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Figure 7.9. Hypothetical daily mass gain of hoarding and nonhoarding birds in winter. The nonhoarder has less predictable food availability and must hedge with larger body fat reserves. There are two cases for hoarders, baseline assumptions (dotted curve) and the three added assumptions described in the text (dashed curve). The most important difference is that under baseline assumptions, predation risk is mass-dependent at low levels of fat, whereas under the three added assumptions, predation risk is mass-dependent only above some limit.
authors had suggested: (1) hoarding takes advantage of ephemeral food supplies; (2) hoarding provides additional food during shortages; (3) hoarding provides an alternative to body fat reserves; (4) hoarding birds can exploit low-cost foraging opportunities—for example, when predators or competitors are scarce. Note that this list does not explicitly mention hoarding as a strategy for counteracting uncertainty in future supplies and costs.
Lucas and Walter discussed two models of optimal hoarding, a harvest rate maximization model and a survival rate maximization model; only the latter was a dynamic state variable model. They assumed that hoards are shortlived (<3 days), as observations of Carolina chickadees in the wild suggest. Lucas and Walter tested their predictions experimentally; the results of the experiments supported the survival rate maximization model. In contrast to simpler rate maximization models, dynamic state variable modeling can predict complex state-dependent decisions; as Lucas and Walter pointed out, "caching behavior is an example of a behavior pattern that is strongly affected by the state of the animal."
In contrast to Carolina chickadees, some passerines from more northern climates use both short-term and long-term hoards (Brodin 1992, 1994a; Brodin and Ekman 1994). Brodin and Clark (1997) modeled this situation. Our model uses three state variables, all measured in kJ: X(t) = body reserves, y(t) = stored seeds retained in memory, and Z(t) = forgotten long-term stores. Newly hoarded seeds first enter the remembered store, Y(t), but when the bird no longer remembers their location, the model transfers these seeds to long-term stores, Z(t). Both stores are subject to loss (decay, pilferage) at a constant rate. The bird can retrieve remembered seeds quickly when needed, but long-term stores merely enrich the natural, background supply.
The model considers fall and winter as two sets of environmental conditions that occur in sequence. Otherwise, the details are fairly similar to those of the models described above: fitness given by survival, mass-dependent costs, stochastic environment. The model bird can decide whether to rest, forage and eat, forage and hoard, or retrieve food from remembered hoards. We used parameter values derived from observations of willow tits in central Sweden.
The model provided several intuitively reasonable predictions. Long-term stores held at the start of winter strongly influence overwinter survival. So the optimal fall strategy builds long-term stores as much as possible. Short-term stores at the start of winter have a smaller influence on overwinter survival. Nevertheless, the optimal strategy in winter includes continual hoarding, because remembered seeds provide a hedge against bad weather, without the need to maintain high levels of body fat. Increasing the capacity for remembered stores, Ymax, had minimal effects on overall fitness, suggesting relatively weak selection for additional memory capacity in willow tits.
Hoarders often lose a portion of their cache to pilferage, and it would seem that this pilferage imposes a cost on hoarding. Lucas et al. (2001) used a dynamic state variable model to investigate the effects ofpilferage on hoarding rates. Intuitively, one might think that high rates of pilferage would make it optimal to store fat instead of caching food. Lucas et al. showed that this is not always the case. Hoarders can compensate for increased pilferage by hoarding at a higher rate as long as the marginal value of caching exceeds that of resting.
Animals can use nocturnal hypothermia to manage their energy budgets. Small birds can reduce their nighttime body temperature to 30°C—38°C (see the end of section 7.4). Willow tits and other small passerines at northern latitudes use hypothermia during cold winter nights. Although one can easily understand how hypothermia saves energy, its costs, if any, remain mysterious. Perhaps the hypothermic state increases predation risk. Alternatively, hypothermia may have long-term physiological consequences that reduce subsequent performance.
