%A Xin Wu,Rong Yuan,Zhaohai Ma %T Analysis on Critical Waves and Non-Critical Waves for Holling-Tanner Predator-Prey System with Nonlocal Diffusion %0 Journal Article %D 2021 %J Acta mathematica scientia,Series A %R %P 1705-1717 %V 41 %N 6 %U {http://121.43.60.238/sxwlxbA/CN/abstract/article_16546.shtml} %8 2021-12-26 %X

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.