One of the biologically unreasonable properties of the original L-V model is that consumers have infinite appetites, violating attribute 4. Hence, the model is often modified by substituting a functional response, usually the prey-dependent cyrtoid response dH / (F + H), for the linear (unsatiated) attack rate dH. We also substitute the logistic prey limitation aH(1 - H/K) = aH - bH² (b = a/K) for the linear prey growth rate aH. Thus, the L-V model with prey-limitation and predator satiation is

Although this model describes attribute 4 it still fails to satisfy condition 3 because attribute 3, intraspecific competition for prey amongst consumers, is not considered by the model.

Population Dynamics with Prey-dependent Satiation

The zero-growth isoclines for the L-V model with prey-dependent satiation are: For the prey,

which is a parabola intercepting the prey axis at the point a / d = K , the equilibrium density if H in the absence of C or the carrying capacity of the environment for prey, and the predator axis at F (see figure below). The maximum of the prey isocline occurs at prey density (K - F) / 2.

The predator isocline is,

which is a constant intercepting the prey axis at mF / (cd - m). Notice that the form of the predator isocline is not affected by the addition of predator satiation; i.e., the isocline is still a vertical straight line as in the original Lotka-Volterra model.

The stability of this model depends on the position of the community equilibrium (the point where the two isoclines cross), being unstable to the left of the peak of the prey isocline (see figure) and stable to the right.

This model was popularized by Rosenzweig and MacArthur (1969) and is the most widely used form of the L-V model. For more on the dynamics of the R-M model check out Alexei Sharov's WWW course at Virginia Tech.

Figure. (Left) Herbivore (H) and carnivore (C) phase space showing zero-growth isoclines for the L-V model with intraspecific competition amongst prey and prey-dependent predator satiation (prey isocline = blue parabola, predator isocline = vertical red line). (Right) Herbivore and carnivore time series plots showing unstable cycles.

Reference

Rosenzweig, M. P. and R. H. MacArthur. 1969. Graphic representation and stability conditions of predator-prey interaction. American Naturalist 97: 209-223.