Traveling wave solutions of a diffusive ratio-dependent Holling-Tanner system with distributed delay

The existence of traveling wave solutions and wave train solutions of a diffusive ratio-dependent predator-prey system with distributed delay is proved. For the case without distributed delay, we first establish the existence of traveling wave solution by using the upper and lower solutions method. Second, we prove the existence of periodic traveling wave train by using the Hopf bifurcation theorem. For the case with distributed delay, we obtain the existence of traveling wave and traveling wave train solutions when the mean delay is sufficiently small via the geometric singular perturbation theory. Our results provide theoretical basis for biological invasion of predator species.