Sylvie Méléard - Stochastic modeling of Darwinian evolution in a chemostat­
Channel: Institut des Hautes Études Scientifiques (IHÉS)
Date: 2014-04-02

Sylvie Méléard (Ecole Polytechnique)
We consider a stochastic model describing the Darwinian evolution of a polymorphic 
population with mutation and selection in a multi-resource chemostat. The interactions between
individuals occur by way of competition for resources whose concentrations depend on the current 
state of the population. Our aim is to model the successive fixations of successful mutants in the 
population and further its diversification on an evolutionary time scale. 
We prove, starting from a birth and death process coupled with a piecewise deterministic Markov 
process, that, when advantageous mutations are rare and the population size large enough, the 
population process behaves on the mutation time scale as a jump process moving between 
successive equilibria. The main idea is a time scale separation. The model explains a possible 
diversification into species well adapted specialized to consume some specific resources.
Essential technical ingredients are the study of a generalized system of ODE's modeling a finite 
number of biological populations in a competitive interaction due to multi-resources and a fine 
description of the invasion and fixation of mutants using branching processes.