no. 13-14
Ordinary Differential Equations
A Lyapunov function for the chemostat with variable yields
[Une fonction de Lyapunov pour le chemostat avec des rendements variables]

Présenté par : Pierre-Louis Lions

Sari, Tewfik1, 2
1 EPI MERE INRIA-INRA, UMR MISTEA, 2, place Viala, 34060 Montpellier, France
2 Laboratoire de mathématiques, informatique et applications, université de Haute Alsace, 4, rue des frères Lumière, 68093 Mulhouse, France
Comptes Rendus. Mathématique, Tome 348 (2010) no. 13-14, pp. 747-751.

Dans cette Note on propose une nouvelle fonction de Lyapunov pour l'étude de la stabilité asymptotique globale dans un modèle mathématique de compétition entre esspèces dans le chemostat. Le modèle inclut des fonctions de croissance monotones ou non monotones, des taux de mortalité différents pour chaque espèce et des taux de rendement variables, fonctions de la concentration en substrat. On obtient, comme corollaires de notre résultat, trois théorèmes de stabilité globale qui ont été considérés dans la litérature.

In this Note, we give a global asymptotic stability result for the competition mathematical model between several species in a chemostat, by using a new Lyapunov function. The model includes both monotone and non-monotone response functions, distinct removal rates for the species and variable yields, depending on the concentration of substrate. We obtain, as corollaries of our result, three global stability theorems which were considered in the literature.

Reçu le :
Accepté le :
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DOI : 10.1016/j.crma.2010.06.008
Sari, Tewfik 1, 2

1 EPI MERE INRIA-INRA, UMR MISTEA, 2, place Viala, 34060 Montpellier, France
2 Laboratoire de mathématiques, informatique et applications, université de Haute Alsace, 4, rue des frères Lumière, 68093 Mulhouse, France 
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Sari, Tewfik. A Lyapunov function for the chemostat with variable yields. Comptes Rendus. Mathématique, Tome 348 (2010) no. 13-14, pp. 747-751. doi : 10.1016/j.crma.2010.06.008. http://www.numdam.org/articles/10.1016/j.crma.2010.06.008/

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