Is a nonlocal diffusion strategy convenient for biological populations in competition?

We study the viability of a nonlocal dispersal strategy in a reaction-diffusion system with a fractional Laplacian operator. We show that there are circumstances-namely, a precise condition on the distribution of the resource-under which the introduction of a new nonlocal dispersal behavior is favored with respect to the local dispersal behavior of the resident population. In particular, we consider the linearization of a biological system that models the interaction of two biological species, one with local and one with nonlocal dispersal, that are competing for the same resource. We give a simple, concrete example of resources for which the equilibrium with only the local population becomes linearly unstable. In a sense, this example shows that nonlocal strategies can invade an environment in which purely local strategies are dominant at the beginning, provided that the resource is sufficiently sparse. Indeed, the example considered presents a high variance of the distribution of the dispersal, thus suggesting that the shortage of resources and their unbalanced supply may be some of the basic environmental factors that favor nonlocal strategies.