Global dynamics of a competition-diffusion system and application to a modified Leslie-Gower model

We investigate the global dynamics of a special case of the classical Lotka-Volterra competition-diffusion system in spatially heterogeneous environment. This model indicates that the evolution of the density of the predator is independent of the density of the prey. Based on the principal spectral theory and the dynamics of the classical single-species logistic model, we obtain the global dynamics of this competition-diffusion system. As an application, under some suitable conditions we use the obtained results to prove the global stability of steady states and the persistence of the two species in a modified Leslie-Gower model with diffusion in heterogeneous environment.