Front dynamics in a two-species competition model driven by Lévy flights

Abstract

A number of recent studies suggest that many biological species follow a Lévy random walk in their search for food. Such a strategy has been shown to be more efficient than classical Brownian motion when resources are scarce. However, current diffusion–reaction models used to describe many ecological systems do not account for the superdiffusive spread of populations due to Lévy flights. We have developed a model to simulate the spatial spread of two species competing for the same resources and driven by Lévy flights. The model is based on the Lotka–Volterra equations and has been obtained by replacing the second-order diffusion operator by a fractional-order one. Consistent with previous known results, theoretical developments and numerical simulations show that fractional-order diffusion leads to an exponential acceleration of the population fronts and a power-law decay of the fronts' leading tail. Depending on the skewness of the fractional derivative, we derive catch-up conditions for different types of fronts. Our results indicate that second-order diffusion–reaction models are not well-suited to simulate the spatial spread of biological species that follow a Lévy random walk as they are inclined to underestimate the speed at which these species propagate.