Point to point traveling wave and periodic traveling wave induced by Hopf bifurcation for a diffusive predator-prey system

Hongyong Zhao; Daiyong Wu

doi:10.3934/dcdss.2020129

Article Contents

2020, Volume 13, Issue 11: 3271-3284. Doi: 10.3934/dcdss.2020129

This issue Previous Article Bogdanov-Takens bifurcation in predator-prey systems Next Article

Point to point traveling wave and periodic traveling wave induced by Hopf bifurcation for a diffusive predator-prey system

Hongyong Zhao^1, , and
Daiyong Wu^1,2,

1.
Department of Mathematics, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, China
2.
Department of Mathematics, Anqing Normal University, Anqing 246133, China

^* Corresponding author: Hongyong Zhao
^* Corresponding author: Hongyong Zhao

Received: November 2018

Revised: March 2019

Early access: November 2019

Published: July 2020

The work is partially supported by the National Natural Science Foundation of China (Nos 11571170, 31570417); the Natural Science Foundation of Anhui Province of China (No 1608085 MA14); the Key Project of Natural Science Research of Anhui Higher Education Institutions of China (No KJ2018A0365)

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Abstract

In this paper, we consider a diffusive Leslie-Gower predator-prey system with prey subject to Allee effect. First, taking into account the diffusion of both species, we obtain the existence of traveling wave solution connecting predator-free constant steady state and coexistence steady state by using the upper and lower solutions method. However, due to the singularity in the predator equation, we need construct a positive suitable lower solution for the prey density. Such a traveling wave solution can model the spatial-temporal process where the predator invades the territory of the prey and they eventually coexist. Second, taking into account two cases: the diffusion of both species and the diffusion of prey-only, we prove the existence of small amplitude periodic traveling wave train solutions by using the Hopf bifurcation theory. Such traveling wave solutions show that the predator invasion leads to the periodic population densities in the coexistence domain.

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Mathematics Subject Classification: Primary: 35K57, 35C07; Secondary: 34C23.

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