Turing patterns induced by self-diffusion in a predator-prey model with schooling behavior in predator and prey. (English) Zbl 1537.35059
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.
MSC:
| 35B36 | Pattern formations in context of PDEs |
| 35B35 | Stability in context of PDEs |
| 35B40 | Asymptotic behavior of solutions to PDEs |
| 35K57 | Reaction-diffusion equations |
| 92D40 | Ecology |
Keywords:
reaction-diffusion predator-prey model; schooling behavior; Turing bifurcation; amplitude equations; pattern formationSoftware:
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