CONVERGENCE OF POPULATION PROCESSES WITH SMALL AND FREQUENT MUTATIONS TO THE CANONICAL EQUATION OF ADAPTIVE DYNAMICS

• 成果类型: Article
• 署名作者: Champagnat, Nicolas; Hass, Vincent
• 署名单位: Universite de Lorraine; Centre National de la Recherche Scientifique (CNRS); Universite Marie et Louis Pasteur
• 刊物名称: ANNALS OF APPLIED PROBABILITY
• ISSN/ISSBN: 1050-5164
• DOI: 10.1214/24-AAP2103
• 发表日期: 2025
• 页码: 1-63
• 关键词: fleming-viot processes; moment equations; EVOLUTION; trait; AGE; sequence; approximations; coevolution; adaptation; strategies

摘要

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 behaviour of the system, where mutations are frequent, in the double simultaneous limit of large population (K-* +infinity) and small mutational effects (sigma K-* 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 (In Applied Stochastic Analysis (1992) 186-209, Springer), 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 and Hass (Stochastic Process. Appl. 166 (2023) 104219).