The stability and bifurcation of homogeneous diffusive predator–prey systems with spatio–temporal delays

Yiwen Tao; Jingli Ren

doi:10.3934/dcdsb.2021038

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2022, Volume 27, Issue 1: 229-243. Doi: 10.3934/dcdsb.2021038

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The stability and bifurcation of homogeneous diffusive predator–prey systems with spatio–temporal delays

Yiwen Tao and
Jingli Ren^,

School of Mathematics and Statistics, Henan Academy of Big Data, Zhengzhou University, Zhengzhou 450001, China

^* Corresponding author: renjl@zzu.edu.cn
^* Corresponding author: renjl@zzu.edu.cn

Received: July 31, 2020

Revised: November 30, 2020

Early access: January 2021

Published: January 2022

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Abstract

In this paper, we consider a generalized predator-prey system described by a reaction-diffusion system with spatio-temporal delays. We study the local stability for the constant equilibria of predator-prey system with the generalized delay kernels. Moreover, using the specific delay kernels, we perform a qualitative analysis of the solutions, including existence, uniqueness, and boundedness of the solutions; global stability, and Hopf bifurcation of the nontrivial equilibria.

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Mathematics Subject Classification: Primary: 35K57.

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Figure 1. Some examples of $ f(u) $ and $ g(u) $. (a). Logistic effect (blue), weak Allee effect (red), strong Allee effect (green). (b). Type I function (blue), Type Ⅱ function (red), Type Ⅲ function (green)

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