Let the demand for prey by individual predators be V so that the total demand by a population of P predators is VP. We expect competition amongst predators to be intense when the total demand for prey is high relative to the total supply of prey H. In other words, the intensity of intraspecific competition should be directly related to the demand/supply ratio, VP/H, where H includes all other prey species available to the predator. Because of this, the maximum per-capita rate of change of the consumer, A_P, should be reduced in inverse proportion to the demand/supply ratio. Assuming a linear effect, then the consumer R-function can be written

where P_t-1 identifies the consumer population as a dynamic variable and all other values are constants. Because H is assumed constant, the equation can be reduced to

or

where C_P = V/H represents the intensity of the "struggle for existence", or intraspecific competition, for a fixed quantity of resources. Because the population is in equilibrium P_t-1 = K_P when R_P = 0, then the equilibrium point can be determined by solving

Substituting 1/K_P for C_P/A_P in the R-function above gives

which is a discrete form of the famous logistic equation of population growth. The equilibrium point of the logistic equation, K_P, is sometimes called the carrying capacity of the environment because it specifies the maximum population that can be sustained indefinitely in a particular environment. Graduate students should be able to follow a more formal derivation of the logistic equation.

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