Propagation dynamics in periodic predator-prey systems with nonlocal dispersal
The propagation dynamics of a time-periodic predator-prey system with nonlocal dispersal was considered in this paper. The authors first studied the existence and nonexistence of periodic traveling waves. Since the solution map of the system lacks compactness with respect to compact open topology, they applied the asymptotic fixed point theorem with the help of the Kuratowski measure of noncompactness to prove the existence of periodic traveling waves. The nonexistence of periodic traveling waves is then established by constructing an auxiliary system and using a comparison argument. Although periodic traveling waves may determine the long time behavior of the system with wave-like initial values, it is very interesting and important to understand how solutions of the system starting from compactly supported initial conditions evolve as time increases. Thus, the authors further investigated the spreading properties of solutions starting from compactly supported initial conditions. It is showed that if predators disperse faster than their prey, then both species spread simultaneously; whereas if the prey diffuses 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. Some of the proofs are inspired by \textit{A. Ducrot} et al. [Calc. Var. Partial Differ. Equ. 58, No. 4, Paper No. 137, 34 p. (2019; Zbl 1418.35230)] for the autonomous predator-prey system with local diffusion. There are certain new ideas in this paper. For example, (i) to prove that the prey is always able to spread outside of predators' range in the case where the prey is faster than predators, they established a priori estimates for solutions to the system; (ii) to obtain the lower estimates on the spreading speed, they generalized the persistence theory in dynamical systems to the initial value problem of the system.
scientific article; zbMATH DE number 7643189
scientific article