On Lotka-Volterra competitive parabolic systems: Exclusion, coexistence and bistability

In this paper, we mainly investigate the population dynamics of a general competitive parabolic system including both diffusion and advection. For such a class of systems, we provide an efficient way to determine the global dynamics by values of competition coefficients b and c. Precisely, a clear picture on the local dynamics of the two semi-trivial steady states is firstly given in terms of critical competition values by the monotonicity of the principal eigenvalue, and then a further determination on the global dynamics in different regions on the b-c plane is presented by establishing the uniqueness or non-existence of coexistence steady states. Our results extend largely those in a previous work [32] by removing two conditions (see ( A 1 ) and ( A 2 ) below).