Random dispersal vs. non-local dispersal

Random dispersal is essentially a localbehavior which describes the movement of organisms betweenadjacent spatial locations. However, the movements andinteractions of some organisms can occur between non-adjacentspatial locations. To address the question about which dispersalstrategy can convey some competitive advantage, we consider amathematical model consisting of one reaction-diffusion equationand one integro-differential equation, in which two competingspecies have the same population dynamics but different dispersalstrategies: the movement of one species is purely by random walkwhile the other species adopts a non-local dispersal strategy.For spatially periodic and heterogeneous environments we showthat (i) for fixed random dispersal rate,if the nonlocal dispersal distance is sufficiently small, thenthe 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, thenthe random disperser can invade the nonlocal disperser but notvice versa.These results suggestthat for spatially periodic heterogeneous environments,the competitive advantage may belong to the species with much lowereffective rate of dispersal. This is in agreement with previous resultsfor 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 hostilesurroundings, 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.