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July  2007, 8(1): 73-93. doi: 10.3934/dcdsb.2007.8.73

Polytopic Lyapunov functions for persistence analysis of competing species

Frédéric Grognard 1, , Frédéric Mazenc 2, and  Alain Rapaport 2,

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  October 2005 Revised  March 2006 Published  April 2007

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.
Keywords: global stability, chemostat., competition, positive systems, Lyapunov function.
Mathematics Subject Classification: 92D25, 34D23, 93D3.
Citation: Frédéric Grognard, Frédéric Mazenc, Alain Rapaport. Polytopic Lyapunov functions for persistence analysis of competing species. Discrete and Continuous Dynamical Systems - B, 2007, 8 (1) : 73-93. doi: 10.3934/dcdsb.2007.8.73
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