Appears in collection : Schlumberger workshop on Topics in Applied Probability

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.