EXISTENCE OF TRAVELING WAVE SOLUTIONS IN A SINGULARLY PERTURBED PREDATOR-PREY EQUATION WITH SPATIAL DIFFUSION

This article deals with the existence of traveling wave solutions of a spatial diffusion predator-prey model that includes the Allee effect on prey and the simplified Holling type III functional response function. Moreover, we assume that the wave speed is much greater than the diffusion rate of the prey and that the growth rate of the prey is much greater than the growth rate of the predator. In this case, the geometric singular perturbation theory is an important tool to analyze such a singularly perturbed equation. The existence of relaxation oscillations, canard cycles, canard explosions and inverse canard explosions phenomena of the system restricted to the critical manifold are illustrated by means of entry-exit functions, Fenichel’s theory and normal form theory. Furthermore, we establish sufficient conditions for the existence of the monotone traveling waves, non-monotone traveling waves and periodic wave trains. The coexistence of two isolated periodic wave trains is also discussed in this article. And some simulation figures are introduced to support these results. Above all, we show the generation and disappearance of the periodic wave trains due to the canard explosion and the inverse canard explosion.