Optimal Resource Consumption, Discrete Dynamics, and Intraspecific Competition

The relation of a single species population to a limiting resource (food) is considered. A notion of optimal consumption is defined whereby an average individual minimizes the net cost (in terms of loss of fitness) of securing sufficient resource for growth and reproduction. A corresponding optimal consumption dynamics for the species/resource interaction is obtained. It is shown that controlled, periodic resource renewal (such as in the experiments with flour beetles of Park et al., 1964) can prevent population crash, and determines both a discrete dynamics (corresponding to population numbers at renewal times) and an optimal continuous dynamics interpolating the values of the discrete dynamics. Furthermore, under mild hypotheses, any single species discrete population dynamics may be construed as arising in this way. Other types of resource renewal are also considered. Specific assumptions about the interaction between the rate of resource renewal, the (optimal) per capita consumption rate, and the cost of maintaining this rate in the face of competition are considered, and a continuum of optimal consumption strategies is delineated. The extremes of this continuum are discussed in relation to dynamic stability, life-history strategies, and the influence of intraspecific competition. Finally, the models constructed here are used to consider the ′stability question": Why do most natural populations appear to be stable?