Optimal Behavior and Consumer Resource Models

Coexisting species often differ in body size, but such differences do not always lead to coexistence based on food size selection. Coexisting species ofgraniv-orous desert rodents often differ in body size (e.g., Brown 1975), yet may overlap almost completely in the sizes of the seeds that they consume (e.g., Lemen 1978). In contrast, coexisting species of Darwin's finches may show distinct differences in both their beak sizes and the seed sizes in their diets (Grant 1986; see section 12.8). Canforaging theory illuminate the causes for such different outcomes?

Imagine two consumer species that compete for two types of food resources. The two species may differ in many respects, including the rates at which they encounter the resource types, the values ofthe different resource types, and the handling time needed to consume a resource item. Encounter rates, values, and handling times all affect rates ofenergy gain and determine diet and patch use decisions (see chap. 1). The species compete through their effects on resource density. The foraging aptitudes of individuals and the foraging choices that they make determine their effects on the resources and their energy gains. What are the conditions for species coexistence that emerge from the species' optimal behaviors? The answer depends on the distribution of the resource types, the nutritional relationship between them, the rates at which the resources are renewed, and the rates at which consumers harvest them. Thus, coexistence in these circumstances is at heart a foraging problem (MacArthur 1972; Tilman 1982).

In such consumer-resource systems, coexistence depends on the resources. If two competitors exploit a single resource, the species whose individuals can subsist on the lowest density of that resource typically outcompetes the other. The threshold density ofa resource at which a consumer species can just survive is referred to as R*. Above R*, the consumer species harvests enough ofthe resource to have a positive population growth rate, and vice versa when the resource is below R*. The expectation is that a population of consumers will grow or decline until its density promotes a resource abundance of R*. The consumer species with the lowest R* outcompetes the other under equilibrial consumer-resource dynamics.

For two consumer species to coexist in these models, there must be more than one resource. Vincent et al. (1996) examined coexistence on two resource types for optimal foragers that can choose both their diet and their habitat use. Their models varied two things: two resources occurred together in the same habitat or in separate habitats, and the resources could be either essential (the resource in shortest supply determines fitness) or perfectly substitutable (both resources contribute additively to fitness). Vincent et al. factorially combined these properties of the environment and resources to create four cases. To find conditions for coexistence, they examined the zero net growth isoclines and the resource depletion vectors.

Zero net growth isoclines are plotted in a state space of resource densities. They represent all combinations of the densities of two resources such that a forager has a zero population growth rate (Tilman 1982). The zero net growth isocline represents the two-resource equivalent of R*. When resource abundances lie above the isocline, the consumer species has a positive growth rate, and vice versa when resource abundances lie below the isocline. At equilibrium, the consumer species should deplete resources to some point along the zero net growth isocline.

The shapes of zero net growth isoclines depend on the resources' nutritional quality and spatial distribution and on the optimal foraging behavior of the consumers. For instance, consider a consumer species opportunistically harvesting two perfectly substitutable resources that occur together. In this case, the consumer species' zero net growth isocline is linear with a negative slope. The R1 and R2 intercepts (densities of resources 1 and 2, respectively) represent the original R*s for the situation in which there is only one resource.

Two perfectly substitutable resources can instead occur apart in different habitats. In this situation, the consumers can seek only one or the other resource at a time. In a situation analogous to the ideal free distribution, the consumers now seek the habitat that offers the highest harvest rate of resources. At equilibrium, the abundance of each resource will be driven down to its R*. Hence the zero net growth isoclines are the horizontal and vertical lines emanating from the R1 and R2 intercepts of the preceding example with two resources occurring together.

For essential resources, growth is limited by the resource in shortest supply. Regardless of whether resources occur together or apart, the isocline resembles an L. The level of each leg of the isocline is set by the density of that resource that yields a harvest rate equal to the foraging costs. Each of the isoclines emerges from the properties of the environment (foods together versus foods apart), the nutritional properties of the foods (substitutable versus essential), and the foraging behavior of the consumers. The consumers in these systems adopt a feeding strategy of opportunism, partial selectivity, or complete selectivity so as to maximize their fitness.

