Global structure of a food chain model with a single mutation in chemostat

Abstract

In this paper we study a food chain system with four reaction–diffusion equations arising from competition of two predators with a single mutation for a single prey in an unstirred chemostat. Due to the introduction of mutation and death of predators, the conservation principle for a standard chemostat model does not hold. In order to better explore the mutation of species we assume that the initial condition for mutant species is zero throughout this paper. We firstly investigate the uniform persistence of such system by the persistence theory. When the reverse rate of mutant species is zero, we study the structure and stability of nonnegative equilibria of the food chain system. The results show that the coexistence occur with small diffusion and small reverse rate of mutant species. Biologically speaking, it implies that small diffusion and small reverse rate of mutant species are sufficiently effective in the survival of mutant species. Finally, our numerical results verified our theoretical results and showed that the mutant species can survive when it has smaller random dispersal rate than the wild-type species.