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Bifurcations and pattern formation in a host–parasitoid model with nonlocal effect

Part of: Qualitative theory Parabolic equations and systems

Published online by Cambridge University Press:  04 March 2024

Chuang Xiang ,
Jicai Huang ,
Min Lu ,
Shigui Ruan Open the ORCID record for Shigui Ruan [Opens in a new window]  and
Hao Wang Open the ORCID record for Hao Wang [Opens in a new window]
Chuang Xiang
Affiliation:
School of Mathematical Sciences, Nanjing Normal University, Nanjing 210023, P. R. China (1604059603@qq.com) School of Mathematics and Statistics, Central China Normal University, Wuhan, Hubei 430079, P. R. China
Jicai Huang
Affiliation:
School of Mathematics and Statistics, Central China Normal University, Wuhan, Hubei 430079, P. R. China (hjc@ccnu.edu.cn, lumin@mails.ccnu.edu.cn)
Min Lu
Affiliation:
School of Mathematics and Statistics, Central China Normal University, Wuhan, Hubei 430079, P. R. China (hjc@ccnu.edu.cn, lumin@mails.ccnu.edu.cn)
Shigui Ruan
Affiliation:
Department of Mathematics, University of Miami, Coral Gables, FL 33146, USA (ruan@math.miami.edu)
Hao Wang
Affiliation:
Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, AB T6G 2G1, Canada (hao8@ualberta.ca)
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Abstract

In this paper, we analyse Turing instability and bifurcations in a host–parasitoid model with nonlocal effect. For a ordinary differential equation model, we provide some preliminary analysis on Hopf bifurcation. For a reaction–diffusion model with local intraspecific prey competition, we first explore the Turing instability of spatially homogeneous steady states. Next, we show that the model can undergo Hopf bifurcation and Turing–Hopf bifurcation, and find that a pair of spatially nonhomogeneous periodic solutions is stable for a (8,0)-mode Turing–Hopf bifurcation and unstable for a (3,0)-mode Turing–Hopf bifurcation. For a reaction–diffusion model with nonlocal intraspecific prey competition, we study the existence of the Hopf bifurcation, double-Hopf bifurcation, Turing bifurcation, and Turing–Hopf bifurcation successively, and find that a spatially nonhomogeneous quasi-periodic solution is unstable for a (0,1)-mode double-Hopf bifurcation. Our results indicate that the model exhibits complex pattern formations, including transient states, monostability, bistability, and tristability. Finally, numerical simulations are provided to illustrate complex dynamics and verify our theoretical results.

Keywords

host–parasitoid modelTuring instabilityHopf bifurcationTuring–Hopf bifurcationdouble-Hopf bifurcationpattern formation

MSC classification

Primary: 34C23: Bifurcation
Secondary: 35K57: Reaction-diffusion equations 92D25: Population dynamics (general)
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , First View , pp. 1 - 40
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Copyright
Copyright © The Author(s), 2024. Published by Cambridge University Press on behalf of The Royal Society of Edinburgh

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