Uniqueness and Complete Dynamics in Heterogeneous Competition-Diffusion Systems

In this paper we study the interactions between diffusion and heterogeneity of the environment in the classical diffusive Lotka--Volterra competition systems. In the weak competition case, we establish the uniqueness, hence the global asymptotic stability, of coexistence steady states under various circumstances, and thereby we obtain a complete understanding of the change in dynamics when one of the interspecific competition coefficients is small.