  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 the present study, we reformulate the problem in the context of adaptive
evolution. We consider a population structured by space and a genetic trait
acting directly on the dispersal (diffusion) rate under the effect of rare
mutations on the genetic trait. We show that, as in simpler models, in the
limit of vanishing mutations, the population concentrates on a single trait
associated to the lowest dispersal rate. We also explain how to compute the
evolution speed towards this evolutionary stable distribution.
  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 the use the maximum principle and
Sobolev-type regularity results. For the trait variable, the concentration to a
Dirac mass requires a different treatment. This is based on the WKB method and
viscosity solutions leading to an effective Hamiltonian (effective fitness of
the population) and a constrained Hamilton-Jacobi equation.
