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Polytopic Lyapunov functions for persistence analysis of competing species

Alain Rapaport 1, 2 Frédéric Grognard 3 Frédéric Mazenc 4
1 MODEMIC - Modelling and Optimisation of the Dynamics of Ecosystems with MICro-organisme
MISTEA - Mathématiques, Informatique et STatistique pour l'Environnement et l'Agronomie, CRISAM - Inria Sophia Antipolis - Méditerranée
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
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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Contributor : Frédéric Grognard Connect in order to contact the contributor
Submitted on : Tuesday, November 29, 2005 - 12:22:29 PM
Last modification on : Wednesday, March 23, 2022 - 12:08:27 PM
Long-term archiving on: : Friday, September 14, 2012 - 4:06:09 PM



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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