Orateur
Description
We present a model for growth in a multi-species population. We consider two types evolving as a logistic branching process with mutation, where one of the types has a selective advantage. We first study the frequency of the disadvantageous type, once the population approaches the carrying capacity. Adapting techniques from Katzenberger, we show that this process converges to a Gillespie-Wright-Fisher diffusion process. We then study the dynamics backward in time: we fix a time horizon at which the population is at carrying capacity and we study the ancestral relations of a sample of individuals. We prove that, provided that the advantageous and disadvantageous branching measures are stochastically ordered, this ancestral line process converges to the moment dual of the limiting diffusion.