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

TL;DR

This paper investigates the conditions under which nonlocal dispersal strategies are advantageous for biological populations competing for resources, highlighting the role of resource distribution and environmental sparsity in favoring such strategies.

Contribution

It provides a mathematical analysis showing when nonlocal dispersal can outperform local strategies, including a concrete example demonstrating instability of local-only populations.

Findings

Nonlocal dispersal can be advantageous in sparse resource environments.

A specific resource distribution can destabilize local-only populations.

High variance in dispersal distribution favors nonlocal strategies.

Abstract

We study the convenience 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 a nonlocal dispersal behavior is favored. 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 become successful even in 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 ingredients that favor nonlocal strategies.