Stability of the chemostat system with a mutation factor

TL;DR

This paper analyzes a chemostat model incorporating mutation effects, demonstrating the existence, stability, and persistence of steady-states, and providing insights into how mutation influences system dynamics and stability.

Contribution

It introduces a mutation-augmented chemostat model, proving stability results and steady-state expansions, which are novel in understanding mutation impacts on such systems.

Findings

Unique locally stable steady-state for all mutation and dilution rates below a threshold

Steady-state expansion in terms of mutation rate

Global asymptotic stability for small dilution rates

Abstract

In this paper, we consider a resource-consumer model taking into account a mutation effect between species (with constant mutation rate). The corresponding mutation operator is a discretization of the Laplacian in such a way that the resulting dynamical system can be viewed as a regular perturbation of the classical chemostat system. We prove the existence of a unique locally stable steady-state for every value of the mutation rate and every value of the dilution rate not exceeding a critical value. In addition, we give an expansion of the steady-state in terms of the mutation rate and we prove a uniform persistence property of the dynamics related to each species. Finally, we show that this equilibrium is globally asymptotically stable for every value of the mutation rate provided that the dilution rate is with small enough values.