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In this article, a stochastic individual-based model describing Darwinianevolution of asexual, phenotypic trait-structured population, is studied. Weconsider a large population with constant population size characterised by aresampling rate modeling competition pressure driving selection and a mutationrate where mutations occur during life. In this model, the population state atfixed time is given as a measure on the space of phenotypes and the evolutionof the population is described by a continuous time, measure-valued Markovprocess. We investigate the asymptotic behavior of the system, where mutationsare frequent, in the double simultaneous limit of large population $(K \to+\infty)$ and small mutational effects $(\sigma \to 0)$ proving convergence toan ODE known as the canonical equation of adaptive dynamics. This result holdsonly for a certain range of $\sigma$ parameters (as a function of $K$) whichmust be small enough but not too small either. The canonical equation describesthe evolution in time of the dominant trait in the population driven by afitness gradient. This result is based on an slow-fast asymptotic analysis. Weuse an averaging method, inspired by (Kurtz, 1992), which exploits a martingaleapproach and compactness-uniqueness arguments. The contribution of the fastcomponent, which converges to the centered Fleming-Viot process, is obtained byaveraging according to its invariant measure, recently characterised in(Champagnat-Hass, 2022).

Convergence of individual-based models with small and frequent mutations to the canonical equation of adaptive dynamics