Global stability of perturbed chemostat systems

TL;DR

This paper investigates the global stability of chemostat systems with perturbations, establishing conditions under which the system remains stable despite exchange interactions between species.

Contribution

It introduces a novel analysis of chemostat stability considering perturbations depending on species, substrate, and a positive parameter, extending existing stability results.

Findings

Existence of a positive threshold for perturbation parameter ensuring stability

Application of Malkin-Gorshin and Smith-Waltman theorems to stability analysis

Numerical simulations confirming theoretical stability conditions

Abstract

This paper is devoted to the analysis of global stability of the chemostat system with a perturbation term representing any type of exchange between species. This conversion term depends on species and substrate concentrations but also on a positive perturbation parameter. After having written the invariant manifold as a union of a family of compact subsets, our main result states that for each subset in this family, there is 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 about perturbations of a globally stable steady state. Properties of steady-states and numerical simulations of the system's asymptotic behavior complete this study for two types of perturbation term between species.