Clark and Dukas (2000), Pravosudov and Lucas (2000), and Welton et al. (2002) have developed dynamic models of optimal hypothermia, assuming that hypothermia increases predation risk. Nocturnal hypothermia may be an adaptation to a temporary deficit of energy reserves, but birds can use hypothermia in other situations as well. As explained below, in extreme conditions, a small bird may use hypothermia to prevent or control the gradual depletion of body reserves over a period of several days. While Pravosudov and Lucas considered hypothermia to be a single drop in temperature, Clark and Dukas considered the depth ofhypothermia to be variable. In these two models, increased predation risk at night or in the morning is the cost ofhypother-mia. Welton et al. included an additional warming-up cost in their model.
Clark and Dukas's model had two decision variables, foraging effort (0 < e(d) < 1) and 0, the energy saved overnight by using hypothermia (0 < 0 < 0max). The model was an elaboration of the basic model we present in equations (7.9—7.18). We will not give all the details of this model here, but only explain how it included hypothermia. In a manner similar to equation (7.12), dawn reserves are given by
Minimal hypothermia, 0minimal, is defined as the nocturnal energy savings needed to prevent overnight starvation:
0 otherwise i.e., 0minimal makes up for any overnight deficit in reserves. Hypothermia is facultative, in the sense that the bird knows the current thermoregulation cost, cth(d), when 0 is chosen. The options on a given night are then to use minimal hypothermia, 0minimal (possibly zero), avoiding starvation and starting the next day with zero body reserves, or to use 0 > 0minimal, starting the next day with positive reserves. The model determines which of these options is optimal in terms of overall survival.
Clark and Dukas modeled the cost of hypothermia through its effect on overnight survival,
where |n is the hypothermic mortality coefficient. How large this cost is depends on the bird's energy reserve strategy as expressed in its dusk reserves,
Xn(d). Exactly how the starvation-predation trade-off involving both reserves and hypothermia is realized depends critically on the distribution ofnocturnal costs, cth(d).
Clark and Dukas (2000) modeled possible spells of bad weather by assuming that such spells were initiated randomly and, once initiated, persisted for a random number of days. This assumption required a second state variable, N(d), the number of bad days in the current spell. This way of looking at winter weather patterns seems realistic and is much simpler computationally than more traditional approaches.
We can think ofhypothermia as an emergency measure that birds use only when nocturnal conditions are unexpectedly severe. Even ifrelatively costly when used, hypothermia may be an adaptive alternative to large, seldom needed energy reserves. On the other hand, birds may need hypothermia to survive in cases in which the maximum reserve capacity is too low to meet nocturnal requirements. A third possibility is that birds use hypothermia only in long spells ofbad weather that affect both the energy supply, f (d), and the nocturnal costs of thermoregulation, cth(d). Hummingbirds are known to combine high metabolism with small body mass. At least in regions with cold nights, as in higher regions ofthe Andes, hummingbirds seem to use nocturnal hypothermia on a regular basis (Hainsworth 1981; Carpenter and Hixon 1988).
As Clark and Dukas varied the food supply during bad weather, they found (1) that hypothermia did not improve fitness (i.e., survival) when the food supply was large; (2) that for intermediate food supplies, the optimal level of hypothermia was minimal (0minimal), but had a large effect on fitness; and (3) that when the food supply was small, the optimal level of hypothermia exceeded 0minimal, with a large effect on fitness. In addition, optimal reserves, X1(d), increased at lower food supply values and were also higher under the no-hypothermia assumption than for optimal hypothermia. These findings suggest that hypothermia serves mainly as an alternative to large body reserves in intermediate environments. As in previous models, the main source ofmortality is predation rather than starvation, expect in very harsh environments. Clark and Dukas (2000) did not consider the interrelation of hoarding and hypothermia, a topic that might be worth pursuing.
Ekman and Lilliendahl (1993) discovered that the dominant bird in a pair of willow tits has lower fat reserves than the subordinate. They suggested that this occurs because the dominant monopolizes the best foraging sites, especially in bad weather; the subordinate has a less secure and more variable food supply. Thus, the subordinate needs to carry extra reserves as a hedge against interruptions in its supply.
