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Champagnat, Nicolas; Méléard, Sylvie; Mirrahimi, Sepideh; Tran, Viet Chi

Filling the gap between individual-based evolutionary models and Hamilton-Jacobi equations. (Des modèles stochastiques d’évolution aux équations de Hamilton-Jacobi.) (English. French summary) Zbl 1526.92035

J. Éc. Polytech., Math. 10, 1247-1275 (2023).
Summary: We consider a stochastic model for the evolution of a discrete population structured by a trait with values on a finite grid of the torus, and with mutation and selection. We focus on a parameter scaling where population is large, individual mutations are small but not rare, and the grid mesh is much smaller than the size of mutation steps. When considering the evolution of the population in long time scales, the contribution of small sub-populations may strongly influence the dynamics. Our main result quantifies the asymptotic dynamics of subpopulation sizes on a logarithmic scale. We establish that under our rescaling, the stochastic discrete process converges to the viscosity solution of a Hamilton-Jacobi equation. The proof makes use of almost sure maximum principles and careful control of the martingale parts.

MSC:

92D15 Problems related to evolution
92D25 Population dynamics (general)
60J85 Applications of branching processes
35F21 Hamilton-Jacobi equations
35D40 Viscosity solutions to PDEs

Keywords:

stochastic birth death models; large population approximation; selection; mutation; viscosity solution; maximum principle; Hamilton-Jacobi equation
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References:

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