Random dispersal vs. non-local dispersal

Random dispersal is essentially a local behavior which describes the movement of organisms between adjacent spatial locations. However, the movements and interactions of some organisms can occur between non-adjacent spatial locations. To address the question about which dispersal strategy can convey some competitive advantage, we consider a mathematical model consisting of one reaction-diffusion equation and one integro-differential equation, in which two competing species have the same population dynamics but different dispersal strategies: the movement of one species is purely by random walk while the other species adopts a non-local dispersal strategy. For spatially periodic and heterogeneous environments we show that (i) for fixed random dispersal rate, if the nonlocal dispersal distance is sufficiently small, then the non-local disperser can invade the random disperser but not vice versa; (ii) for fixed nonlocal dispersal distance, if the random dispersal rate becomes sufficiently small, then the random disperser can invade the nonlocal disperser but not vice versa. These results suggest that for spatially periodic heterogeneous environments, the competitive advantage may belong to the species with much lower effective rate of dispersal. This is in agreement with previous results for the evolution of random dispersal [9, 13] that the slower disperser has an advantage. Nevertheless, if random dispersal strategy with either zero Dirichlet or zero Neumann boundary condition is compared with non-local dispersal strategy with hostile surroundings, the species with much lower effective rate of dispersal may not have the competitive advantage. Numerical results will be presented to shed light on the global dynamics of the system for general values of non-local interaction distance and also to point to future research directions.