Spreading Speeds and Traveling Waves of Nonlocal Monostable Equations in Time and Space Periodic Habitats

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 $u^{\ast }(t,x)$. In this paper, we show that such an equation has a spatial spreading speed $c^{\ast }(\xi )$ in the direction of any given unit vector $\xi$. A variational characterization of $c^{\ast }(\xi )$ 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 $u^{\ast }(\cdot ,\cdot )$ and $0$ propagating in any given direction of $\xi$ with speed $c>c^{\ast }(\xi )$.