Comptes Rendus
no. 5
Partial Differential Equations
Models of population dynamics under the influence of external perturbations: mathematical results
[Analyse de modèles de dynamique de populations sous l'influence de perturbations externes]

Présenté par : Haïm Brezis

Mickaël D. Chekroun1 ; Lionel J. Roques2
1 Environmental Research and Teaching Institute, École normale supérieure, 24, rue Lhomond, Paris cedex 05, France
2 INRA, unité de biométrie, domaine Saint-Paul, site agroparc, 84914 Avignon cedex 9, France
Comptes Rendus. Mathématique, Volume 343 (2006) no. 5, pp. 307-310.

Cette Note a pour objet l'étude des états stationnaires et du comportement asymptotique d'équations de réaction-diffusion avec coefficients hétérogènes en espace, auxquelles nous ajoutons un terme de perturbation stationnaire ou périodique en temps. Nos résultats peuvent s'interpreter en termes de prélèvement maximal supportable par une population. Nous soulignons cet aspect à l'aide d'un calcul numérique.

In this note, we describe the stationary equilibria and the asymptotic behaviour of an heterogeneous logistic reaction-diffusion equation under the influence of autonomous or time-periodic forcing terms. We show that the study of the asymptotic behaviour in the time-periodic forcing case can be reduced to the autonomous one, the last one being described in function of the ‘size’ of the external perturbation. Our results can be interpreted in terms of maximal sustainable yields from populations. We briefly discuss this last aspect through a numerical computation.

Accepté le :
Publié le :
DOI : 10.1016/j.crma.2006.07.012
Mickaël D. Chekroun 1 ; Lionel J. Roques 2

1 Environmental Research and Teaching Institute, École normale supérieure, 24, rue Lhomond, Paris cedex 05, France
2 INRA, unité de biométrie, domaine Saint-Paul, site agroparc, 84914 Avignon cedex 9, France 
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Mickaël D. Chekroun; Lionel J. Roques. Models of population dynamics under the influence of external perturbations: mathematical results. Comptes Rendus. Mathématique, Volume 343 (2006) no. 5, pp. 307-310. doi : 10.1016/j.crma.2006.07.012. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2006.07.012/

[1] H. Berestycki; F. Hamel; L. Roques Analysis of the periodically fragmented environment model: I–Species persistence, J. Math. Biol., Volume 51 (2005) no. 1, pp. 75-113

[2] M. Chekroun, L. Roques, Spatialized harvesting models. The influence of seasonal variations, in preparation

[3] R.A. Fisher The advance of advantageous genes, Ann. Eugenics, Volume 7 (1937), pp. 335-369

[4] J.K. Hale; S.M. Verduyn Lunel Averaging in infinite dimensions, J. Integral Equations Appl., Volume 2 (1990) no. 4, pp. 463-494

[5] A. Pazy Semigroup of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag, 1983

[6] L. Roques, M. Chekroun, Harvesting models in heterogeneous environments, Preprint, 2006

[7] G.R. Sell; Y. You Dynamics of Evolutionary Equations, Springer-Verlag, 2002

[8] N. Shigesada; K. Kawasaki; E. Teramoto Traveling periodic waves in heterogeneous environments, Theor. Population Biol., Volume 30 (1986), pp. 143-160

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