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In this paper, the global dynamics of two-species Lotka-Volterra competitionmodels with nonlocal dispersals is studied. Under the assumption that dispersalkernels are symmetric, we prove that except for very special situations, localstability of semi-trivial steady states implies global stability, while whenboth semi-trivial steady states are locally unstable, the positive steady stateexists and is globally stable. Moreover, our results cover the case thatcompetition coefficients are location-dependent and dispersal strategies aremixture of local and nonlocal dispersals.

Global dynamics of competition models with nonlocal dispersals I: Symmetric kernels