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

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→+∞) 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 (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).