From branching processes to the Ewens sampling formula, it is remarkable how models originally from population genetics have also featured prominently in other areas of probability. The aim of this workshop is to present some recent progress in the probabilistic modelling of population genetics as well as various interactions with other areas. Three experts will talk about their recent works. There will also be opportunities for discussion with them.
13:00-14:00 Amandine Veber (CNRS, University of Paris Cité)
Growth properties of the infinite-parent spatial Lambda-Fleming-Viot process
The « infinite-parent » spatial Lambda-Fleming-Viot process is a model for spatially expanding populations in a two dimensional continuum, in which empty areas are filled with ghost individuals. This model can be seen as a continuous-space version of the Eden growth model, and it comes with a dual process that allows us to trace back the origins of a sample of individuals taken from the current population. In this talk, we shall focus on the growth properties of the area covered by real individuals. With the help of a simple toy model, we shall also investigate how the fluctuations at the front edge lead to a much larger speed of growth of the occupied region than that predicted by simple first-moment estimates.
Joint work with Apolline Louvet (Ecole Polytechnique and University of Paris Cité)
14:00-15:00 Josué Corujo (University of Strasbourg)
Spectrum and ergodicity of a neutral multi-allelic Moran model
We will present some recent results related to a neutral multi-allelic Moran model, which is a finite continuous-time Markov process. For this process, it is assumed that the individuals can be of different types (among a finite set) and they interact according to two mechanisms: a mutation process where they mutate independently of each other according to an irreducible rate matrix, and a Moran type reproduction process, where two individuals are uniformly chosen, one dies and the other is duplicated. During this talk we will discuss some recent results for the spectrum of the generator of the neutral multi-allelic Moran process. We will show explicit expressions for its eigenvalues in terms of the eigenvalues of the rate matrix that drives the mutation process. Our approach does not require that the mutation process be reversible, or even diagonalisable. Additionally, we will discuss some applications of these results to the study of the speed of convergence to stationarity of the Moran process with a general mutation scheme. Under some non-restrictive hypotheses, we can prove a lower bound for the mixing time of the multi-allelic Moran process. Then we focus on the case where the mutation scheme satisfies the so-called "parent independent" condition, where (and only where) the neutral Moran model becomes reversible. In this latter case, we can go further by proving the existence of a cutoff phenomenon for the convergence to stationarity.https://arxiv.org/abs/2010.08809
This presentation is based on a recently submitted work, for which a preprint is available at
15:00-16:00 Sarah Pennington (University of Bath)
Genealogies in bistable waves
Consider a diploid population (one in which each individual carries two copies of each gene) living in one spatial dimension. Suppose a particular gene appears in two forms (alleles) A and a, and that individuals carrying AA have a higher fitness than aa individuals, while Aa individuals have a lower fitness than both AA and aa individuals. The proportion of advantageous A alleles expands through the population approximately according to a travelling wave. We can prove that on a suitable timescale, the genealogy of a sample of A alleles taken from near the wavefront converges to a Kingman coalescent as the population density goes to infinity. This contrasts with the case of directional selection in which the corresponding limit is thought to be the Bolthausen-Sznitman coalescent.
Joint work with Alison Etheridge.
Organising Group: RSS Applied Probability Section
Organisers: Minmin Wang
& Emma Horton