Spatio-temporal dynamics of nonlocal dispersal systems in time-space periodic habitats

This paper is concerned with the spatio-temporal dynamics of nonlocal dispersal systems with monostable and time-space periodic nonlinearity. Firstly, when the dispersal kernels are all light-tailed, we obtain the existence and variational characterization of the linear spreading speed; while the accelerated propagation happens if one species has a long-tailed dispersal kernel, and the accelerated spreading rate can be determined by the principle eigenvalue of the linearized system and the tail of the maximum of kernels. Secondly, we establish the existence and non-existence of traveling waves and semi-transition-waves in cooperative case and non-cooperative, respectively. Lastly, we apply these analytic results to a man-environment-man model and conduct some numerical simulations.