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This paper is devoted to the investigation of spatial spreading speeds and traveling wave solutions of monostable evolution equations with nonlocal dispersal in time and space periodic habitats. It has been shown in an earlier work by the first two authors of the current paper that such an equation has a unique time and space periodic positive stable solution u * (t, x). In this paper, we show that such an equation has a spatial spreading speed c * (ξ) in the direction of any given unit vector ξ. A variational characterization of c * (ξ) is given. Under the assumption that the nonlocal dispersal operator associated to the linearization of the monostable equation at the trivial solution 0 has a principal eigenvalue, we also show that the monostable equation has a continuous periodic traveling wave solution connecting u * (·, ·) and 0 propagating in any given direction of ξ with speed c > c * (ξ).

Abstract. The current paper is concerned with the existence of spreading speeds and linear determinacy for two species competition systems with nonlocal dispersal in time and space periodic habitats. The notion of spreading speed intervals for such a system is first introduced via the natural features of spreading speeds. The existence and lower bounds of spreading speed intervals are then established. When the periodic dependence of the habitat is only on the time variable, the existence of a single spreading speed is proved. It also shows that, under certain conditions, the spreading speed interval in any direction is a singleton, and, moreover, the linear determinacy holds.

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