# Polytopic Lyapunov functions for persistence analysis of competing species

1 MODEMIC - Modelling and Optimisation of the Dynamics of Ecosystems with MICro-organisme
CRISAM - Inria Sophia Antipolis - Méditerranée , MISTEA - Mathématiques, Informatique et STatistique pour l'Environnement et l'Agronomie
3 COMORE - Modeling and control of renewable resources
LOV - Laboratoire d'océanographie de Villefranche, CRISAM - Inria Sophia Antipolis - Méditerranée
4 DISCO - Dynamical Interconnected Systems in COmplex Environments
L2S - Laboratoire des signaux et systèmes, Inria Saclay - Ile de France, SUPELEC, CNRS - Centre National de la Recherche Scientifique : UMR8506
Abstract : We show that stability of the equilibrium of a family of interconnected scalar systems can be proved by using a sum of monotonic ${\mathcal C}^0$ functions as Lyapunov function. We prove this result in the general framework of nonlinear systems and then in the special case of Kolmogorov systems. As an application, it is then used to show that intra-specific competition can explain coexistence of several species in a chemostat where they compete for a single substrate. This invalidates the Competitive Exclusion Principle, that states that in the classical case (without this intra-specific competition), it is indeed known that only one of the species will survive.
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https://hal.inria.fr/inria-00000873
Contributor : Frédéric Grognard <>
Submitted on : Tuesday, November 29, 2005 - 12:22:29 PM
Last modification on : Saturday, May 25, 2019 - 1:43:02 AM
Long-term archiving on : Friday, September 14, 2012 - 4:06:09 PM

### Citation

Alain Rapaport, Frédéric Grognard, Frédéric Mazenc. Polytopic Lyapunov functions for persistence analysis of competing species. Discrete and Continuous Dynamical Systems - Series B, American Institute of Mathematical Sciences, 2007, 8 (1), pp.73-93. ⟨10.3934/dcdsb.2007.8.73⟩. ⟨inria-00000873⟩

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