Probabilistic and deterministic analysis of the evolution : influence of a spatial structure and a mating preference.

Abstract : We study the spatial and evolutionary dynamics of a population by using probabilistic and deterministic tools. In the first part of this thesis, we are concerned with the influence of a heterogeneous environment on the evolution of species. The population is modeled by an individual-based process with some interactions and which describes the birth, the death, the mutation and the spatial diffusion of each individual. The rates of those events depend on the characteristics of the individuals : their phenotypic trait and their spatial location. First, we study the system of partial differential equations that describes the spatial and demographic dynamics of a population composed of two traits in a large population limit. We characterize precisely the conditions of extinction and long time survival for this population. Secondly, we study the initial individual-based model under two asymptotic: large population and rare mutations such as demographic and mutational timescales are separated. Thus, when a mutant appears, the resident population has reached its demographic balance. We characterize the survival probability of the population descended from this mutant. Then, by studyingthe process in the mutational scale, we prove that the microscopic process converges to a jump process which describes the successive fixations of the most advantaged traits and the spatial distribution of populations carrying these traits. We then extend the model to introduce mutualistic interactions between two species. We study this model in a limit of large population. We also give numerical results and a detailed biological behavior analysis around two issues: the co-evolution of phenotypic and spatial niches of mutualistic species and the invasion dynamics of a homogeneous space by these species. In the second part of this thesis, we develop a probabilistic model to study the effect of the sexual preference on the speciation. Here, the population is structured on two patches and the individuals, characterized by a trait, are ecologically and demographically similar and differ only in their sexual preferences: two individuals of the same trait are more likely to reproduce than two individuals of distinct traits. We show that in the absence of any other ecological differences, the sexual preferences lead to reproductive isolation between the two patches.
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Submitted on : Sunday, February 26, 2017 - 4:36:07 PM
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Hélène Leman. Probabilistic and deterministic analysis of the evolution : influence of a spatial structure and a mating preference.. Probability [math.PR]. Université Paris-Saclay, 2016. English. ⟨NNT : 2016SACLX026⟩. ⟨tel-01389668v2⟩

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