Size-structured competition models with periodic coefficients.

Abstract

A system of ordinary differential equations describing the dynamics of two size structured species competing for a single unstructured resource in a chemostat is derived, under the assumption that all physiological parameters of the species are periodic functions of time. The existence of nontrivial periodic solutions of those systems is considered. Under fairly general conditions, using global bifurcation techniques, we show that a continuum of solutions bifurcate from a noncritical periodic solution of the reduced system. The positivity and stability of the bifurcating branch solutions are studied. Application to mathematical ecology is given by considering several specific cases where all system parameters are constant except some selected parameters which are taken to be small amplitude oscillations around a positive mean value. Those specific cases are studied both analytically and numerically. It is also shown, that the average size of individuals of each species is governed by a nonautonomous logistic equation whose solution under fairly general conditions approaches the unique positive periodic solution of the classical periodic logistic equation. The relationship between competitive success, coexisting species and species size is discussed. Some biological considerations are addressed including the Size-Efficiency-Hypothesis, the effects of seasonality and food limitation on the coexistence of both species.