Population evolution in mutualistic Lotka-Volterra system with spatial diffusion

We consider the population dynamics of two species described by the mutualistic Lotka-Volterra model with a +/+ interaction in the presence of spatial diffusions. The results demonstrate that diffusion does not affect the system's stability but it brings two situations: one is a win-win situation where both species propagate with the same largest speed; in the other situation the aggressive species has two propagating wave fronts and the other species travels with a single slow wave front. Our model may help to understand the evolution of mutualism.