Abstract
In this paper, we study the propagation dynamics of a time-periodic predator-prey system with nonlocal dispersal. We first establish the existence and nonexistence of periodic traveling waves and then investigate the spreading properties of solutions starting from compactly supported initial conditions. Roughly speaking, we show that if predators disperse faster than the prey, then both species spread simultaneously; whereas if the prey disperses faster than predators, then there exist two separate invasion fronts, one front occurs as the prey invades open habitats, and the other front appears when predators catch up the prey. We emphasize that one needs to find some new techniques to treat nonlocal predator-prey systems due to the presence of the time dependence of nonlinearity, the lack of compactness of the nonlocal dispersal operators and the lack of the comparison principle for predator-prey systems.
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