**Benoît Perthame**

(Université P. et M. Curie, INRIA-Rocquencourt and Institut Universitaire de France, France)

Adaptive evolution: a population view

The two processes of mutations and selection, proposed by C. Darwin, can be written in mathematical words. In a very simple, general and idealized description, the environment can be considered as a nutrient shared by all the population. This alllows certain individuals, characterized by a 'phenotypical trait', to multiply faster because they are better adapted to use the environment. This leads to select the 'fittest trait' in the population. On the other hand, the new-born individuals undergo small variations of the trait under the effect of mutations. In these circumstances, is it possible to observ 'speciation' and to describe the dynamical evolution of the 'fittest' traits?

We will give a class of self-contained mathematical models of such population dynamics and show that an asymptotic view allows us to formalize precisely the concepts of monomorphic or polymorphic populations. We can describe the evolution of the 'fittest traits' and various forms of branching points. We will also show that numerical solutions are consistant with individual based stochastic simulations.

Eventhough the model is very simple, its analysis relates to remarkable recent progresses of nonlinear analysis.