where m is a deterioration factor (e.g., decay) andpN is the probability that a food item that was not stored will remain available.

If a hoarder stores food in a location where any member of the group is equally likely to find it, the hoarder will be at a disadvantage. If stored supplies are communal property, the individuals that refrain from assuming the costs of hoarding will gain the same benefit from the stored supplies as others. Even if population size is decreasing due to a lack of stored food, a hoarder will always do worse than a nonhoarder will.

The probability that a hoarder will retrieve its own cache, pH, can be divided into two probabilities: the probability that the cache will be found, pf, and the probability that a stored item will remain until retrieval,pr (Moreno etal. 1981):

Figure 7.6. In a small group of foragers, as the cost of hoarding increases from 0 to 1 (= death), the ESS will change. If the cost of hoarding is low, hoarding may be the ESS even if there is no recovery advantage for hoarders. (After Smulders 1998.)

Figure 7.6. In a small group of foragers, as the cost of hoarding increases from 0 to 1 (= death), the ESS will change. If the cost of hoarding is low, hoarding may be the ESS even if there is no recovery advantage for hoarders. (After Smulders 1998.)

With substitution into equation (7.6), this gives pf > PNm + C. (7.8)

A low probability that food that is not stored will remain where it is found, pN, will decrease the right-hand term and may make hoarding worthwhile even if retrieval probability is low.

In a game theoretical model of the "playing against the field" type (Maynard Smith 1982), Smulders (1998) showed that under certain conditions, hoarding may evolve in species that forage in small groups even when all members of a foraging group share caches equally. If hoarding provides a net benefit, a rare hoarding mutant can always invade a pure population of nonhoarders. On the other hand, nonhoarders will have the highest fitness in groups that contain both strategies because they do not pay the costs of hoarding. Even if nonhoarders have higher fitness within the group, the global fitness of nonhoarders will be low if other groups do not contain hoarders. The reason is that nonhoarders must pilfer from hoarders to achieve their higher fitness. The result is a mixed ESS (fig. 7.6).

Modeling hoarding effort as a function of dominance rank, Brodin et al. (2001) showed that differences in hoarding behavior in a population may depend on rank. Dominants can steal caches more easily from subordinates than vice versa, so dominants should be less willing to pay the costs ofhoard-ing. Thus, under the right conditions, subordinates should hoard more than dominants. According to this model, dominant group members should not hoard at all unless they have a recovery advantage. Without a recovery advantage, dominants should specialize in pilfering. In addition, compared with that of subordinates, the dominants' investment in hoarding should increase as the environment gets harsher (fig. 7.7).

Neither the graphical paradigm nor analytic models consider how an organism might alter the two quantities, resource supply and energy expenditure, over the year. In our modeling approach, we consider strategies that either alter the natural resource supply by means of energy storage, or alter energy expenditure by methods that reduce energy use, or both. Our approach allows animals to store energy reserves either internally—for example, as body fat—or externally—as food caches or hoards. Methods of reducing energy expenditure include wintering in the tropics, physiological changes, the use ofhypothermia, and hibernation.

Such problems require dynamic state variable models (Houston and Mc-Namara 1999; Clark and Mangel 2000), which explicitly involve temporal dynamics, including seasonal and stochastic effects, and also consider dynamic state variables, including the current levels of energy reserves, both internal and external. One can include other important aspects of energy storage and use, such as predation risk and social interaction, in these models. Behavioral ecologists have applied the dynamic state variable approach to many aspects of winter survival strategies for small birds in northern climates; we review this literature below.

State variables are essential components of these models. They characterize the organism's internal state and, at the same time, model the effects of short-term decisions on fitness. In models of energy storage typically, state variables may be fat reserves, gut contents, or cache sizes. These models determine optimal (i.e., fitness-maximizing) time- and state-dependent behavioral strategies. In the simplest cases, such models treat individual fitness

Figure 7.7. In a model by Brodin et al., animals of different rank experienced a period of food surplus (e.g., autumn) followed by a period of food scarcity (e.g., winter). The figure shows the ratio between optimal hoarding by a dominant animal, hD\ and optimal hoarding by a subordinate animal, hS*, as a function of unstored food available during the period of food scarcity, h0. Decreasing background food (h0) will have the same effect as making the environment colder, since both changes increase energy expenditure. (After Brodin etal. 2001.)

Figure 7.7. In a model by Brodin et al., animals of different rank experienced a period of food surplus (e.g., autumn) followed by a period of food scarcity (e.g., winter). The figure shows the ratio between optimal hoarding by a dominant animal, hD\ and optimal hoarding by a subordinate animal, hS*, as a function of unstored food available during the period of food scarcity, h0. Decreasing background food (h0) will have the same effect as making the environment colder, since both changes increase energy expenditure. (After Brodin etal. 2001.)

maximization, but game theoretical models may be required where fitness is frequency-dependent.

Dynamic game theoretical models can be quite complicated (e.g., Houston and McNamara 1999), so various simplifying assumptions are often employed. For example, the dynamic aspect may be suppressed, or a dominantsubordinate social structure may be assumed (e.g., Clark and Ekman 1995; Brodin et al. 2001).

