Adaptive evolution and concentrations in parabolic PDEs

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.