An integro-PDE model for evolution of random dispersal

We consider an integro-PDE model for a population structured by the spatial variables and a trait variable which is the diffusion rate. Competition for resource is local in spatial variables, but nonlocal in the trait variable. We focus on the asymptotic profile of positive steady state solutions. Our result shows that in the limit of small mutation rate, the solution remains regular in the spatial variables and yet concentrates in the trait variable and forms a Dirac mass supported at the lowest diffusion rate. Hastings [16] and Dockery et al. [14] showed that for two competing species in spatially heterogeneous but temporally constant environment, the slower diffuser always prevails, if all other things are held equal. Our result suggests that their findings may well hold for arbitrarily many or even a continuum of traits.