It is well-known that spatio-temporal delays and nonlocal dispersal exist inevitablly in nature. Recently, many researchers considered the effect of the spatio-temporal delays and nonlocal dispersal to the differential equations. They constructed the nonlocal dispersal equations with spatio-temporal delays which are more real to illustrate the realistic problems. However, the study of the nonlocal dispersal equations with spatio-temporal delays should be considered not only that the nonlocal dispersal term will make the maximum principle fail, but also the spatio-temporal delayed term will make the mathematical research more difficult.In this paper, we mainly consider the entire solutions of the following non-local diffusion equation with spatio-temporal delays where f is bistable type. Here the entire solutions are defined in the whole space and for all t ā R. In this paper, we study existence, uniqueness and the large-time behavior of the traveling waves at first and establish a new comparison principle. By using two traveling waves which are coming from opposite direction, we establish the Cauchy problem with initial time t=-n and construct the entire solutions by the uniformly convergence of the solutions of the Cauchy problem when nā'ā. Moreover, we also obtain the asymptotic behavior and some qualitative properties of the entire solutions by constructing sup-solutions and sub-solutions. |