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

TL;DR

This paper proves that under certain conditions, a stochastic model of asexual population evolution with frequent mutations converges to the canonical equation of adaptive dynamics, linking individual-level processes to population-level evolution.

Contribution

It establishes the convergence of a measure-valued Markov process to the canonical equation of adaptive dynamics in a double limit of large population and small mutations, using a novel averaging approach.

Findings

Convergence to the canonical equation under specific mutation size conditions.

Use of averaging and martingale methods for asymptotic analysis.

Characterization of the fast component as a Fleming-Viot process.

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 $(K \to +\infty)$ and small mutational effects $(\sigma_{K} \to 0)$ proving convergence to an ODE known as the canonical equation of adaptive dynamics. This result holds only for a certain range of $\sigma_{K}$ 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).