Comptes Rendus
no. 23-24
Ordinary Differential Equations/Dynamical Systems
Convergence to equilibrium in competitive Lotka–Volterra and chemostat systems
[Convergence vers l'équilibre pour des systèmes compétitifs de Lotka–Volterra et du Chémostat]

Présenté par : Gérard Iooss

Nicolas Champagnat1 ; Pierre-Emmanuel Jabin1, 2 ; Gaël Raoul3
1 TOSCA project-team, INRIA Sophia Antipolis – Méditerranée, 2004 rte des Lucioles, B.P. 93, 06902 Sophia Antipolis Cedex, France
2 Laboratoire J.-A. Dieudonné, Université de Nice – Sophia Antipolis, Parc Valrose, 06108 Nice Cedex 02, France
3 DAMTP, CMS, University of Cambridge, Wilberforce Road, Cambridge CB3 0WA, United Kingdom
Comptes Rendus. Mathématique, Volume 348 (2010) no. 23-24, pp. 1267-1272.

Nous étudions un système généralisé d'équations différentielles modélisant un nombre fini de populations biologiques en interaction compétitive. En adaptant les techniques de Jabin et Raoul [8] et de Champagnat et Jabin (2010) [2], nous prouvons la convergence vers un unique équilibre stable.

We study a generalized system of ODE's modeling a finite number of biological populations in a competitive interaction. We adapt the techniques in Jabin and Raoul [8] and Champagnat and Jabin (2010) [2] to prove the convergence to a unique stable equilibrium.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2010.11.001
Nicolas Champagnat 1 ; Pierre-Emmanuel Jabin 1, 2 ; Gaël Raoul 3

1 TOSCA project-team, INRIA Sophia Antipolis – Méditerranée, 2004 rte des Lucioles, B.P. 93, 06902 Sophia Antipolis Cedex, France
2 Laboratoire J.-A. Dieudonné, Université de Nice – Sophia Antipolis, Parc Valrose, 06108 Nice Cedex 02, France
3 DAMTP, CMS, University of Cambridge, Wilberforce Road, Cambridge CB3 0WA, United Kingdom 
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     author = {Nicolas Champagnat and Pierre-Emmanuel Jabin and Ga\"el Raoul},
     title = {Convergence to equilibrium in competitive {Lotka{\textendash}Volterra} and chemostat systems},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {1267--1272},
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     doi = {10.1016/j.crma.2010.11.001},
     language = {en},
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Nicolas Champagnat; Pierre-Emmanuel Jabin; Gaël Raoul. Convergence to equilibrium in competitive Lotka–Volterra and chemostat systems. Comptes Rendus. Mathématique, Volume 348 (2010) no. 23-24, pp. 1267-1272. doi : 10.1016/j.crma.2010.11.001. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2010.11.001/

[1] N. Champagnat, S. Méléard, Polymorphic evolution sequence and evolutionary branching, Probab. Theor. Relat. Fields (2010), , in press. | DOI

[2] N. Champagnat, P.E. Jabin, The evolutionary limit for models of populations interacting competitively with many resources, preprint, 2010.

[3] O. Diekmann A beginner's guide to adaptive dynamics, Banach Center Publications, Volume 63 (2004), pp. 47-86

[4] O. Diekmann; P.E. Jabin; S. Mischler; B. Perthame The dynamics of adaptation: An illuminating example and a Hamilton–Jacobi approach, Theor. Popul. Biol., Volume 67 (2005), pp. 257-271

[5] K. Gopalsamy Global asymptotic stability in Volterra's population systems, J. Math. Biology, Volume 19 (1984), pp. 157-168

[6] M.W. Hirsch Systems of differential equations which are competitive or cooperative. III. Competing species, Nonlinearity, Volume 1 (1988) no. 1, pp. 51-71

[7] J. Hofbauer; K. Sigmund Evolutionary Games and Population Dynamics, Cambridge University Press, Cambridge, 1998

[8] P.E. Jabin, G. Raoul, Selection dynamics with competition, J. Math. Biol., in press.

[9] K. Krisztina; S. Kovács Qualitative behavior of n-dimensional ratio-dependent predator–prey systems, Appl. Math. Comput., Volume 199 (2008) no. 2, pp. 535-546

[10] J.A.J. Metz; S.A.H. Geritz; G. Meszéna; F.A.J. Jacobs; J.S. van Heerwaasden Adaptive dynamics: a geometrical study of the consequences of nearly faithful reproduction (S.J. van Strien; S.M. Verduyn Lunel, eds.), Stochastic and Spatial Structures of Dynamical Systems, North-Holland, Amsterdam, 1996, pp. 183-231

[11] B. Perthame Transport Equations in Biology, Frontiers in Mathematics, Birkhäuser, 2007

[12] H.L. Smith; P. Waltman The Theory of the Chemostat, Dynamics of Microbial Competition, Cambridge Studies in Mathematical Biology, vol. 13, Cambridge University Press, 1995

[13] M.L. Zeeman Hopf bifurcations in competitive three-dimensional Lotka–Volterra systems, Dynam. Stability Systems, Volume 8 (1993) no. 3, pp. 189-217

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