Spreading speeds for monostable equations with nonlocal dispersal in space periodic habitats

The current paper is devoted to the study of spatial spreading dynamics of monostable equations with nonlocal dispersal in spatially periodic habitats. In particular, the existence and characterization of spreading speeds is considered. First, a principal eigenvalue theory for nonlocal dispersal operators with space periodic dependence is developed, which plays an important role in the study of spreading speeds of nonlocal periodic monostable equations and is also of independent interest. In terms of the principal eigenvalue theory it is then shown that the monostable equation with nonlocal dispersal has a spreading speed in every direction in the following cases: the nonlocal dispersal is nearly local; the periodic habitat is nearly globally homogeneous or it is nearly homogeneous in a region where it is most conducive to population growth in the zero-limit population. Moreover, a variational principle for the spreading speeds is established.