Evolution of cross-diffusion and self-diffusion

This article is concerned with the evolution of certain types of density-dependent dispersal strategy in the context of two competing species with identical population dynamics and same random dispersal rates. Such density-dependent movement, often referred to as cross-diffusion and self-diffusion, assumes that the movement rate of each species depends on the density of both species and that the transition probability from one place to its neighbourhood depends solely on the arrival spot (independent of the departure spot). Our results suggest that for a one-dimensional homogeneous habitat, if the gradients of two cross- and self-diffusion coefficients have the same direction, the species with the smaller gradient will win, i.e. the dispersal strategy with the smaller gradient of cross- and self-diffusion coefficient will evolve. In particular, it suggests that the species with constant cross- and self-diffusion coefficients may have competitive advantage over species with non-constant cross- and self-diffusion coefficients. However, if the two gradients have opposite directions, neither of the two dispersal strategies wins as these two species can coexist.