Dynamics: Specialized Predators

Predators can be subdivided into specialists, adapted to feed on one or a limited number of prey, and generalists, which feed on many species of prey. Specialists usually have life cycles that are synchronized to those of their major prey species and behaviors tuned to finding the habitats of their prey. In this section we study the dynamics of specialized consumers. Because all ratio-dependent models have similar dynamic properties, we will use the simplest to them, the logistic individual survival model.

Consider the general logistic R-function for any member of a food chain

remembering that

R_i = the realized per-capita rate of population change for a given physical environment with given density of species X_i, resources X_i_-1, and predators X_i₊₁.

a_i = the maximum per-capita rate of population change in a given environment when population density is very small, food and space are unlimited and predators are absent.

b_i = the coefficient of intra-specific competition for fixed resources, usually some kind of spatial resource.

c_i = the coefficient of intra-specific competition for depletable resources, usually food from the lower trophic level.

d_i = the per-capita demand of predators for food, with respect to the species we are considering.

w_i = the relative abundance of other sources of food for the species we are considering, in units relative to species X_i_-1.

w_i+1 = the relative abundance of other sources of food for the predators, in units relative to species X_i.

In a completely specialized food chain, the abundance of alternative food is zero, so that w_i = w_i+1 = 0 and the R-function for an herbivore (H) with the density of plants (P) constant, is

However, as P is constant, then b_h, c_h, and P can be combined into a single parameter and the R-function for the herbivore population simplifies to

where b_h now subsumes the three constants. The R-function for the carnivore (C) has a similar configuration

Assuming that predators do not compete for fixed resources (b_c = 0), and setting R_h = R_c = 0, then the zero growth isoclines for the two species are (omitting the time subscripts for convenience)

prey isocline

predator isocline

The isoclines are drawn in H,C phase space below. Notice that the isocline structure is identical to that of a Lotka-Volterra model with ratio-dependent functional responses. In fact, as mentioned earlier, all models driven by ratio-dependent demand/supply relationships have similar isocline structures.

Figure. (Left) Herbivore (H) and carnivore (C) phase space showing zero-growth isoclines for the logistic individual survival model (prey isocline = blue, predator isocline = red) with parameters a_h = 1, b_h = 0.003, a_c = 0.5, b_c = 0, c_c = 6, d_c = 4, w_c = 0.

(Right) Time series plot of the herbivore (blue) and carnivore (red) populations showing the damped-stable cyclical approach to equilibrium.