A Lyapunov function for the chemostat with variable yields

Tewfik Sari

doi:10.1016/j.crma.2010.06.008

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

Tewfik Sari ^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

Comptes Rendus. Mathématique, Volume 348 (2010) no. 13-14, pp. 747-751.

Résumé
Abstract

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 : 2010-03-01
Accepté le : 2010-06-03
Publié le : 2010-06-23

DOI : 10.1016/j.crma.2010.06.008

Affiliations des auteurs :

Tewfik Sari ^{1, 2}

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     year = {2010},
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     language = {en},
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Tewfik Sari. A Lyapunov function for the chemostat with variable yields. Comptes Rendus. Mathématique, Volume 348 (2010) no. 13-14, pp. 747-751. doi : 10.1016/j.crma.2010.06.008. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2010.06.008/

Bibliographie
Cité par

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[2] G.J. Butler; G.S.K. Wolkowicz A mathematical model of the chemostat with a general class of functions describing nutrient uptake, SIAM Journal on Applied Mathematics, Volume 45 (1985), pp. 138-151

[3] S.B. Hsu Limiting behavior for competing species, SIAM Journal on Applied Mathematics, Volume 34 (1978), pp. 760-763

[4] S.B. Hsu; S.P. Hubbell; P. Waltman A mathematical theory for single nutrient competition in continuous culture of micro-organisms, SIAM Journal on Applied Mathematics, Volume 32 (1977), pp. 366-383

[5] P. de Leenheer; B. Li; H.L. Smith Competition in the chemostat: Some remarks, Canadian Applied Mathematics Quarterly, Volume 11 (2003) no. 3, pp. 229-248

[6] B. Li Global asymptotic behavior of the chemostat: General response functions and differential removal rates, SIAM Journal on Applied Mathematics, Volume 59 (1998), pp. 411-422

[7] J. Monod La technique de culture continue. Théorie et applications, Ann. Inst. Pasteur, Volume 79 (1950), pp. 390-410

[8] S.S. Pilyugin; P. Waltman Multiple limit cycles in the chemostat with variable yields, Mathematical Biosciences, Volume 182 (2003), pp. 151-166

[9] T. Sari Global dynamics of the chemostat with variable yields, 2010 | HAL

[10] T. Sari; F. Mazenc Global dynamics of the chemostat with different removal rates and variable yields, 2009 | HAL

[11] H.L. Smith; P. Waltman The Theory of the Chemostat, Dynamics of Microbial Competition, Cambridge University Press, 1995

[12] G.S.K. Wolkowicz; Z. Lu Global dynamics of a mathematical model of competition in the chemostat: General response functions and differential death rates, SIAM Journal on Applied Mathematics, Volume 52 (1992), pp. 222-233

[13] G.S.K. Wolkowicz; H. Xia Global asymptotic behavior of a chemostat model with discrete delays, SIAM Journal on Applied Mathematics, Volume 57 (1997), pp. 1019-1043

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