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}

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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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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Térence Bayen; Henri Cazenave-Lacroutz; Jérôme Coville Stability of the chemostat system including a linear coupling between species, Discrete and Continuous Dynamical Systems. Series B, Volume 28 (2023) no. 3, pp. 2104-2129 | DOI:10.3934/dcdsb.2022160 | Zbl:1508.34048
Wen Liu; Yanling Li Positive solution and stability of the unstirred chemostat with variable yield, Mathematical Methods in the Applied Sciences, Volume 44 (2021) no. 14, pp. 11754-11762 | DOI:10.1002/mma.7376 | Zbl:1473.35337
R. Fekih-Salem; Alain Rapaport; T. Sari Emergence of coexistence and limit cycles in the chemostat model with flocculation for a general class of functional responses, Applied Mathematical Modelling, Volume 40 (2016) no. 17-18, pp. 7656-7677 | DOI:10.1016/j.apm.2016.03.028 | Zbl:1471.92223
Nahla Abdellatif; Radhouane Fekih-Salem; Tewfik Sari Competition for a single resource and coexistence of several species in the chemostat, Mathematical Biosciences and Engineering, Volume 13 (2016) no. 4, pp. 631-652 | DOI:10.3934/mbe.2016012 | Zbl:1352.92115
Rubayyi T. Alqahtani; Mark I. Nelson; Annette L. Worthy Analysis of a Chemostat Model with Variable Yield Coefficient and Substrate Inhibition: Contois Growth Kinetics, Chemical Engineering Communications, Volume 202 (2015) no. 3, p. 332 | DOI:10.1080/00986445.2013.836630
Tewfik Sari Competitive exclusion for chemostat equations with variable yields, Acta Applicandae Mathematicae, Volume 123 (2013) no. 1, pp. 201-219 | DOI:10.1007/s10440-012-9761-8 | Zbl:1402.92433
M. I. Nelson; T. Nicholls; N. Hamzah A biological process subject to noncompetitive substrate inhibition in a generalized flow reactor, The ANZIAM Journal, Volume 54 (2013) no. 4, pp. 273-290 | DOI:10.1017/s1446181113000126 | Zbl:1275.92027
Radhouane Fekih Salem; Tewfik Sari; Nahla Abdellatif On a model of competition and coexistence in the chemostat, ARIMA. Revue Africaine de la Recherche en Informatique et Mathématiques Appliqués, Volume 14 (2011), pp. 15-30 | DOI:10.46298/arima.1953 | Zbl:8019134
Global dynamics of the chemostat with different removal rates and variable yields, Mathematical Biosciences and Engineering, Volume 8 (2011) no. 3, p. 827 | DOI:10.3934/mbe.2011.8.827

Cité par 9 documents. Sources : Crossref, zbMATH

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