(Notice the seminar is shifted one hour later this week)
We perform an asymptotic analysis of models of population dynamics with a fractional Laplacian. An approach based on a WKB ansatz and Hamilton-Jacobi equations has been developed since the 80’s to study qualitative behavior of reaction-diffusion equations. This approach has allowed to describe propagation phenomena in dispersion models, and Dirac concentrations in selection-mutation models.
In this work we extend this approach to reaction-diffusion equations with fractional diffusion. In particular, by performing a long time/long range rescaling of the fractional Fisher-KPP equation, we rediscover the exponential speed of propagation of the population and show that the only role of the fractional Laplacian in determining this speed is at the initial layer where it determines the thickness of the tails of the solution.
This is a joint work with Sylvie Méléard.