https://hal.science/hal-04034027Champagnat, NicolasNicolasChampagnatBIGS - Biology, genetics and statistics - Inria Nancy - Grand Est - Inria - Institut National de Recherche en Informatique et en Automatique - IECL - Institut Élie Cartan de Lorraine - UL - Université de Lorraine - CNRS - Centre National de la Recherche ScientifiqueIECL - Institut Élie Cartan de Lorraine - UL - Université de Lorraine - CNRS - Centre National de la Recherche ScientifiqueHass, VincentVincentHassBIGS - Biology, genetics and statistics - Inria Nancy - Grand Est - Inria - Institut National de Recherche en Informatique et en Automatique - IECL - Institut Élie Cartan de Lorraine - UL - Université de Lorraine - CNRS - Centre National de la Recherche ScientifiqueIECL - Institut Élie Cartan de Lorraine - UL - Université de Lorraine - CNRS - Centre National de la Recherche ScientifiqueConvergence of individual-based models with small and frequent mutations to the canonical equation of adaptive dynamicsHAL CCSD2023Adaptive dynamicsCanonical equationIndividual-based modelMeasure-valued Markov processSlow-fast asymptotic analysisAveraging methodCentered Fleming-Viot process[MATH.MATH-PR] Mathematics [math]/Probability [math.PR]HASS, Vincent2023-03-17 11:57:012023-03-24 14:53:312023-03-17 14:10:28enPreprints, Working Papers, ...application/pdf1In 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 (σ → 0) proving convergence to an ODE known as the canonical equation of adaptive dynamics. This result holds only for a certain range of σ 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).