Séance Séminaire

Séminaire ACSIOM

Concentration and singular waves in a nonlocal reaction-diffusion equation

I consider a reaction-diffusion equation modelling the propagation of a species that possesses a continuum of phenotypic traits. The spatial dynamics of the individuals is modelled by a diffusion process, and the population undergoes a reproduction-mutation-competition dynamics at each spatial point, which is modelled by a nonlocal operator acting on the bounded domain representing the phenotypic space. Under some conditions on the fitness function, the mutation rate and the dimensionality of the domain, a concentration phenomenon is known to happen for the linearized equation, meaning that a singular measure part exists in the principal eigenfunction. I will discuss the validity of this phenomenon for the full (nonlinear) equation, with a particular attention to homogeneous stationary states and traveling waves. In particular, I will talk about the techniques used to construct weak (possibly singular) traveling waves.