Stochastic and deterministic models for age-structured populations with genetically variable traits

Régis Ferrière^1, 2 ; Viet Chi Tran³

¹ Laboratoire Ecologie & Evolution, CNRS UMR 7625, Ecole Normale Supérieure, 46 rue d'Ulm, 75230 Paris Cedex 05, France
² Department of Ecology and Evolutionary Biology, University of Arizona, Tucson AZ 85721, USA
³ Laboratoire Paul Painlevé, UFR de Mathématiques, Université des Sciences et Techniques Lille 1, 59655 Villeneuve d'Ascq Cedex, France.

ESAIM. Proceedings, Tome 27 (2009), pp. 289-310

Résumé

Understanding how stochastic and non-linear deterministic processes interact is a major challenge in population dynamics theory. After a short review, we introduce a stochastic individual-centered particle model to describe the evolution in continuous time of a population with (continuous) age and trait structures. The individuals reproduce asexually, age, interact and die. The 'trait' is an individual heritable property (d-dimensional vector) that may influence birth and death rates and interactions between individuals, and vary by mutation. In a large population limit, the random process converges to the solution of a Gurtin-McCamy type PDE. We show that the random model has a long time behavior that differs from its deterministic limit. However, the results on the limiting PDE and large deviation techniques à la Freidlin-Wentzell provide estimates of the extinction time and a better understanding of the long time behavior of the stochastic process. This has applications to the theory of adaptive dynamics used in evolutionary biology. We present simulations for two biological problems involving life-history trait evolution when body size is plastic and individual growth is taken into account.

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DOI : 10.1051/proc/2009033