Persistance et vitesse d'extinction pour des modèles de populations stochastiques multitypes en temps discret.

Abstract : This thesis is devoted to the mathematical study of stochastic modelds of structured populations dynamics.In the first chapter, we introduce a discrete time stochastic process taking into account various ecological interactions between individuals, such as competition, migration, mutation, or predation. We first prove a ``law of large numbers'': where we show that if the initial population tends to infinity, then, on any finite interval of time, the stochastic process converges in probability to an underlying deterministic process. We also quantify the discrepancy between these two processes by a kind of ``central limit theorem''. Finally, we give a criterion of persistence/extinction in order to determine the long time behavior of the process. This criterion highlights a critical case which will be studied in more detail in the following chapters.In the second chapter, we give a criterion for the possible unlimited growth in the critical case mentioned above. We apply this criterion to the example of a source-sink metapopulation with two patches of type source, textit{i.e.} the population of each patch goes to extinction if we do not take into account the migration. We prove that there is a possible survival of the metapopulation.In the third chapter, we focus on the behavior of our critical process when it tends to infinity. We prove a convergence in distribution of the scaled process to a gamma distribution, and in a more general framework, by also rescaling time, we obtain a distribution limit of a function of our process to the solution of a stochastic differential equation called a squared Bessel process.In the fourth and last chapter, we study hitting times of some compact sets when our process does not tend to infinity. We give nearly optimal bounds for the tail of these hitting times. If the process goes to extinction almost surely, we deduce from these bounds precise estimates of the tail of the extinction time. Moreover, if the process is a Markov chain, we give a criterion of null recurrence or positive recurrence and in the latter case, we obtain a subgeometric convergence of its transition kernel to its invariant probability measure.
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Etienne Adam. Persistance et vitesse d'extinction pour des modèles de populations stochastiques multitypes en temps discret.. Probabilités [math.PR]. Université Paris-Saclay, 2016. Français. ⟨NNT : 2016SACLX019⟩. ⟨tel-01476864⟩

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