The Lotka–Volterra predator–prey model has a curious (and unrealistic) property: the cycle’s amplitude (the magnitude by which predator and prey numbers rise and fall) depends on the initial numbers of predators and prey. If the initial numbers shift even slightly, the amplitude of the cycle changes. Partly for this reason, the model has never been successfully applied to any predator–prey system, either in the laboratory or in the field.
The unrealistic behavior of the model results from two assumptions of the model that we know to be wrong. First, in the absence of predators, it is assumed that the prey population grows exponentially indefinitely (since when P = 0, dN/dt = rN). Second, the rate at which predators remove prey (the term aNP in the top line of Equation 12.1) is not realistic. Use of this term implies that individual predators remove prey at a rate of aNP/N = aP, a rate that continues to increase as P increases, no matter how large P becomes. In effect, this feature of the Lotka–Volterra model implies that an individual predator can never get full.
These two assumptions have been corrected in more realistic models, such as the Rosenzweig and MacArthur (1963) model:
All the parameters in this model are the same as those in the Lotka–Volterra model, save for the addition of the terms K and b. Unlike the Lotka–Volterra model, when there are no predators in the Rosenzweig and MacArthur model, the prey population does not grow exponentially—instead, it grows according to the logistic growth equation (dN/dt = rN(1–N/K), where K represents the carrying capacity of the prey population.
The other new parameter, b, controls how rapidly an individual predator becomes full. The per capita rate at which predators eat prey has been changed from its form in the Lotka–Volterra model (aP) to the following:
In this equation, when the number of prey (N) becomes very large, the rate at which prey are removed by an individual predator approaches a constant value (a); you can check this statement by replacing “N” with a series of increasingly large numbers. Thus, unlike the situation implied by the assumptions of the Lotka–Volterra model, in the Rosenzweig and MacArthur model, individual predators can become full.
As in the Lotka–Volterra predator–prey model, population cycles can result from the Rosenzweig and MacArthur (1963) model. However, the amplitude of these cycles do not depend on the initial numbers of predators and prey. Thus, by assuming that the prey population grows logistically and that individual predators can become full, the Rosenzweig and MacArthur model gets rids of the unrealistic property that the amplitudes of the predator and prey population cycles depend on the initial conditions. Furthermore, as described in Turchin (2003), the Rosenzweig–MacArthur model has been successfully applied to real systems.