The population sizes of consumers and their foraging strategies result in resource depletion. The harvesting of resources and the renewal of resources result in a new equilibrium abundance of resources that is lower than it would be in the absence of harvesting. The depletion vector of a consumer species gives all combinations of equilibrium abundances of two resources that will occur as the population size of that consumer increases. The depletion vector starts at high values for R1 and R2 when the population of consumers is zero, then declines as the number of consumers increases. It is positively sloped if the consumers harvest some of both resources. It is horizontal if the consumers harvest only R1, and it is vertical if the consumers harvest only R2. Depletion vectors can be plotted in the same state space as zero net growth isoclines (Tilman 1982).

As with zero net growth isoclines, the distribution and characteristics of the resources and the foraging behavior of the consumers determine the position and shape of the depletion vectors. For substitutable resources occurring in the same habitat, the slope of the depletion vector is determined by the consumer's rates ofencountering the two resources, a1 and a2, and the abundances ofthe two resources, R1 and R2. The slope at any point in the state space of R1 and R2 is given by a2R2/a1R1. For substitutable resources that occur in separate habitat patches, the optimal habitat selection behavior of the consumers becomes paramount in determining the shape of the depletion vector. In this case, the consumers should balance their activity between habitats so that the fitness values oftheir harvest rates are equal (assuming equal costs offoraging between habitats): e1a1R1/(1 + a^1 R1) = e2a2R2/(1 + a2h2R2). This behavior produces a linear depletion vector whose slope is influenced by the consumer's energetic gain from the resource, e, rate of encountering the resource, a, and handling time for the resource, h. For essential resources, the ratio of the contribution of each resource to the consumer's fitness determines the slope of the depletion vector (Vincent et al. 1996).

The intersection of a consumer species' zero net growth isocline with its depletion vector determines the equilibrium abundance of resources at the equilibrium population size of that consumer species. We can find the conditions for coexistence by combining the zero net growth isoclines and depletion vectors of two different species. As a first condition, coexistence requires that the zero net growth isoclines cross. If not, one species (the one with the zero net growth isocline closest to the origin) can always outcompete the other by depleting resources to a point where the second species can no longer exploit them profitably. When zero net growth isoclines cross, different resources limit each species. Coexistence also requires that each species consume more of the resource that most limits its own growth; that is, the species with the shallower zero net growth isocline must have a depletion vector that increases less steeply (fig. 12.4).

Traits that affect foraging aptitudes also help determine the zero net growth isocline and the depletion vectors, and hence conditions for coexistence. Tra-offs in those traits among the consumer species cause zero net growth isoclines to cross. Zero net growth isoclines typically include the coefficients for encounter rate (a), handling time (h), and conversion efficiency (e) of resources, but depletion vectors often have only one of these coefficients. Hence, it is often the trade-offs among the coefficients in the depletion vectors that determine when two consumer species can and cannot coexist. When substitutable resources co-occur in the same patch, the relevant trade-off for coexistence requires differences between the two consumer species in their rate of encountering each resource. One consumer must have a higher rate of encountering resource 1, while the other consumer species must have a higher rate of encountering resource 2. For coexistence on essential resources, the two consumer species must have a trade-off in their conversion efficiencies (es). In this case, the

- zero net growth isocline

Figure 12.4. Zero net growth isoclines and resource depletion vectors for two species, 1 and 2, plotted in a state space of resource density (R1, R2). Consumers are limited by the resource in shortest supply. Typically, for coexistence, zero net growth isoclines (labeled 1 and 2 for species 1 and species 2, respectively) must cross, and the resource supply point (the maximum amount of the two resources in the absence of consumption) must lie in a region bounded by the two resource depletion vectors (labeled C1 and C2 for consumption by species 1 and species 2, respectively). The regions of each panel where both species coexist orwhere one species orthe otherwins out in competition when the resource supply point lies within that region are labeled. (After Vincent et al. 1996.)

