Acta mathematica scientia,Series A ›› 2021, Vol. 41 ›› Issue (6): 1705-1717.

Previous Articles     Next Articles

Analysis on Critical Waves and Non-Critical Waves for Holling-Tanner Predator-Prey System with Nonlocal Diffusion

Xin Wu1,Rong Yuan2,Zhaohai Ma3,*()   

  1. 1 School of Sciences, East China Jiaotong University, Nanchang 330013
    2 School of Mathematical Sciences, Beijing Normal University, Beijing 100875
    3 School of Science, China University of Geosciences, Beijing 100083
  • Received:2020-03-30 Online:2021-12-26 Published:2021-12-02
  • Contact: Zhaohai Ma;
  • Supported by:
    the NSFC(11771044);the NSFC(11871007);the NSFC(12001502);the NSF of Jiangxi Province(20202BABL211003);the Science and Technology Project of Jiangxi Education Department(GJJ180354);and the Fundamental Research Funds for the Central University(2652019015)


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

CLC Number: 

  • O175