Thackeray 427

### Abstract or Additional Information

The evolution of dispersal is a classical question in evolutionary ecology, which has been widely studied with several mathematical models. The main question is to define the fittest dispersal rate for a population in a bounded domain, and, more recently, for traveling waves in the full space. In 2015, Perthame and Souganidis introduced a novel approach to study the evolution of unconditional dispersal. They considered an integro-PDE model for a population structured by the spatial variables and a (continuous) trait variable which is the random diffusion rate, and showed that, in the limit of vanishing mutations, the population concentrates on a single trait associated to the lowest dispersal rate. The mathematical interest stems from the asymptotic analysis which requires a completely different treatment of the different variables. For the space variable, the ellipticity leads to regularity results. For the trait variable, the concentration to a Dirac mass requires a different treatment. In our talk, we will talk about some recent developments, and the mathematical approach based on the WKB method and viscosity solutions leading to a constrained Hamilton-Jacobi equation.