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
CRISAM - Inria Sophia Antipolis - Méditerranée , MISTEA - Mathématiques, Informatique et STatistique pour l'Environnement et l'Agronomie
2 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.
Keywords : Lyapunov function global stability positive systems competition chemostat
Document type :
Journal articles
Complete list of metadatas
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

File

polytopicdef.pdf

Identifiers

Collections

INRIA | INRA | INSMI | SUPELEC | UPMC | INSU | SUP_LSS | AGROPOLIS | L2S | TDS-MACS | SORBONNE-UNIVERSITE | LOV | SU-SCIENCES | UMS-829

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⟩

Export

BibTeX TEI DC DCterms EndNote

Share

Metrics

Record views

660

Files downloads

402

Contact


archive-ouverte@inria.fr