Traveling wave phenomena in a nonlocal dispersal predator-prey system with the Beddington-DeAngelis functional response and harvesting

Zhihong Zhao; Yan Li; Zhaosheng Feng; Zhihong Zhao; Yan Li; Zhaosheng Feng

doi:10.3934/mbe.2021084

Mathematical Biosciences and Engineering

2021, Volume 18, Issue 2: 1629-1652. doi: 10.3934/mbe.2021084

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Traveling wave phenomena in a nonlocal dispersal predator-prey system with the Beddington-DeAngelis functional response and harvesting

1.
School of Mathematics and Physics, University of Science & Technology Beijing, Beijing 100083, China
2.
School of Mathematical and Statistical Sciences, University of Texas Rio Grande Valley, Edinburg, Texas 78539, USA

Received: 07 October 2020 Accepted: 27 January 2021 Published: 02 February 2021

This paper is devoted to studying the existence and nonexistence of traveling wave solution for a nonlocal dispersal delayed predator-prey system with the Beddington-DeAngelis functional response and harvesting. By constructing the suitable upper-lower solutions and applying Schauder's fixed point theorem, we show that there exists a positive constant $ c^* $ such that the system possesses a traveling wave solution for any given $ c > c^* $. Moreover, the asymptotic behavior of traveling wave solution at infinity is obtained by the contracting rectangles method. The existence of traveling wave solution for $ c = c^* $ is established by means of Corduneanu's theorem. The nonexistence of traveling wave solution in the case of $ c < c^* $ is also discussed.
- predator-prey model,
- Beddington-DeAngelis functional response,
- traveling wave solution,
- nonlocal dispersal equation,
- upper-lower solutions,
- asymptotic behavior
Citation: Zhihong Zhao, Yan Li, Zhaosheng Feng. Traveling wave phenomena in a nonlocal dispersal predator-prey system with the Beddington-DeAngelis functional response and harvesting[J]. Mathematical Biosciences and Engineering, 2021, 18(2): 1629-1652. doi: 10.3934/mbe.2021084

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Abstract

This paper is devoted to studying the existence and nonexistence of traveling wave solution for a nonlocal dispersal delayed predator-prey system with the Beddington-DeAngelis functional response and harvesting. By constructing the suitable upper-lower solutions and applying Schauder's fixed point theorem, we show that there exists a positive constant $ c^* $ such that the system possesses a traveling wave solution for any given $ c > c^* $. Moreover, the asymptotic behavior of traveling wave solution at infinity is obtained by the contracting rectangles method. The existence of traveling wave solution for $ c = c^* $ is established by means of Corduneanu's theorem. The nonexistence of traveling wave solution in the case of $ c < c^* $ is also discussed.

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