Size Structured Competition Model with Periodic Coefficients

A model of Diekmann et al. [14] for two size-structured populations competing for a single resource is extended in this paper to allow for periodic variation in parameter values (having a common period). The hyperbolic partial differential equation model is reduced to systems of ordinary differential equations describing the dynamic of two size-structured species competing for a single unstructured resource. The existence of nontrivial periodic solutions of those systems is considered. Under fairly general conditions, global bifurcation techniques as used by Gushing are used to show the existence of a continuum of solutions that bifurcate from a noncritical periodic solution of the reduced single species system. The positivity and the stability of the bifurcating branch solutions are studied. The stability is shown to depend on the stability of the solutions of the reduced system and the direction of bifurcation. The results and techniques can be generalized to the case of n, n ≥ 1, size-structured species competing for a single resource. 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 periodic solution of the classical periodic logistic equation.