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

TL;DR

This paper studies the long-term behavior of population models with traits, using Hamilton-Jacobi equations to analyze survival and growth under various branching regimes and assumptions.

Contribution

It extends Hamilton-Jacobi methods to more general trait spaces and weaker assumptions, linking stochastic and deterministic dynamics in population models.

Findings

Characterized survival sets using Hamilton-Jacobi equations.

Proved convergence to classical Hamilton-Jacobi equations in super-critical regimes.

Established asymptotic equivalence of stochastic and deterministic models.

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.