Adaptation in a stochastic multi-resources chemostat model

We are interested in modeling the Darwinian evolution resulting from the interplay of phenotypic variation and natural selection through ecological interactions, in the specific scales of the biological framework of adaptive dynamics. Adaptive dynamics so far has been put on a rigorous footing only for direct competition models (Lotka-Volterra models) involving a competition kernel which describes the competition pressure from one individual to another one. We extend this to a multi-resources chemostat model, where the competition between individuals results from the sharing of several resources which have their own dynamics. Starting from a stochastic birth and death process model, we prove that, when advantageous mutations are rare, the population behaves on the mutational time scale as a jump process moving between equilibrium states (the polymorphic evolution sequence of the adaptive dynamics literature). An essential technical ingredient is the study of the long time behavior of a chemostat multi-resources dynamical system. In the small mutational steps limit this process in turn gives rise to a differential equation in phenotype space called canonical equation of adaptive dynamics. From this canonical equation and still assuming small mutation steps, we prove a rigorous characterization of the evolutionary branching points. MSC 2000 subject classification: 92D25, 60J80, 37N25, 92D15, 60J75 Key-words: Mutation-selection individual-based model; fitness of invasion; adaptive dynamics; long time behavior of dynamical systems; polymorphic evolution sequence; multiresources chemostat systems; evolutionary branching; piecewise-deterministic Markov processes. ∗Université de Lorraine, Institut Elie Cartan de Lorraine, UMR 7502, Vandœuvre-lès-Nancy, F-54506, France; E-mail: [email protected] †CNRS, Institut Elie Cartan de Lorraine, UMR 7502, Vandœuvre-lès-Nancy, F-54506, France ‡Inria, TOSCA, Villers-lès-Nancy, F-54600, France §Cscamm and Dpt. of Mathematics, University of Maryland, College Park, MD 20742 USA, Email: [email protected] ¶CMAP, Ecole Polytechnique, CNRS, route de Saclay, 91128 Palaiseau Cedex-France; E-mail: [email protected] 1 ar X iv :1 30 2. 05 52 v1 [ m at h. PR ] 3 F eb 2 01 3