Abstract
We consider the evolutive chemostat model with a continuous phenotypic trait of the micro-organisms that impacts both the specific growth rate and the yield conversion, and whose evolution is driven by a mutation kernel. We study the evolutionary attractors (EAs) for a general class of non-linear growth functions, without assuming that the dynamics of the resource is fast compared to the growth. We generalize the well-known reproduction number as a function of the trait, and show a concentration of the asymptotic state on values of the trait that maximize the reproduction number. However, in the case of multiple maxima, we show that the system can have a unique globally stable EA, depending on a total order defined on the trait values.