This talk aims to study the role of migration as a key driver of the evolution of spatially structured populations. To this end, we consider a metapopulation setting where the adaptive dynamics at the level of each isolated patch is modelled by a Moran model, which describes the evolution of a quantitative trait in a population of fixed size by two main mechanisms : trait resampling and mutations. Migrations are added in order to take into account interactions between patches and the question we would like to answer is: how do these migrations influence the long term evolution of the population at the level of a single patch and at the level of the entire metapopulation.
For this purpose, we study several scaling limits of the model. Assuming rare mutations and migrations, we adapt a technique from Champagnat & Lambert (2007) in order to get a mean-field network of Trait Substitution Sequence (TSS) describing long-term successive dominant traits in each patch. We derive a propagation of chaos as the metapopulation becomes large. Patches are therefore i.i.d copies of each other, with a TSS described by a McKean-Vlasov pure jump process. In the limit where mutations have small effects and migration is further slowed down accordingly, we obtain the convergence of the TSS, in the new migration timescale, to the solution of a stochastic differential equation which can be referred to as a new canonical equation of adaptive dynamics. This equation includes an advection term representing selection, a diffusive term due to genetic drift, and a jump term, representing the effect of migration, to a state distributed according to its own law.
Joint work with: Amaury Lambert, Hélène Leman and Hélène Morlon.
