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Hopf bifurcation for a delayed predator–prey diffusion system with Dirichlet boundary condition

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  • Ma, Zhan-Ping
  • Huo, Hai-Feng
  • Xiang, Hong

Abstract

A delayed predator–prey diffusion system with Beddington–DeAngelis functional response under Dirichlet boundary condition is investigated. The existence and stability of the positive spatially nonhomogeneous steady-state solution are obtained via the implicit function theorem. Moreover, taking feedback time delay τ as the bifurcation parameter, Hopf bifurcation near the positive steady-state solution is proved to occur at a sequence of critical values, we can show that feedback time delay can induce nonhomogeneous periodic oscillatory patterns. The direction of Hopf bifurcation is forward when parameter m in model (1.2) is sufficiently large. Numerical simulations and numerical solutions are presented to illustrate our theoretical results.

Suggested Citation

  • Ma, Zhan-Ping & Huo, Hai-Feng & Xiang, Hong, 2017. "Hopf bifurcation for a delayed predator–prey diffusion system with Dirichlet boundary condition," Applied Mathematics and Computation, Elsevier, vol. 311(C), pages 1-18.
    • Handle: RePEc:eee:apmaco:v:311:y:2017:i:c:p:1-18
    DOI: 10.1016/j.amc.2017.05.012
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    Cited by:

    1. Chakraborty, Bhaskar & Ghorai, Santu & Bairagi, Nandadulal, 2020. "Reaction-diffusion predator-prey-parasite system and spatiotemporal complexity," Applied Mathematics and Computation, Elsevier, vol. 386(C).
    2. Yuanxian Hui & Yunfeng Liu & Zhong Zhao, 2022. "Hopf Bifurcation in a Delayed Equation with Diffusion Driven by Carrying Capacity," Mathematics, MDPI, vol. 10(14), pages 1-16, July.

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