  This paper is concerned with the approximations of random dispersal
operators/equations by nonlocal dispersal operators/equations. It first proves
that the solutions of properly rescaled nonlocal dispersal initial-boundary
value problems converge to the solutions of the corresponding random dispersal
initial-boundary value problems. Next, it proves that the principal spectrum
points of nonlocal dispersal operators with properly rescaled kernels converge
to the principal eigenvalues of the corresponding random dispersal operators.
Finally, it proves that the unique positive time periodic solutions of nonlocal
dispersal KPP equations with properly rescaled kernels converge to the unique
positive time periodic solutions of the corresponding random dispersal KPP
equations.
