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 integrodifferential 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 either hostile surroundings or spatially periodic and heterogeneous environments we show that the species with random dispersal can not invade when rare, while the species with non-local dispersal and small non-local interaction distance can invade when rare. These results suggest that for hostile surroundings or spatially periodic heterogeneous environments, non-local dispersal can be preferred over random dispersal. Nevertheless, for spatially heterogeneous environments if random dispersal strategy with zero Neumann boundary condition is compared with non-local dispersal strategy with hostile surroundings, each of the two species can invade when rare and both species can coexist. The biological meaning behind is that the zero-flux boundary condition can somehow help counterbalance the disadvantage caused by local dispersal. 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.