Traveling waves in a nonlocal dispersal predator-prey model

Yu-Xia Hao; Wan-Tong Li; Fei-Ying Yang

doi:10.3934/dcdss.2020340

Article Contents

2021, Volume 14, Issue 9: 3113-3139. Doi: 10.3934/dcdss.2020340

This issue Previous Article Traveling wave fronts in a diffusive and competitive Lotka-Volterra system Next Article Uniform polynomial stability of second order integro-differential equations in Hilbert spaces with positive definite kernels

Traveling waves in a nonlocal dispersal predator-prey model

School of Mathematics and Statistics, Lanzhou University, Gansu, Lanzhou 730000, People's Republic of China

^* Corresponding author: Wan-Tong Li
^* Corresponding author: Wan-Tong Li

Received: April 30, 2019

Revised: September 30, 2019

Early access: April 2020

Published: September 2021

The second author is supported by NSF of China grant 11731005, 11671180 and the third author is supported by NSF of China grant 11601205

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Abstract

This paper is concerned with the traveling wave solutions for a class of predator-prey model with nonlocal dispersal. By adopting the truncation method, we use Schauder's fixed-point theorem to obtain the existence of traveling waves connecting the semi-trivial equilibrium to non-trivial leftover concentrations for $ c>c_{*} $, in which $ c_* $ is the minimal wave speed. Meanwhile, through the limiting approach and the delicate analysis, we establish the existence of traveling wave solutions with the critical speed. Finally, we show the nonexistence of traveling waves for $ 0<c<c_{*} $ by the characteristic equation corresponding to the linearization of original system at the semi-trivial equilibrium. Throughout the whole paper, we need to overcome the difficulties brought by the nonlocal dispersal and the non-preserving of system itself.

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Mathematics Subject Classification: Primary: 35K57, 35R20; Secondary: 92D25.

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