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...
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MDPI AG
2023-01-01
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author | Wenqi Zhang Dan Jin Ruizhi Yang |
author_facet | Wenqi Zhang Dan Jin Ruizhi Yang |
author_sort | Wenqi Zhang |
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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. |
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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 |
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