In a series of papers, McNamara and Houston and their co-workers have used dynamic state variable models to study various aspects of avian fat regulation during winter (McNamara and Houston 1990; Houston and McNamara 1993; Bednekoff and Houston 1994a, 1994b; McNamara et al. 1994; Houston et al. 1997). These models have considered (1) optimal winter fat regulation strategies; (2) the sensitivity of overwinter survival, and of daily fat levels, to ecological parameters, such as seasonal changes in day length and ambient temperature; and (3) the relative risks of starvation and predation in bird populations during winter. As always in modeling, the modeler seeks an improved understanding of complex natural phenomena. The construction, analysis, and testing of models adds intellectual rigor to what is otherwise often little more than speculation.

The framework of McNamara and Houston's models includes the following basic assumptions:

1. Birds store energy reserves as body fat.

2. Birds die of starvation if their energy reserves fall to zero at any time.

3. Daily food intake and overnight metabolic costs vary stochastically.

4. Metabolic costs during daytime activities increase with body mass.

5. Predation risk while foraging also increases with body mass.

6. Natural selection has favored fat regulation strategies that maximize the probability of overwinter survival.

The starvation-predation trade-off discussed in section 7.4 is essential in these models. As Houston and McNamara (1993) point out, the fact that many birds maintain lower levels of fat reserves in winter than they could probably reflects the importance of this trade-off, rather than the occurrence of resource shortages. Lima (1986) made this point previously.

To begin, we will describe a basic dynamic state variable model of optimal energy reserves for a generic "small bird in winter" (McNamara and Houston 1990; Bednekoff and Houston 1994b). This basic model can be extended in many ways.

Time periods d(d = 1, 2, . . . , D) correspond to the days that make up the winter, with D being the last day. The state variable X denotes a bird's energy reserves in kJ and has a present value of x. X0(d), then, is the bird's energy reserves at dawn on day d. The daily foraging effort, £(d), represents the fraction of daylight hours spent actively foraging (0 < e(d) < 1). This is also the decision variable; the foraging bird can "decide" how much time it should spend foraging. The rest of the time, 1 — £, is then spent resting. Food intake on day d is £(d) f (d), wheref (d) is a random variable (measured in kJ of usable energy). Thus, food intake depends on both foraging effort and environmental stochasticity.

Total daytime metabolic costs are c(x, e) = c1(x)£ + c2(x)(1 — £), x = X0(d), (7.9)

where q (x), c2 (x) are activity and resting costs, respectively. Since c is a function of x, these costs depend on body mass; for example, fatter birds may have higher metabolic rates.

Let X1(d) be the bird's reserves at dusk, after a day's foraging. The expression

X1(d) = X0(d) + £(d) f (d) — c [X0(d), £(d)] (7.10)

relates reserves at dusk, X^d), to reserves earlier that morning, X0(d). Fat reserves cannot be negative or immensely large. The inequality

constrains both X0 and X1, where Xmax is the maximum capacity for body reserves. If reserves at dusk in equation (7.10) fall to zero or less, then starvation kills the bird.

Overnight metabolic costs are cn(d); for simplicity, these costs are random and serially uncorrelated, with stationary distribution. In a more realistic (and complex) model, we could instead assume that an unusually cold night was likely to be followed by another cold night. Before the next day (d + 1), the bird will metabolize fat reserves. This is described by the overnight state dynamics,

subject to the constraints of equation (7.11); as before, the bird dies of starvation if reserves the next morning, X0(d + 1), fall to zero.

While foraging, the bird faces a predation risk, |l(x), that depends on body mass; the daily survival probability S [cf. equation (7.2)] then depends on both body mass and foraging effort:

Hence, we have two sources of mortality, predation and starvation.

The fitness currency is the probability of survival to the end of winter at d = D. We define the function

F(x, d) = max Pr(bird survives from day d to D), given X^(d) = x, (7.14)

where the maximization refers to the choice ofhow much ofthe day the bird should spend foraging, £ = £(x, d). Since our model allows foraging decisions to depend on state, the bird can control its reserve level, for example, by increasing foraging activities when X0(d) is low and vice versa. As in all dynamic state variable models, we work backward from the last time period; in our case, the end of winter. We can find the probability of survival on the last day of winter (day D) from the assumption that the bird dies of starvation if X0(d) = x = 0. This gives us the terminal condition,

That is, the bird survives the winter if it has more than zero reserves on day D, and dies otherwise.

The dynamic programming equation for the day before the end ofwinter (for d < D) follows from the definition of F(x, d) and the model specifications (cf. Clark and Mangel 2000):

We now have an equation that can get us through the whole winter, since the function at day d connects to itself at day d + 1. The first step in our backward iteration would then be to calculate fitness for the penultimate day ofwinter, D — 1, when equation (7.16A) would become

On the right-hand side we have F(x', D), which is either 1 or 0 [eq. (7.15)], depending on the value of x'. We can calculate x' from equation (7.10), which gives the change in fat reserves over the day, and equation (7.12), which gives the overnight energy loss. If we write these together, we get

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