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.
10th Anniversary Speaker Series 2007