A number of recent studies have shown that mobility patterns for both humans and biological species can be quite complex and exhibit a scale-free dynamics, characteristic of Levy flights. Levy-flight patterns have been observed in the dispersion of bank notes, in human mobility patterns derived from mobile phone data as well as in the foraging patterns of a numbers of animal species. However, current reaction-diffusion models used to describe the spread of humans and other biological species do not account for the superdiffusive effect due to Levy flights. This could result in higher spreading speeds than predicted by classical models. We have considered two-species reaction-diffusion models driven by Levy flights. That family of models is based on the Lotka-Volterra equations and has been obtained by replacing the second-order diffusion operator by a fractional-order one. Depending on the parameter values, it can be used to represent the interaction between susceptible and infective populations in an epidemics model or the interactions between ecological species competing for the same resources. 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 can derive catch-up conditions for different types of fronts. Our results confirm that second-order reaction-diffusion models are not well-suited to simulate the spatial spread of modern epidemics and biological species that follow a Levy random walk as they are inclined to underestimate the propagation speeds at which they spread
