Appears in collection : 2022 - T1 - WS3 - Mathematical models in ecology and evolution

Joint work with Yves Le Jan.

Our goal is to study the genetic composition of a population in which each individual has 2 parents, who contribute equally to the genome of their o˙spring. We use a bi-parental Moran model, which is characterized by its fixed number N of individuals. We fix an individual and consider the proportions of the genomes of all individuals living n time steps later, that come from this individual. When n goes to in˝nity, these proportions all converge almost surely towards the same random variable. When N then goes to infinity, this random variable multiplied by N (i.e. the stationary weight of any ancestor in the whole population) converges in law towards the mixture of a Dirac measure in 0 and an exponential law with parameter 1/2, and the weights of a ˝nite number of ancestors are independent. As a consequence, we obtain that the sequence of increasing weights of all ancestors, when properly rescaled, converges to the function −2 ln(2(1 − u)) for u superior at 1/2.