Abstract: In this paper, a two-species Lotka-Volterra competition-diffusion model with homo-geneous Neumann boundary conditions is considered. The effect of spatial heterogeneity and spatial homogeneity of environment on two competing species and their different competition abilities are studied. In particular, we consider the distribution of resources is heterogeneous for one species but homogeneous for another species with the different total resources in the weak competition. It turns out to be that some parameters play very important roles in this model. The existence, the stability of coexistence state of system is considered, and hence the unique coexistence state and the globally asymptotically stable of the coexistence state of system, and any semi-trivial solution of system can be established under some suitable conditions. Moreover, some limiting behaviors of coexistence state as the dispersal rates are also studied. Our results show that the dynamics of system is very complicated for some general parameters. The proposed method of analysis is based on spectral analysis and monotone dynamical systems theory.

Key words: spatial variation, globally asymptotically stable, competition, coexistence states