Polytopic Lyapunov functions for persistence analysis of competing species

Frédéric Grognard; Frédéric Mazenc; Alain Rapaport

doi:10.3934/dcdsb.2007.8.73

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2007, Volume 8, Issue 1: 73-93. Doi: 10.3934/dcdsb.2007.8.73

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Polytopic Lyapunov functions for persistence analysis of competing species

1.
INRIA Sophia-Antipolis, COMORE Project-team, 2004 route des lucioles, BP 93, 06902 Sophia-Antipolis Cedex
2.
INRA-INRIA, MERE Project-team, UMR Analyse des systemès et biométrie, 2, place Viala, 34060 Montpellier, France

Received: September 30, 2005

Revised: February 28, 2006

Published: December 2006

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Abstract

We show that stability of the equilibrium of a family of interconnected scalar systems can be proved by using a sum of monotonic $C^0$ functions as a 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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Mathematics Subject Classification: 92D25, 34D23, 93D30.

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