Clark and Ekman (1995) modeled dominant-subordinate fattening strategies as a one-sided dynamic game. The dominant maximizes its fitness independently of the subordinate. The subordinate then maximizes its fitness, subject to the condition that the dominant always excludes it from foraging in the dominant's current foraging habitat. Clark and Ekman considered three habitats: H0 was a refuge with no food and no predators, H1 provided an inferior intake rate but was relatively safe (e.g., interior branches of trees), and H2 provided a high intake rate, but was risky (e.g., outer branches).
Clark and Ekman assumed that food in the mediocre but safer H1 habitat was sufficient to cover metabolic costs on normal days, but not on cold days (which occurred with a certain probability p). Thus, both dominant and subordinate had to use the more dangerous H2 habitat at least part ofthe time. The model predicted that the dominant would use the mediocre H1 habitat most of the time, thereby excluding the subordinate from this preferred habitat. Consequently, the subordinate experienced higher risk of predation than the dominant. In addition, the dominant excluded the subordinate from the rich high-risk habitat (H2) on cold days. The model thus predicted that the subordinate would carry greater fat reserves than the dominant as a hedge against inadequate food on cold days.
The model formalized the hypothesis of Ekman and Lilliendahl (1993) and explained how the observed differential between dominant and subordinate survival rates could arise. Clark and Ekman also considered how changes in the food supply in the mediocre H2 habitat affected their predictions. At low levels of food supply, both birds suffered increased mortality rates. In addition, the dominant switched to carrying the same level of reserves as the subordinate. Farther north, where winter conditions are harsher than in Ekman and Lilliendahl's field area, dominants carry as much or even more fat than subordinates (Koivula et al. 1995; Verhulst and Hogstad 1996).
Organisms must store energy because they will experience periods when energy expenditure exceeds energy intake. Animals can store energy either internally, as fat or carbohydrates, or externally, as hoarded food. Alternatively, animals can reduce energy expenditure during periods when energy intake is not possible. Hibernation and temporary hypothermia provide examples of this strategy.
Energy storage incurs both costs and benefits, and the trade-off between these makes storing behaviors well suited for optimization modeling. Stored energy not only permits energy use when expenditure exceeds intake, but it may also hedge against unpredictable variation in intake rates. For example, hoarded food may provide insurance against rare food shortages even if the hoarder never consumes it. In order to build energy stores, animals must find more food than they need for immediate consumption. This added effort may increase energy expenditure and predation risk. Large fat reserves may increase mortality, both because fat stresses the heart and vascular system and because it decreases the animal's ability to escape from an attacking predator (especially for birds). Hoarded supplies may incur costs if hoarders must defend them, or if hoarders forget their locations.
Behavioral ecologists have modeled the regulation of energy reserves in several ways, including game theoretical models, rate maximization models, and other analytic models. However, most models on this topic have been dynamic state variable models, which are better suited for the complexity of energy storage problems. Simple analytic models cannot simultaneously incorporate phenomena such as temporal dynamics, stochastic effects, nonlinear fitness effects, and predation effects. Small birds in winter offer an especially appealing modeling problem, since they face a delicate trade-offbetween predation and starvation that nonflying animals do not face. At the same time, their high metabolic rates further complicate their energy storage problems.
Vander Wall's book (1990) offers the most comprehensive summary of food hoarding. It is now over sixteen years old, but still contains a valuable summary and an impressive list of references. Most reviews of energy storage focus on birds, but Vander Wall covers food hoarding in all animals, including insects. Kallander and Smith (1990), who reviewed avian food hoarding in the same year, concentrated on the evolutionary aspects of hoarding. Witter and Cuthill (1993) and, more recently, Pravosudov and Grubb (1997) cover fat storage by birds. Cuthill and Houston (1997) provide a more general perspective on energy acquisition and storage. Blem (1990) gives a physiological perspective on fat storage in birds. Bulmer (1994) gives an overview of the theory of evolutionary ecology. Houston and McNamara (1999) and Clark and Mangel (2000) explain the modeling techniques and concepts that we have used in this chapter.
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Modern Foraging Theory
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