Stability and Bifurcation Analysis on a Predator–Prey System with the Weak Allee Effect

In this paper, the dynamics of a predator-prey system with the weak Allee effect is considered. The sufficient conditions for the existence of Hopf bifurcation and stability switches induced by delay are investigated. By using the theory of normal form and center manifold, an explicit expression, wh...

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Published in:Mathematics
Main Authors: Jianming Zhang, Lijun Zhang, Yuzhen Bai
Format: Article
Language:English
Published: MDPI AG 2019-05-01
Subjects:
weak Allee effect
predator-prey system
Hopf bifurcations
stability switches
Online Access:https://www.mdpi.com/2227-7390/7/5/432
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author Jianming Zhang
Lijun Zhang
Yuzhen Bai
author_facet Jianming Zhang
Lijun Zhang
Yuzhen Bai
author_sort Jianming Zhang
collection DOAJ
container_title Mathematics
description In this paper, the dynamics of a predator-prey system with the weak Allee effect is considered. The sufficient conditions for the existence of Hopf bifurcation and stability switches induced by delay are investigated. By using the theory of normal form and center manifold, an explicit expression, which can be applied to determine the direction of the Hopf bifurcations and the stability of the bifurcating periodic solutions, are obtained. Numerical simulations are performed to illustrate the theoretical analysis results.
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spelling doaj-art-e9c79c809e4c40c180785fa2b3f4e6132025-08-19T20:43:47ZengMDPI AGMathematics2227-73902019-05-017543210.3390/math7050432math7050432Stability and Bifurcation Analysis on a Predator–Prey System with the Weak Allee EffectJianming Zhang0Lijun Zhang1Yuzhen Bai2Department of Mathematics, School of Science, Zhejiang Sci-Tech University, Hangzhou 310018, ChinaCollege of Mathematics and Systems Science, Shandong University of Science and Technology Qingdao 266590, ChinaSchool of Mathematical Sciences, Qufu Normal University, Qufu 273165, ChinaIn this paper, the dynamics of a predator-prey system with the weak Allee effect is considered. The sufficient conditions for the existence of Hopf bifurcation and stability switches induced by delay are investigated. By using the theory of normal form and center manifold, an explicit expression, which can be applied to determine the direction of the Hopf bifurcations and the stability of the bifurcating periodic solutions, are obtained. Numerical simulations are performed to illustrate the theoretical analysis results.https://www.mdpi.com/2227-7390/7/5/432weak Allee effectpredator-prey systemHopf bifurcationsstability switches
spellingShingle Jianming Zhang
Lijun Zhang
Yuzhen Bai
Stability and Bifurcation Analysis on a Predator–Prey System with the Weak Allee Effect
weak Allee effect
predator-prey system
Hopf bifurcations
stability switches
title Stability and Bifurcation Analysis on a Predator–Prey System with the Weak Allee Effect
title_full Stability and Bifurcation Analysis on a Predator–Prey System with the Weak Allee Effect
title_fullStr Stability and Bifurcation Analysis on a Predator–Prey System with the Weak Allee Effect
title_full_unstemmed Stability and Bifurcation Analysis on a Predator–Prey System with the Weak Allee Effect
title_short Stability and Bifurcation Analysis on a Predator–Prey System with the Weak Allee Effect
title_sort stability and bifurcation analysis on a predator prey system with the weak allee effect
topic weak Allee effect
predator-prey system
Hopf bifurcations
stability switches
url https://www.mdpi.com/2227-7390/7/5/432
work_keys_str_mv AT jianmingzhang stabilityandbifurcationanalysisonapredatorpreysystemwiththeweakalleeeffect
AT lijunzhang stabilityandbifurcationanalysisonapredatorpreysystemwiththeweakalleeeffect
AT yuzhenbai stabilityandbifurcationanalysisonapredatorpreysystemwiththeweakalleeeffect

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