Invasive competition with Fokker-Planck diffusion and noise

Species competition in a fluctuating environment is often modelled with stochastic reaction-diffusion equations. In most cases the movement of individuals is described as Fickian diffusion. However, in heterogeneous environments this is not the first choice. Recently, it has been shown that Fokker-Planck diffusion describes the movement of species in a more realistic way. Fickian diffusion always leads to spatially uniform stationary distributions whereas the Fokker-Planck diffusion generates nonuniform solutions according to the heterogeneity of the environment and the corresponding spatial variation of diffusion. Species accumulate in regions of low diffusivity and tend to lower their densities in areas of high diffusivity. In the present paper, the impact of Fokker-Planck diffusion is studied with particular consideration of changing spatio-temporal population patterns during the competitive invasion of a spatially heterogeneous, populated habitat. The standard Lotka-Volterra competition model is applied to describe the resident-invader interaction. The resident is assumed to be adapted to the heterogeneous living conditions, i.e., its motion is modelled as space-dependent Fokker-Planck diffusion. The invader's diffusion is taken as neutral Fickian. Furthermore, it is shown that multiplicative environmental noise can either foster or hinder the invasion.