Begin with the R-function for a population of consumers competing for fixed resources (the 3rd principle)

where A_P is the maximum per-capita rate of change of a consumer population, P_t-1 is the density of the consumer population, and C_P measures the degree of intraspecific competition for fixed resources. Remember that this equation applies only to non-reactive factors.

We can extend this model to include reactive factors by considering the circular causal pathway between populations of consumer and its reactive resources. Here we require expressions for the effect of each population on the per-capita rate of change of the other. Remembering that the equation for the 3^rd principle can be written

where H is the quantity of resources available to the population. Now H can be split into the density of a particular resource population, say N_t-1, plus a constant density of all other resources, F, to give

Finally, we can integrate this equation into the first to provide a model for a consumer utilizing both passive and reactive resources

For consistency, the per-capita consumer demand for resources, V, has been replaced by the coefficient of intraspecific competition, C_PN . These parameters are directly related because the larger the demand for resources the greater will be intraspecific competition for those resources. The time subscripts identify both P_t-1 and N_t-1 as dynamic variables in this equation.

The R-function for the prey population is derived as follows: First assume that the prey population is regulated by intraspecific competition for non-reactive resources when predators are absent from the environment. Under these conditions, the R-function for the prey in the absence of predation is defined by the familiar logistic equation

When predators are present, however, the per-capita rate of change of the prey population needs to be reduced by the death rate due to predation (2nd principle)

where intraspecific competition for predator-free-space, C_NP, replaces the killing power of the predator W.

Notice that both equations are very similar in that the per-capita rates of change are linear functions of the same predator/prey ratio, P_t-1 / (N_t-1 + F). Models such as these are sometimes called ratio-dependent.

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