Global stability of perturbed chemostat systems

Abstract

This paper is devoted to the analysis of the global stability of the chemostat system with a perturbation term representing a general form of exchange between species. This conversion term depends not only on species and substrate concentrations, but also on a positive perturbation parameter. After expressing the invariant manifold as a union of a family of compact subsets, our main result states that for each subset in this family, there exists a positive threshold for the perturbation parameter below which the system is globally asymptotically stable in the corresponding subset. Our approach relies on the Malkin-Gorshin Theorem and on a Theorem by Smith and Waltman concerning perturbations of a globally stable steady-state. Properties of the steady-states and numerical simulations of the system’s asymptotic behavior complement this study for two types of perturbation terms between the species.