To understand phenomena such as the Arctic lynx-hare cycle discussed in the prologue, one needs population models. When abundances are great enough to be treated as continuous rather than discrete variables (see box 11.1), one uses differential equations (see also chap. 13), such as dP
dt the predator half of a predator-prey model. The variables are P (predator density, say, of lynx [density is the number of individuals per unit area]), R (resource or prey density, say, of hares), and t (time). The expression dP/dt is the instantaneous rate ofchange in P; one can think ofthis as the difference in P(dP) over a small time interval (dt). The biology enters into how one relates this change in density to foraging and other factors. The quantity B(R) is a function describing the rate at which prey are captured and consumed (the predator's "functional response"; Holling 1959a) as a function of prey abundance. To relate foraging rates to predator population dynamics, one must determine how foraging affects predator birth and death rates. In this example, we assume that feeding influences births in a simple fashion, in that b is a conversion factor translating the rate of prey consumption by an individual predator into predator births. To finish this mathematical representation of predator demography, we also must account for deaths. Here we simply assume that predators die at a constant per capita mortality rate, m.
To complete the model, we need an expression for prey dynamics (e.g., hares):
The quantity G(R) describes how the prey population grows in the absence of predation. For instance, a hare population might grow according to the classic logistic expression G(R) = rR(1 — R/K). The quantity r is the species' intrinsic growth rate (the rate at which it grows when rare enough to grow exponentially), and K is "carrying capacity," the prey abundance at equilibrium with births matching deaths. In describing the predator population, B(R) expresses the rate at which an individual predator consumes prey as a function of prey abundance. Therefore, the total mortality imposed by predators on the prey population is PB(R), which one must subtract from the prey's inherent growth to give the net growth shown in equation (11.2).
So far, we have said nothing specific about foraging. However, we can build assumptions about behavior into the detailed form ofthe functional response. Usually, B(R) will increase with R, or at least not decrease; feeding rates typically rise with increasing prey numbers. (Sometimes this assumption does not hold, for example, if groups of prey defend themselves against predators, but we assume that this is not the case.) A classic predator-prey model arises if we make the following simplifying assumptions about foraging: that a predator searches at a constant rate a while foraging in a nondepleting patch, that each prey requires a fixed time h for the predator to handle it (during which other prey cannot be encountered), and that each consumed prey is worth a constant amount, b. Holling's "disc" equation describes the rate at which the predator consumes prey (Holling 1959a; Murdoch and Oaten 1975; Hassell 1978, 2000), which translates into a predator recruitment term of
(the familiar saturating "type II" functional response). Figure 11.1 shows an example of this functional response in the context ofthe classic optimal diet model (see below and chap. 1). A crucial feature of this functional response is that predators become saturated with prey when prey numbers are large.
Figure 11.1. A graphical rendition of the classic optimal diet model, assuming sequential prey encounter and fixed handling times. The saturating curves represent the expected foraging yield of a consumer when it specializes on a particular resource (or prey type), of abundance Ri (resource 1 in A and resource 2 in B). The dashed lines represent the expected rate of yield resulting from having captured an item of type i (which equals the net benefit, bi, divided by the handling time, hi). The maximum gain rate from feeding exclusively on resource i (when it is very abundant) is bi/hi. Resource 1 is of higher quality than resource 2. (A) If resource 1 is sufficiently abundant, the expected yield from capturing and consuming a single item of type 2 is less than the consumer can achieve by ignoring that item and continuing to search for type 1; this implies thatthe consumer should specialize on resource 1 atabundances greaterthan the intersection shown and generalize at lower abundances of Rt. (B) Here, the consumer should always consume resource 1, because even the maximal foragingyield it can obtain from resource 2 is less than the yield it can obtain from a single encountered item of resource 1. As the graph shows, changing the abundance of resource 2 does not change this relationship.
Figure 11.1. A graphical rendition of the classic optimal diet model, assuming sequential prey encounter and fixed handling times. The saturating curves represent the expected foraging yield of a consumer when it specializes on a particular resource (or prey type), of abundance Ri (resource 1 in A and resource 2 in B). The dashed lines represent the expected rate of yield resulting from having captured an item of type i (which equals the net benefit, bi, divided by the handling time, hi). The maximum gain rate from feeding exclusively on resource i (when it is very abundant) is bi/hi. Resource 1 is of higher quality than resource 2. (A) If resource 1 is sufficiently abundant, the expected yield from capturing and consuming a single item of type 2 is less than the consumer can achieve by ignoring that item and continuing to search for type 1; this implies thatthe consumer should specialize on resource 1 atabundances greaterthan the intersection shown and generalize at lower abundances of Rt. (B) Here, the consumer should always consume resource 1, because even the maximal foragingyield it can obtain from resource 2 is less than the yield it can obtain from a single encountered item of resource 1. As the graph shows, changing the abundance of resource 2 does not change this relationship.
With multiple prey types, all parameters are indexed by prey type i = 1, ..., n. This «-prey-type extension of the disc equation produces the following harvest rate by a nonselective predator:
Ebi aiRi
A large theoretical literature takes this functional response as a given and uses it to analyze questions of predator-prey dynamics. For instance, saturating functional responses can permit prey to escape limitation by predators temporarily and can generate sustained predator-prey cycles such as the hare-lynx cycle. Model predators allowed to choose between prey types ("optimal foragers," for short) can exhibit very different functional forms relating feeding rates to prey density. For instance, Abrams (e.g., 1982, 1987) examined the functional responses of optimally foraging consumers for a wide range of ecological scenarios. Figure 11.2 shows an example of the nontraditional functional responses that can emerge when an optimal forager attacks two prey containing different ratios of two required nutrients (e.g., nitrogen and phosphorus). The rate ofconsumption ofresource (prey type) 1 increases with the abundance of resource 1, but with abrupt thresholds between levels of feeding. Figure 11.2B shows how the rate of consumption of resource 1 varies with the abundance of the alternative resource. The functional response shows threshold responses, and despite an overall decline in attacks on resource 1 with increasing abundance ofresource 2, some situations exist in which an increase in resource 2 leads to increased attacks on resource 1. These threshold responses, when integrated into a population model, would generate abrupt changes in population dynamics. Jeschke et al. (2002) provide a useful review of the wide range of functional response forms that ecologists have proposed and show how to incorporate digestive satiation as well as handling time constraints.
The above model of predator-prey dynamics illustrates a "top-down" approach to linking foraging and population dynamics (Schmitz 2001; Bolker et al. 2003). This approach takes an existing population model and refines one or more of its components in light of some idea about how the average consumer's foraging affects average birth or death rates. For instance, MacArthur and Pianka (1966) constructed a model of how predators select among prey when prey differ in caloric value and handling time. Several investigators interested in the effects of foraging on aspects of population dynamics have used the MacArthur and Pianka model to address issues such as indirect interactions between prey species (e.g., Holt 1977, 1983; Gleeson and Wilson 1986). Modelers call this approach "mean-field" modeling: the resulting equations describe how average (mean) predator foraging rates vary as a function of average (mean) prey densities, with a minimal specification of biological details. Mean-field models do not capture all ofthe complexity ofreal populations; because of their simplicity, however, they often generate very clear and testable predictions and clarify crucial conceptual issues.
Nonetheless, in many circumstances, considering average individuals ignores critical features of ecological systems, features that become apparent when one closely examines the behavior of individuals. Individual foragers ill y
un tf it
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