Abstract:

In the current paper we improve the recent results established in [2] concerning the traveling wave solutions for a Holling-Tanner predator-prey system. It is shown that there is a $c^*>0$ such that for every $c>c^*$, this system has a traveling wave solution $(u(\xi), v(\xi))$ with speed $c$ connecting the constant steady states $(1, 0)$ and $(\frac{1}{1+\beta}, \frac{1}{1+\beta})$ under the technical assumptions $\limsup\limits_{\xi\rightarrow+\infty}u(\xi) < 1$ and $\liminf\limits_{\xi\rightarrow+\infty}v(\xi)>0$. Here we do not assume these assumptions and obtain the existence of traveling waves for every $c>c^*$ by some analysis techniques. Moreover, we deal with the open problem in [2] and complete the study of traveling waves with the critical wave speed $c^*$ by the approximating method. We also point out that both the nonlocal dispersal and coupling of the system in the model bring some difficulties in the study of traveling wave solutions.

Key words: Critical waves, Non-critical waves, Predator-prey system, Nonlocal diffusion, Reaction diffusion equation