not support this prediction: most birds that die in winter

Figure 7.8. The effect of foraging effort (or fat reserves) on survival probability in wintering birds. Total survival (solid curve) depends on the combined effects of starvation risk (dashed curve) and predation risk (dotted curve) on survival. The optimal foraging effort (or optimal level of fat reserves) will be where the marginal values of these risks are equal.

Figure 7.8. The effect of foraging effort (or fat reserves) on survival probability in wintering birds. Total survival (solid curve) depends on the combined effects of starvation risk (dashed curve) and predation risk (dotted curve) on survival. The optimal foraging effort (or optimal level of fat reserves) will be where the marginal values of these risks are equal.

die of predation, not starvation (Jansson et al. 1981). Only unusually extreme conditions result in starvation, but predation risk is always present and unavoidable. McNamara and Houston (1987b, 1990) provided a theoretical explanation for this observation. Total mortality, |It(x), at the level of fat reserves x is the sum of mortality from both starvation (S) and predation (P):

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