A discrete size-structured competition model.

Matrix difference equations have been used to model the discrete time dynamics of a variety of populations whose individual members have been categorized into a finite number of classes based on, for example, age, size, or stage. Examples of such models include Leslie's age-structured model and the Usher model, a size-classified model which has been applied to trees, corals, sea turtles, copepods, and fish. These matrix difference equations can incorporate virtually any type of nonlinearity arising from the density dependence of fertility and survival rates and transition probabilities between classes. Under a fairly general set of assumptions, it can be shown that the normalized class distribution vector equilibrates, and thus an asymptotic or limiting equation for total population size can be derived. In this research we assume the existence of a dynamically modeled resource in limited supply for which the members of the species compete, either exploitatively or through interference. The existence and stability of population size equlibria or cycles is then studied by means of bifurcation theory. Several biological considerations are addressed, including the Size-Efficiency Hypothesis of Brooks and Dodson, the effects of changes in individual physiological parameters on the size and competitive success of a species, and the effects of delays on the viability of a species.