## Adaptive evolution and concentrations in parabolic PDEs

- Date: 04/26/2007

Benoit Perthame (Ecole Normale Superieure)

University of Alberta

Living systems are subject to constant evolution. Their environment can

be considered as a nutrient shared by all the population. This allows

certain individuals, characterized by a ‘physiological trait’, to

expand faster because they are better adapted to the environment. This

leads to select the ‘best adapted trait’ in the population (singular

point of the system). On the other hand, the new-born population

undergoes small variance on the trait under the effect of genetic

mutations. In these circumstances, is it possible to describe the

dynamical evolution of the current trait?

We will give a mathematical model of such dynamics, based on parabolic

equations, and show that an asymptotic method allows us to formalize

precisely the concepts of monomorphic or polymorphic population. Then,

we can describe the evolution of the ‘best adapted trait’ and

eventually to compute branchings which lead to the cohabitation of two

different populations. In the regular regime, we obtain a canonical

equation where the drift is given by a nonlinear problem.

The asymptotic method leads to evaluate the weight and position of a

moving Dirac mass describing the population. We will show that a

Hamilton-Jacobi equation with constraints naturally describes this

asymptotic. Some more theoretical questions as uniqueness for the

limiting H.-J. equation will also be addressed.

This work is a collaboration with O. Diekmann, P.-E. Jabin, S.

Mischler, S. Cuadrado, J. Carrillo, S. Genieys, M. Gauduchon and G.

Barles.

10th Anniversary Speaker Series 2007