Propagation dynamics of nonlocal dispersal equations with bistable nonlinearity in spatially periodic media

This paper is concerned with the new types of entire solutions other than traveling waves of nonlocal dispersal equations with bistable nonlinearity in spatially periodic media. We first establish the existence of pulsating fronts with small periods by the implicit function theorem, and then give the asymptotic behavior near infinity of the pulsating fronts (if they exist). Finally, we investigate the existence and qualitative properties of new types of entire solutions (defined for all $ (x, t)\in\mathbb{R}^2 $) by the sub- and super-solutions method and the comparison principle. Throughout the whole paper, we need to overcome the difficulties brought by the nonlocal dispersal and the spatial heterogeneity.