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Preprints, Working Papers, ... Year : 2023

Convergence of population processes with small and frequent mutations to the canonical equation of adaptive dynamics

Abstract

In this article, a stochastic individual-based model describing Darwinian evolution of asexual, phenotypic trait-structured population, is studied. We consider a large population with constant population size characterised by a resampling rate modeling competition pressure driving selection and a mutation rate where mutations occur during life. In this model, the population state at fixed time is given as a measure on the space of phenotypes and the evolution of the population is described by a continuous time, measure-valued Markov process. We investigate the asymptotic behavior of the system, where mutations are frequent, in the double simultaneous limit of large population (K → +∞) and small mutational effects (σK → 0) proving convergence to an ODE known as the canonical equation of adaptive dynamics. This result holds only for a certain range of σK parameters (as a function of K) which must be small enough but not too small either. The canonical equation describes the evolution in time of the dominant trait in the population driven by a fitness gradient. This result is based on an slow-fast asymptotic analysis. We use an averaging method, inspired by (Kurtz, 1992), which exploits a martingale approach and compactness-uniqueness arguments. The contribution of the fast component, which converges to the centered Fleming-Viot process, is obtained by averaging according to its invariant measure, recently characterised in (Champagnat-Hass, 2022).
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Dates and versions

hal-04034027 , version 1 (17-03-2023)
hal-04034027 , version 2 (25-05-2023)
hal-04034027 , version 3 (02-02-2024)

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  • HAL Id : hal-04034027 , version 3

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Nicolas Champagnat, Vincent Hass. Convergence of population processes with small and frequent mutations to the canonical equation of adaptive dynamics. 2023. ⟨hal-04034027v3⟩
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