Propagation Phenomena of a Time‐Periodic Leslie–Gower Predator–Prey System With Nonlocal Dispersal in Shifting Habitats

ABSTRACT In this paper, we investigate the propagation dynamics for a nonlocal dispersal Leslie–Gower predator–prey system in time‐periodic shifting habitats. Applying the asymptotic fixed‐point theory and upper–lower solution technique, we first establish the existence of three types of time‐periodic forced waves that connect from the trivial state to trivial, semi‐trivial, and coexistence states, respectively. Long‐term behavior of these forced waves will illustrate different dynamics of the species invasion fronts. Next, we study the spreading properties of the time‐periodic model. We provide some conditions to guarantee the extinction and persistence for each species individually. Indeed, we point out that the predators can achieve persistent propagation even in the absence of prey, which corresponds to the semi‐trivial state. Furthermore, basing on these single‐species persistence results and using the persistence theory for dynamical systems, we derive the sufficient conditions for species coexistence.