Upcoming events

3:15 pm: Julie Touniaire (Besançon) - Stochastic neutral fractions and the effective population size

Population genetics aims to explain observed genetic diversity through past evolutionary forces. In the neutral setting, i.e., in the absence of natural selection and ecological constraints, diversity arises solely from demographic fluctuations. In this simplified framework, the allelic composition of a population converges, in the large-population limit, to the Wright–Fisher diffusion. This Wright–Fisher model is a purely genetic model, and a key question is how ecological constraints (such as population structure) may influence genetic composition. In this context, the ‘effective population size’, defined as the size of a Wright–Fisher population experiencing the same level of genetic drift as the population under study, plays a central role.

In this talk, I will introduce a stochastic differential equation with an infinite divisibility property to model the dynamics of general structured populations. This property allows the population to be decomposed into neutral allelic fractions. When demographic fluctuations are small, a fast–slow principle yields a general expression for the effective population size in structured settings.

This is joint work with R. Forien, E. Schertzer, and Z. Talyigas.

4:15 pm: Coffee break

4:45 pm: Viet Chi Tran (Lille) - From stochastic individual-based models to Hamilton-Jacobi PDEs

We study the evolution of a population with a phenotypic trait structure, where the dynamics is ruled by births, deaths and mutations. We are interested in following populations in logarithmic scales of size and time and derive a limiting Hamilton-Jacobi equation (with state constraints) from the stochastic individual based model. The limiting partial differential equation takes into account possible extinction events of the system on certain regions of the trait space. The proof emphasizes the links with the theory of large deviations.