Hopf Bifurcation in a Predator–Prey Model with Memory Effect in Predator and Anti-Predator Behaviour in Prey

In this paper, a diffusive predator–prey model with a memory effect in predator and anti-predator behaviour in prey is studied. The stability of the coexisting equilibrium and the existence of Hopf bifurcation are analysed by analysing the distribution of characteristic roots. The property of Hopf b...

Full description

Bibliographic Details
Main Authors: Wenqi Zhang, Dan Jin, Ruizhi Yang
Format: Article
Language:English
Published: MDPI AG 2023-01-01
Series:Mathematics
Subjects:
Online Access:https://www.mdpi.com/2227-7390/11/3/556
_version_ 1797623871112740864
author Wenqi Zhang
Dan Jin
Ruizhi Yang
author_facet Wenqi Zhang
Dan Jin
Ruizhi Yang
author_sort Wenqi Zhang
collection DOAJ
description In this paper, a diffusive predator–prey model with a memory effect in predator and anti-predator behaviour in prey is studied. The stability of the coexisting equilibrium and the existence of Hopf bifurcation are analysed by analysing the distribution of characteristic roots. The property of Hopf bifurcation is investigated by the theory of the centre manifold and normal form method. Through the numerical simulations, it is observed that the anti-predator behaviour parameter <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>η</mi></semantics></math></inline-formula>, the memory-based diffusion coefficient parameter <i>d</i>, and memory delay <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>τ</mi></semantics></math></inline-formula> can affect the stability of the coexisting equilibrium under some parameters and cause the spatially inhomogeneous oscillation of prey and predator’s densities.
first_indexed 2024-03-11T09:34:57Z
format Article
id doaj.art-0025908e2b564053b36b5f03ca774fcf
institution Directory Open Access Journal
issn 2227-7390
language English
last_indexed 2024-03-11T09:34:57Z
publishDate 2023-01-01
publisher MDPI AG
record_format Article
series Mathematics
spelling doaj.art-0025908e2b564053b36b5f03ca774fcf2023-11-16T17:21:17ZengMDPI AGMathematics2227-73902023-01-0111355610.3390/math11030556Hopf Bifurcation in a Predator–Prey Model with Memory Effect in Predator and Anti-Predator Behaviour in PreyWenqi Zhang0Dan Jin1Ruizhi Yang2Department of Mathematics, Northeast Forestry University, Harbin 150040, ChinaDepartment of Mathematics, Northeast Forestry University, Harbin 150040, ChinaDepartment of Mathematics, Northeast Forestry University, Harbin 150040, ChinaIn this paper, a diffusive predator–prey model with a memory effect in predator and anti-predator behaviour in prey is studied. The stability of the coexisting equilibrium and the existence of Hopf bifurcation are analysed by analysing the distribution of characteristic roots. The property of Hopf bifurcation is investigated by the theory of the centre manifold and normal form method. Through the numerical simulations, it is observed that the anti-predator behaviour parameter <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>η</mi></semantics></math></inline-formula>, the memory-based diffusion coefficient parameter <i>d</i>, and memory delay <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>τ</mi></semantics></math></inline-formula> can affect the stability of the coexisting equilibrium under some parameters and cause the spatially inhomogeneous oscillation of prey and predator’s densities.https://www.mdpi.com/2227-7390/11/3/556memory effectanti-predator behaviourdelayHopf bifurcation
spellingShingle Wenqi Zhang
Dan Jin
Ruizhi Yang
Hopf Bifurcation in a Predator–Prey Model with Memory Effect in Predator and Anti-Predator Behaviour in Prey
Mathematics
memory effect
anti-predator behaviour
delay
Hopf bifurcation
title Hopf Bifurcation in a Predator–Prey Model with Memory Effect in Predator and Anti-Predator Behaviour in Prey
title_full Hopf Bifurcation in a Predator–Prey Model with Memory Effect in Predator and Anti-Predator Behaviour in Prey
title_fullStr Hopf Bifurcation in a Predator–Prey Model with Memory Effect in Predator and Anti-Predator Behaviour in Prey
title_full_unstemmed Hopf Bifurcation in a Predator–Prey Model with Memory Effect in Predator and Anti-Predator Behaviour in Prey
title_short Hopf Bifurcation in a Predator–Prey Model with Memory Effect in Predator and Anti-Predator Behaviour in Prey
title_sort hopf bifurcation in a predator prey model with memory effect in predator and anti predator behaviour in prey
topic memory effect
anti-predator behaviour
delay
Hopf bifurcation
url https://www.mdpi.com/2227-7390/11/3/556
work_keys_str_mv AT wenqizhang hopfbifurcationinapredatorpreymodelwithmemoryeffectinpredatorandantipredatorbehaviourinprey
AT danjin hopfbifurcationinapredatorpreymodelwithmemoryeffectinpredatorandantipredatorbehaviourinprey
AT ruizhiyang hopfbifurcationinapredatorpreymodelwithmemoryeffectinpredatorandantipredatorbehaviourinprey