  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.
