Ph.D. Thesis

Ph.D. Thesis on applied mathematics, under the direction of Sylvie Méléard (Paris X) and the biological co-direction of Régis Ferrière (ENS Paris), defended on the 6th december 2004 in the University of Nanterre (Paris 10).

Summary:

This thesis is interested in the probabilistic study of ecological models belonging to the recent theory of adaptive dynamics. After having presented and generalized the scope and the biological heuristics of these models, we obtain a microscopic justification of a jump process modelizing evolution from a measure-valued interacting particle system describing the population dynamics at the individual level. This is a time scale separation result based on two asymptotics: rare mutations and large population. Then, we obtain an ordinary differential equation known as the canonical equation of adaptive dynamics by applying an asymptotic of small jumps to the preceding process. This limit leads us to introduce a diffusion model of evolution as a diffusion approximation of the jump process, which coefficients present bad regularity properties: discontinuous drift and degenerate diffusion parameter at the same points. We then study the weak existence, uniqueness in law and strong Markov property for this process, which are linked to the question whether this diffusion can reach particular isolated points of the space in finite time or not. Finally, we prove a large deviations principle for these degenerate diffusions, allowing to study the problem of diffusion exit from an attracting domain, which is a fundamental biological question.

Key-words:

models of evolution; adaptives dynamics; interacting particle systems; measure-valued processes; time scale separation; convergence; jump processes; degenerate diffusions; large deviations; problem of diffusion exit from a domain; polarity of isolated points; weak existence; uniqueness in law; strong Markov property.