A Stochastic Adaptive Dynamics Model for Bacterial Populations with Mutation, Dormancy and Transfer

Blath, Jochen; Paul, Tobias; Tóbiás, András

doi:10.48550/arxiv.2105.09228

Cited by 1 publication

(1 citation statement)

References 21 publications

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“…Scaling limits of individual-based models on a discrete trait space with rare mutations and large population, and allowing to deal with negligible populations and local extinction, were proposed in [13,4,9,10,3]. These references focus on population sizes of the order of K β and characterize the asymptotic dynamics of the exponent β.…”

Section: Introduction and Presentation Of The Modelmentioning

confidence: 99%

Filling the gap between individual-based evolutionary models and Hamilton-Jacobi equations

Champagnat¹,

Mirrahimi²,

Tran³

2022

Preprint

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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. Traits are vertically inherited unless a mutation occurs, and influence the birth and death rates. We focus on a parameter scaling where population is large, individual mutations are small but not rare, and the grid mesh for the trait values is much smaller than the size of mutation steps. When considering the evolution of the population in a long time scale, the contribution of small sub-populations may strongly influence the dynamics. Our main result quantifies the asymptotic dynamics of sub-population sizes on a logarithmic scale. We establish that under the parameter scaling the logarithm of the stochastic population size process, conveniently normalized, converges to the unique viscosity solution of a Hamilton-Jacobi equation. The proof makes use of almost sure maximum principles and careful controls of the martingale parts.

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