Global dynamics of the chemostat with different removal rates and variable yields

Tewfik Sari; Frederic Mazenc; Tewfik Sari; Frederic Mazenc

doi:10.3934/mbe.2011.8.827

Mathematical Biosciences and Engineering

2011, Volume 8, Issue 3: 827-840. doi: 10.3934/mbe.2011.8.827

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Global dynamics of the chemostat with different removal rates and variable yields

1.
Université de Haute Alsace, Mulhouse
2.
Projet INRIA DISCO, CNRS-SUPELEC, 3 Rue Joliot Curie, 91192, Gif-sur-Yvette

Received: 01 April 2010 Accepted: 29 June 2018 Published: 01 June 2011
MSC : Primary: 92A15, 92A17; Secondary: 34C15, 34C35.

In this paper, we consider a competition model between $n$ species in a chemostat including both monotone and non-monotone growth functions, distinct removal rates and variable yields. We show that only the species with the lowest break-even concentration survives, provided that additional technical conditions on the growth functions and yields are satisfied. We construct a Lyapunov function which reduces to the Lyapunov function used by S. B. Hsu [SIAM J. Appl. Math., 34 (1978), pp. 760-763] in the Monod case when the growth functions are of Michaelis-Menten type and the yields are constant. Various applications are given including linear, quadratic and cubic yields.
- Lyapunov function,
- competitive exclusion principle,
- Chemostat,
- global asymptotic stability,
- variable yield model.
Citation: Tewfik Sari, Frederic Mazenc. Global dynamics of the chemostat with different removal rates and variable yields[J]. Mathematical Biosciences and Engineering, 2011, 8(3): 827-840. doi: 10.3934/mbe.2011.8.827

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Abstract

In this paper, we consider a competition model between $n$ species in a chemostat including both monotone and non-monotone growth functions, distinct removal rates and variable yields. We show that only the species with the lowest break-even concentration survives, provided that additional technical conditions on the growth functions and yields are satisfied. We construct a Lyapunov function which reduces to the Lyapunov function used by S. B. Hsu [SIAM J. Appl. Math., 34 (1978), pp. 760-763] in the Monod case when the growth functions are of Michaelis-Menten type and the yields are constant. Various applications are given including linear, quadratic and cubic yields.
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