We now turn our attention to cooperation and assume, for convenience, that we are talking about a prey species. We can make the reasonable assumption that the per-capita rate of change of the prey will be inversely proportional to the number of predators searching for that particular species and directly proportional to the number of prey species present in the environment. For example, assuming the predator population is constant, then increasing the prey population will decrease the number killed by predators and, therefore, increase the prey per-capita rate of change until all predators are satiated, after which the prey per-capita rate of change will become constant. This is best viewed on the graph showing the per-capita rate of change of the prey population as prey density changes (Figure 2). As we see, the prey R-function rises with prey density up to a maximum value, the reproductive potential of the species. It passes through R_N = 0 at U where it creates an unstable (extinction) threshold; i.e., when the prey population is below this threshold it declines to extinction. There are, of course, many other possible causes of an R-function increasing with density; for instance, there may be an increase in mating as populations rise from very low densities, as hunting groups become larger, as defensive groups increase in size, and so on. We will discuss the cooperative process in more detail when we focus on the mathematics of the fifth principle. However, the main point is that survival and birth rates increase with rising population density when cooperative processes dominate. This is the opposite of competition, in that cooperation is usually manifested at low densities while competition is usually a high-density phenomenon. They tend to therefore act sequentially to create a humped shaped R-function, with declines in per-capita rates of change occurring at low and high densities, and the highest R-values in between.

We can express the effect of the second principle on the per-capita rate of change of the prey species with a mathematical form very similar to the third principle (Berryman 1999)

R_N = A_N(1 – MP /N_t-1) . (2)

Where R_N is the per-capita rate of change of the prey, A_N is the maximum per-capita rate of change of the prey in a given environment, M is a proportionality constant, P is the constant density of predators, and N_t is the density of prey at time t. Notice that the parameter which changes the curvature of the R-function is left out of this model but, of course, it can be added if necessary. Notice also that MP = U, so that the number of predators determines the level of the unstable threshold (Figure 2). A number of different forms have been proposed for the second principle but this one is most similar to the third principle (which is supported by data).

Figure 2. Operation of the cooperative process causes R_N to increase with population density up to the maximum per-capita rate of change of the species A_N. U is an unstable threshold on the line R_N = 0.

Berryman, A. A. 1999. Principles of population dynamics and their application. Stanley Thornes (Garland Science, Taylor & Francis).