Dirichlet problem for a diffusive logistic population model with two delays

Xuejun Pan; Hongying Shu; Yuming Chen

doi:10.3934/dcdss.2020134

Article Contents

2020, Volume 13, Issue 11: 3139-3155. Doi: 10.3934/dcdss.2020134

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Dirichlet problem for a diffusive logistic population model with two delays

1.
Tongji Zhejiang College, Jiaxing, Zhejiang 314051, China
2.
School of Mathematics and Information Science, Shaanxi Normal University, Xi'an 710062, China
3.
Department of Mathematics, Wilfrid Laurier University, Waterloo, ON N2L 3C5, Canada

^* Corresponding author
^* Corresponding author

Received: November 30, 2018

Revised: January 31, 2019

Early access: November 2019

Published: July 2020

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Abstract

In this paper, we investigate a diffusive logistic equation with non-zero Dirichlet boundary condition and two delays. We first exclude the existence of positive heterogeneous steady states, which implies the uniqueness of constant positive steady state. Then, we analyze the local stability and local Hopf bifurcation at the positive steady state. We show that multiple delays can induce multiple stability switches. Furthermore, we prove global stability of the positive steady state under certain conditions and obtain global Hopf bifurcation results. Our theoretical results are illustrated with numerical simulations.

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Mathematics Subject Classification: Primary: 92D25, 35K57; Secondary: 34K18.

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Figure 1. The phase portrait of (9)

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Figure 2. The root distribution of (14)

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Figure 3. Left: $ u_* $ is stable at $ \tau = 5 $. Right: $ u_* $ is unstable at $ \tau = 12 $

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Figure 4. Left: $ u_* $ is stable at $ \tau = 18 $. Right: $ u_* $ is unstable at $ \tau = 30 $

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