Global stability of perturbed chemostat systems

Claudia Alvarez-Latuz; Terence Bayen; Jerome Coville

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arXiv · 2501.08011

Global stability of perturbed chemostat systems

Claudia Alvarez-Latuz·Terence Bayen·Jerome Coville

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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.

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Claudia Alvarez-Latuz, Terence Bayen, Jerome Coville. 2025-01-14. Global stability of perturbed chemostat systems. https://arxiv.org/abs/2501.08011

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