Competitive exclusion for chemostat equations with variable yields

TL;DR

This paper analyzes the global dynamics of a chemostat model with variable yields, extending previous work by constructing Lyapunov functions to study stability and exclude periodic orbits.

Contribution

It introduces Lyapunov functions tailored for chemostat models with variable yields, generalizing prior methods that assumed proportional growth rates.

Findings

Proves global stability of equilibrium points.

Extends Lyapunov function techniques to more general chemostat models.

Provides conditions for the nonexistence of periodic orbits.

Abstract

In this paper, we study the global dynamics of a chemostat model with a single nutrient and several competing species. Growth rates are not required to be proportional to food uptakes. The model was studied by Fiedler and Hsu [J. Math. Biol. (2009) 59:233-253]. These authors prove the nonexistence of periodic orbits, by means of a multi-dimensional Bendixon-Dulac criterion. Our approach is based on the construction of Lyapunov functions. The Lyapunov functions extend those used by Hsu [SIAM J. Appl. Math. (1978) 34:760-763] and by Wolkowicz and Lu [SIAM J. Appl. Math. (1997) 57:1019-1043] in the case when growth rates are proportional to food uptakes.