Dynamics of a Spatio-Temporal Slow-Fast Predator-Prey System with Reproductive Allee Effect and a Generalist Predator

In this paper, we consider the dynamics of a slow-fast predator-prey system in space and time, influenced by the reproductive Allee effect in the prey and the presence of a generalist predator. Using geometric singular perturbation theory (GSPT), entry-exit function and normal form theory, we first investigate the non-spatial case, which exhibits rich dynamical phenomena, including the existence of relaxation oscillations, canard cycles, heteroclinic and homoclinic orbits. We then consider the corresponding spatio-temporal model and reveal different kinds of traveling wave solutions, including monotone and non-monotone traveling fronts, traveling pulses, and periodic wave trains. Especially, the existence and uniqueness of the large-amplitude periodic traveling wave train, as well as the traveling front connecting the periodic wave train with the boundary equilibrium, are established. In addition, we show how the corresponding spatial pattern changes in the presence of different timescales. We demonstrate that the system can exhibit Turing instability under certain non-negative diffusion rates. Further, we present possible patterns like spatio-temporal chaos generated by the predator-prey system with diffusion. Simulation results are carried out to verify the behaviors of both the temporal and spatio-temporal models. These results reveal the interplay between the reproductive Allee effect and different timescales, which may lead to a regime shift.