Modeling Predator–Prey Dynamics
265
are the familiar type I, II, III, and IV functional responses (figure 8.3) proposed by Holling (1966). Among arthropods most functional responses fit a
type II or III (Hassell 1978).
Although some have claimed that mammals often have type III functional
responses, apparently due to learning (Holling 1966; Maynard Smith 1974),
Messier (1994) and Dale et al. (1994) present evidence that wolves preying on
moose and caribou (Rangifer tarandus) better fit a type II response.
But the rate of prey capture is much more complex than just a dependence
on prey density (i.e., a dependence on the physical environment, vulnerability
of prey, condition of the predator, prey group size, and a number of other variables). Indeed, much of the theory of optimal foraging (Stephens and Krebs
1986; Fryxell and Lundberg 1994) deals with understanding adaptations to
factors that influence the rate of prey capture, and much of this theory is relevant to the development of sound models for functional response. Students of
herbivory (Spalinger et al. 1988) appear to have a more mechanistic and
enlightened perspective on the structure of the functional response than those
studying predation.
The numerical response is usually modeled as a simple multiple of the functional response, so the numerical response assumes the same shape as the functional response. Indeed, there is an empirical basis for this relationship (Emlen
1984) that is especially well documented among invertebrates. But vertebrate
examples also exist. For example, Maker (1970) found a logistic-shaped plot
(type III) of the density of pomarine jaeger (Stercorarius pomarinus) nests as a
function of the density of brown lemmings (Lemmus trimucronatus) in Alaska.
Messier (1994) found what appeared to be a type II curve for both the functional response and numerical responses of wolves preying on moose (figure
8.4).
Numerical response is defined in different ways. As noted earlier a numerical response can be defined to predict the response in population growth rate
for the predator afforded by the killing of prey. Alternatively, a numerical
response may be defined to be the number of predators at equilibrium at a given
prey population size (Holling 1959; Messier 1994). The latter definition is convenient because when this quantity is multiplied by the functional response it
yields the total number of prey consumed for a given prey abundance. Dividing this quantity by prey population size yields the predation rate (figure 8.4).
As an example of the Kolmogorov equations, we will consider specifically
the pair of equations that Caughley (1976) offered as a plant–herbivore model:
dV/dt = rV (1 – V/K ) – Va(exp(–c 1 V ))
(8.13)
265
are the familiar type I, II, III, and IV functional responses (figure 8.3) proposed by Holling (1966). Among arthropods most functional responses fit a
type II or III (Hassell 1978).
Although some have claimed that mammals often have type III functional
responses, apparently due to learning (Holling 1966; Maynard Smith 1974),
Messier (1994) and Dale et al. (1994) present evidence that wolves preying on
moose and caribou (Rangifer tarandus) better fit a type II response.
But the rate of prey capture is much more complex than just a dependence
on prey density (i.e., a dependence on the physical environment, vulnerability
of prey, condition of the predator, prey group size, and a number of other variables). Indeed, much of the theory of optimal foraging (Stephens and Krebs
1986; Fryxell and Lundberg 1994) deals with understanding adaptations to
factors that influence the rate of prey capture, and much of this theory is relevant to the development of sound models for functional response. Students of
herbivory (Spalinger et al. 1988) appear to have a more mechanistic and
enlightened perspective on the structure of the functional response than those
studying predation.
The numerical response is usually modeled as a simple multiple of the functional response, so the numerical response assumes the same shape as the functional response. Indeed, there is an empirical basis for this relationship (Emlen
1984) that is especially well documented among invertebrates. But vertebrate
examples also exist. For example, Maker (1970) found a logistic-shaped plot
(type III) of the density of pomarine jaeger (Stercorarius pomarinus) nests as a
function of the density of brown lemmings (Lemmus trimucronatus) in Alaska.
Messier (1994) found what appeared to be a type II curve for both the functional response and numerical responses of wolves preying on moose (figure
8.4).
Numerical response is defined in different ways. As noted earlier a numerical response can be defined to predict the response in population growth rate
for the predator afforded by the killing of prey. Alternatively, a numerical
response may be defined to be the number of predators at equilibrium at a given
prey population size (Holling 1959; Messier 1994). The latter definition is convenient because when this quantity is multiplied by the functional response it
yields the total number of prey consumed for a given prey abundance. Dividing this quantity by prey population size yields the predation rate (figure 8.4).
As an example of the Kolmogorov equations, we will consider specifically
the pair of equations that Caughley (1976) offered as a plant–herbivore model:
dV/dt = rV (1 – V/K ) – Va(exp(–c 1 V ))
(8.13)