  In this paper, the global dynamics of two-species Lotka-Volterra competition
models with nonlocal dispersals is studied. Under the assumption that dispersal
kernels are symmetric, we prove that except for very special situations, local
stability of semi-trivial steady states implies global stability, while when
both semi-trivial steady states are locally unstable, the positive steady state
exists and is globally stable. Moreover, our results cover the case that
competition coefficients are location-dependent and dispersal strategies are
mixture of local and nonlocal dispersals.
