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

► The spatial spread of species following a Lévy motion can be simulated by a fractional-order diffusion-reaction model. ► Fractional-order diffusion leads to exponentially accelerating fronts and a power-law decay of the fronts leading tail. ► Catch-up conditions can be derived in the case of two species competing for the same resources. ► Second-order diffusion-reaction are likely to underestimate the speed at which species following a Lévy motion propagate.