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

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

2008-09-22

arxiv.org/abs/0809.3767v2

ESAIM: Proceedings 27 (2009) 289--310

Mathematics

Probability

This work is a proceeding of the CANUM 2008 conference

Scientific paper

10.1051/proc/2009033

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 \textit{\`a 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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