  In this paper, we gives a complete classification of the global dynamics of
two- species Lotka-Volterra competition models with nonlocal dispersals: where
K, P represent nonlocal operators, under the assumptions that the nonlo- cal
operators are symmetric, the models admit two semi-trivial steady states and
0<bc<1. In particular, when both semi-trivial steady states are locally stable,
it is proved that there exist infinitely many steady states and the solution
with non- negative and nontrivial initial data converges to some steady state.
Furthermore, we generalize these results to the case that competition
coefficients are location-dependent and dispersal strategies are mixture of
local and nonlocal dispersals.

Analysis of PDEs