Concentration phenomena in an integro-PDE model for evolution of conditional dispersal

In order to study the evolution of conditional dispersal, we extend the Perthame-Souganidis mutation-selection model and consider an integro-PDE model for a population structured by the spatial variables and one trait variable. We assume that both the diffusion rate and advection rate are functions of the trait variable, which lies within a short interval I. Competition for resource is local in spatial variables, but nonlocal in the trait variable. Under proper conditions on the invasion fitness gradient, we show that in the limit of small mutation rate, the positive steady state solution will concentrate in the trait variable and forms the following: (i) a Dirac mass supported at one end of I; (ii) or a Dirac mass supported at the interior of I; (iii) or two Dirac masses supported at both ends of I, respectively. While cases (i) and (ii) imply the evolutionary stability of a single strategy, case (iii) suggests that when no single strategy can be evolutionarily stable, it is possible that two peculiar strategies as a pair can be evolutionarily stable and resist the invasion of any other strategy in our context.