Abstract
This paper is concerned with the propagation dynamics of nonlocal dispersal monostable equations in time-space periodic habitats. We first show that such an equation admits a single spreading speed in every direction under certain conditions and then give several spreading properties in terms of spreading speeds such as asymptotic spreading ray speeds and asymptotic spreading sets. Furthermore, we consider the dependence of the spreading speed on the dispersal rate and reaction term and prove that taking the temporal average or spatial average can decrease the spreading speed. Finally, we employ the viscosity vanishing method to establish the existence of time-space periodic traveling fronts with the critical speed in every direction under the partially temporally homogeneous case and partially nearly flat case, which solves partially the open problem raised by Rawal, Shen, and Zhang (Discrete Contin. Dyn. Syst.35 (2015) 1609â1640).
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