非局部扩散Holling-Tanner捕食者-食饵系统的临界与非临界行波解分析

数学物理学报 ›› 2021, Vol. 41 ›› Issue (6): 1705-1717.

非局部扩散Holling-Tanner捕食者-食饵系统的临界与非临界行波解分析

吴鑫¹,袁荣²,马兆海^3,*()

¹ 华东交通大学理学院南昌 330013
² 北京师范大学数学科学学院北京 100875
³ 中国地质大学(北京)数理学院北京 100083

收稿日期:2020-03-30 出版日期:2021-12-26 发布日期:2021-12-02
通讯作者: 马兆海 E-mail:zhaohaima@mail.bnu.edu.cn; zhaohaima@cugb.edu.cn
基金资助:
国家自然科学基金(11771044);国家自然科学基金(11871007);国家自然科学基金(12001502);江西省自然科学基金(20202BABL211003);江西省教育厅科技项目(GJJ180354);中央高校基本科研基金(2652019015)

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

Xin Wu¹,Rong Yuan²,Zhaohai Ma^3,*()

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 E-mail:zhaohaima@mail.bnu.edu.cn; zhaohaima@cugb.edu.cn
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)

摘要/Abstract

摘要：

该文改进了文献[2]关于Holling-Tanner捕食者-食饵系统行波解的最新结果.结果表明：存在常数$c^*>0$使得对任意的$c>c^*$，在假设条件$\limsup\limits_{\xi\rightarrow+\infty}u（\xi） < 1$和$\liminf\limits_{\xi\rightarrow+\infty}v（\xi）>0$下，该系统有一个波速为$c$的连接常数稳态解$（1，0）$和$（\frac{1}{1+\beta}，\frac{1}{1+\beta}）$的行波解$（u（\xi），v（\xi））$.该文去掉这些技术假设，并通过一些分析技术得到$c>c^*$时行波解的存在性.进而利用逼近方法得到临界行波解的存在性，从而解决了文献[2]中的公开性问题.值得指出的是模型中系统的耦合性与非局部扩散都给行波解的研究带来了困难.

关键词: 临界行波解, 非临界行波解, 捕食者-食饵系统, 非局部扩散, 反应扩散方程

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

中图分类号:

O175

吴鑫,袁荣,马兆海. 非局部扩散Holling-Tanner捕食者-食饵系统的临界与非临界行波解分析[J]. 数学物理学报, 2021, 41(6): 1705-1717.

Xin Wu,Rong Yuan,Zhaohai Ma. Analysis on Critical Waves and Non-Critical Waves for Holling-Tanner Predator-Prey System with Nonlocal Diffusion[J]. Acta mathematica scientia,Series A, 2021, 41(6): 1705-1717.

参考文献

1	Chen Y Y , Guo J S , Yao C H . Traveling wave solutions for a continuous and discrete diffusive predator-prey model. J Math Anal Appl, 2017, 445, 212- 239
2	Cheng H M , Yuan R . Traveling wave solutions for a continuous and discrete diffusive predator-prey model. Appl Math Comput, 2018, 338, 12- 24
3	Ducrot A . Convergence to generalized transition waves for some Holling-Tanner prey-predator reaction-diffusion system. J Math Pures Appl, 2013, 100, 1- 15
4	Jin Y , Zhao X Q . Spatial dynamics of a periodic population model with dispersal. Nonlinearity, 2009, 222, 1167- 1189
5	李海银. 密度制约且具有时滞捕食-被捕食模型的Hopf分支. 数学物理学报, 2019, 39A (2): 358- 371
5	Li H Y . Hopf bifurcation of delayed density-dependent predator-prey model. Acta Math Sci, 2019, 39A (2): 358- 371
6	Liang X , Zhao X Q . Asymptotic speeds of spread and traveling waves for monotone semiflows with applications. Commun Pure Appl Math, 2007, 60, 1- 40
7	Murray J D . Mathematical Biology Ⅱ: Spatial Models and Biomedical Applications. New York: Springer, 2003
8	Renshaw E . Modelling Biological Populations in Space and Time. Cambridge: Cambridge University Press, 1991
9	Saez E , Gonzalez-Olivares E . Dynamics of a predator-prey model. SIAM J Appl Math, 1999, 59, 1867- 1878
10	史红波, 李万同, 林国. 一类修正的Leslie-Gower型扩散捕食系统的定性分析. 数学物理学报, 2011, 31A (5): 1403- 1415
10	Shi H B , Li W T , Lin G . Qualitative analysis of a modified Leslie-Gower predator-prey system with diffusion. Acta Math Sci, 2011, 31A (5): 1403- 1415
11	Su T , Zhang G B . Invasion traveling waves for a discrete diffusive ratio-dependent predator-prey model. Acta Math Sci, 2020, 40B (5): 1459- 1476
12	Tanner J T . The stability and the intrinsic growth rates of prey and predator populations. Ecology, 1975, 56, 855- 867
13	王雪臣, 魏俊杰. 时滞在具有Ivlev型功能反应函数的捕食者-食饵扩散系统中的作用. 数学物理学报, 2014, 34A (3): 234- 250
13	Wang X C , Wei J J . The effect of delay on a diffusive predator-prey system with Ivlev-type functional response. Acta Math Sci, 2014, 34A (2): 234- 250
14	Wollkind D J , Collings J B , Logan J A . Metastability in a temperature-dependent model system for predator-prey mite outbreak interactions on fruit flies. Bull Math Biol, 1988, 50, 379- 409
15	Zhang G B , Li W T , Wang Z C . Spreading speeds and traveling waves for nonlocal dispersal equation with degenerate monostable nonlinearity. J Differential Equations, 2012, 252, 5096- 5124
16	Zhao Z H , Li R , Zhao X K , Feng Z S . Traveling wave solutions of a nonlocal dispersal predator-prey model with spatiotemporal delay. Z Angew Math Phys, 2018, 69, Article number: 146

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