Turing patterns induced by self-diffusion in a predator–prey model with schooling behavior in predator and prey

This article considers a reaction–diffusion predator–prey model with schooling behavior both in predator and prey species and subject to the homogeneous Neumann boundary condition on a square domain. With the help of the standard linearized analysis, the spatially homogeneous Hopf bifurcation curve...

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Bibliographic Details
Published in:Nonlinear dynamics 2021-09, Vol.105 (4), p.3731-3747
Main Authors: Zhou, Yan, Yan, Xiang-Ping, Zhang, Cun-Hua
Format: Article
Language:English
Subjects:
Amplitudes
Automotive Engineering
Behavior
Boundary conditions
Classical Mechanics
Control
Domains
Dynamical Systems
Engineering
Hopf bifurcation
Mathematical models
Mathematics
Mechanical Engineering
Original Paper
Predation
Predator-prey simulation
Predators
Self diffusion
Stability
Steady state
Vibration
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Summary:This article considers a reaction–diffusion predator–prey model with schooling behavior both in predator and prey species and subject to the homogeneous Neumann boundary condition on a square domain. With the help of the standard linearized analysis, the spatially homogeneous Hopf bifurcation curve and the Turing bifurcation curve of the unique constant positive steady state are obtained. These curves divide the existence domain of the constant positive steady state of the model into the stable, the Hopf unstable, the Turing unstable and the Hopf–Turing unstable regions. When the parameters are in the Turing unstable domain and near the Turing bifurcation curve, by applying the multiple-scale analysis and the successive approximations, the amplitude equations of the system near the constant steady state are derived. Meanwhile, the classification and stability of the patterns of the system are presented in terms of the existence and stability of the stationary solutions of the derived amplitude equations. Numerical simulations show that the presented model can exhibit complicated dynamical behaviors and may help us better understand the interaction between two species.
ISSN:0924-090X
1573-269X
DOI:10.1007/s11071-021-06743-2