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arXiv · 2602.20825

Asymptotic behavior of some stochastic models in population dynamics: a Hamilton-Jacobi approach

Anouar Jeddi
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Abstract

In this paper, we investigate the asymptotic behavior of individual-based models describing the evolution of a population structured by a real trait, subject to selection and mutation. We consider two different sets of assumptions: first, the case of critical or subcritical branching population processes in a regime combining a discretization of the trait space, small mutations, large time and large initial population size, where we are able to characterize using a Hamilton-Jacobi approach, the survival set of the population, and the asymptotic of the logarithmic scaling of subpopulation sizes. Second, we generalize by a direct method the convergence to the classical Hamilton-Jacobi equation obtained in the super-critical branching regime considered in \cite{CMMT} to a more general trait space and under weaker assumptions. Moreover, we establish that the stochastic and the deterministic dynamics are asymptotically equivalent in large population.

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Anouar Jeddi. 2026-02-24. Asymptotic behavior of some stochastic models in population dynamics: a Hamilton-Jacobi approach. https://arxiv.org/abs/2602.20825

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