A discrete nonlinear model of state-structured populations.

Abstract

We consider in this dissertation a general class of nonlinear matrix models for stage-structured populations competing for a single (unstructured) resource. Individuals in such a population are characterized by a scalar variable (e.g., age, size) and are classified into a finite number of classes. The models are treated with sufficient generality so that transitions between any two classes are permitted (the sole constraint being that all newborns lie in the same class). The nonlinearities, which arise from the density dependence of fertility rates, survival rates and transition probabilities between classes, are introduced by assuming that the class-specific birth rates and survival rates are functions of the total population density and the number of individuals in higher or lower ranking classes. This assumption enables us to derive a scalar difference equation for the total population size. The proposed models are used to study intra-competition and intra-predation (cannibalism) populations with constant or varying resources. In both cases, we study the existence and stability of equilibria for the total population size when by means the bifurcation parameter π, known as the inherent net reproduction value, varies. The intra-specific competition models include contest and scramble competitions as two extreme cases. We show that contest competition is always "more stable" than scramble competition in the sense that it yields higher equilibrium levels and larger stability regions (all other factors being identical). We also show, under certain restrictions, that near π = 1 scramble competition has higher equilibrium resilience than contest competition. The size-structured model for the dynamics of a cannibalistic population is derived under the assumption that cannibals attack only smaller victims, as is generally the case in the biological world. By incorporating the positive-negative feedback mechanism resulting from cannibalism, our analysis yields many dynamical features that have been attributed to cannibalism in the literature, including density self-regulation, a "life-boat strategy" phenomenon and multiple stable positive equilibrium states and hysteresis.