Figure 12.4. Zero net growth isoclines and resource depletion vectors for two species, 1 and 2, plotted in a state space of resource density (R1, R2). Consumers are limited by the resource in shortest supply. Typically, for coexistence, zero net growth isoclines (labeled 1 and 2 for species 1 and species 2, respectively) must cross, and the resource supply point (the maximum amount of the two resources in the absence of consumption) must lie in a region bounded by the two resource depletion vectors (labeled C1 and C2 for consumption by species 1 and species 2, respectively). The regions of each panel where both species coexist orwhere one species orthe otherwins out in competition when the resource supply point lies within that region are labeled. (After Vincent et al. 1996.)

species with the lower ratio of conversion efficiency of resource 1 relative to resource 2 should also leave the higher amount of resource 1 when it stops foraging (highest Ri*). In contrast, when resources occur in separate habitats, coexistence can result from trade-offs between the two consumer species in encounter rate (a), handling time (h), or conversion efficiency (contributing to e). Thus, for organisms following the rules of optimal diet and habitat selection models, the distribution and nutritional quality of resources limits the kinds of trade-offs and mechanisms that promote species coexistence. In general, habitat selection offers more opportunities for coexistence than opportunistic feeding on co-occurring foods because a larger suite of trade-offs satisfy the conditions for coexistence under habitat selection than under overlapping diet choice.

How, then, does this apply to the desert rodents and the finches? In both cases, the coexisting species consume seeds of various sizes that co-occur in patches. They are most likely exploiting substitutable resources that co-occur.

In this situation, coexistence by diet choice requires a trade-offin encounter rates with the different food types. The ability to encounter large seeds must come at the expense of the ability to encounter small ones. The desert rodents often forage on buried seeds that are encountered by olfaction. Any characteristic that improves their ability to smell large seeds also probably improves their ability to smell small seeds, so the required trade-off does not exist. We must look elsewhere for a mechanism of coexistence (see section 12.8). For the finches, encounter rates with small versus large seeds do not appear to vary between small- versus large-beaked birds. But small-beaked foragers cannot generate enough force to crack open large seeds with thick coverings. Effectively, it is as if they do not encounter such seeds, resulting in a trade-off of encounter rates according to beak size and seed size and providing the necessary conditions for coexistence.

12.8 Mechanisms of Species Coexistence of Optimal Foragers

Consider the two gerbils, G.pyramidum and G. a. allenbyi, discussed previously. Recall that these species show distinct patterns of habitat selection, but they do not coexist due to habitat selection. Might the foraging abilities of the two gerbil species and salient features of their environment reveal the mechanism by which they coexist? In regard to the foraging abilities of the gerbils, the same field experiments that showed the smaller G. a. allenbyi always to be a more efficient forager than the larger G. pyramidum also suggested that G. pyramidum often arrives at resource patches first. In addition, G.pyramidum can handle food items more quickly and feeds faster at high seed densities (Kotler and Brown 1990). Isoleg analysis suggests that G. pyramidum dominates G. a. allenbyi via interference competition (Abramsky et al. 1990). On the other hand, G. a. allenbyi has evolved an especially low metabolic rate (Linder 1987) that should reduce its energetic costs of foraging. In regard to the gerbils' environment, predictable afternoon winds redistribute seeds and renew seed patches daily (Ben-Natan et al. 2004). The aptitudes of the gerbils and the daily renewal ofseeds open the possibility that these gerbils partition resource variability, with G. pyramidum using its ability to interfere and harvest seeds quickly to monopolize and deplete rich resource patches early in the night. Later, G. a. allenbyi, by virtue of its especially low energetic cost of foraging, can forage profitably on what remains (Brown, Kotler, and Mitchell 1994). The result is temporal partitioning.

To test this mechanism, Kotler, Brown, and colleagues conducted two experiments. In the first, Kotler, Brown, and Subach (1993) looked for temporal partitioning. They set out groups of six seed trays at the beginning of

Table 12.1 Temporal partitioning in two gerbil species, Gerbillus andersoni allenbyi and Gerbillus pyramidum

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