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Stability and Hopf bifurcation for a delayed cooperation diffusion system with Dirichlet boundary conditions

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  • Li, Wan-Tong
  • Yan, Xiang-Ping
  • Zhang, Cun-Hua

Abstract

This paper is concerned with a delayed cooperation diffusion system with Dirichlet boundary conditions. By applying the implicit function theorem, the normal form theory and the center manifold reduction, the asymptotic stability of positive equilibrium and Hopf bifurcation are investigated. It is shown that an increase in delay will destabilize the positive equilibrium and lead to the occurrence of a supercritical Hopf bifurcation when the delay crosses through a sequence of critical values. Based on the normal form theory and the center manifold reduction for partial functional differential equations (PFDEs), we find that the bifurcating periodic solution occurring from the first Hopf bifurcation point is stable on the center manifold and those occurring from the other bifurcation points are unstable. Finally, some numerical simulations are given to illustrate our results.

Suggested Citation

  • Li, Wan-Tong & Yan, Xiang-Ping & Zhang, Cun-Hua, 2008. "Stability and Hopf bifurcation for a delayed cooperation diffusion system with Dirichlet boundary conditions," Chaos, Solitons & Fractals, Elsevier, vol. 38(1), pages 227-237.
    • Handle: RePEc:eee:chsofr:v:38:y:2008:i:1:p:227-237
    DOI: 10.1016/j.chaos.2006.11.015
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    References listed on IDEAS

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    1. Berezowski, M. & Fudała, E., 2006. "Bifurcation analysis of the statics and dynamics of a logistic model with two delays," Chaos, Solitons & Fractals, Elsevier, vol. 28(2), pages 543-554.
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    Cited by:

    1. Jankovic, Masha & Petrovskii, Sergei & Banerjee, Malay, 2016. "Delay driven spatiotemporal chaos in single species population dynamics models," Theoretical Population Biology, Elsevier, vol. 110(C), pages 51-62.
    2. Vsevolod G. Sorokin & Andrei V. Vyazmin, 2022. "Nonlinear Reaction–Diffusion Equations with Delay: Partial Survey, Exact Solutions, Test Problems, and Numerical Integration," Mathematics, MDPI, vol. 10(11), pages 1-39, May.
    3. Gökçe, Aytül & Yazar, Samire & Sekerci, Yadigar, 2022. "Stability of spatial patterns in a diffusive oxygen–plankton model with time lag effect," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 194(C), pages 109-123.
    4. Hu, Guang-Ping & Li, Wan-Tong & Yan, Xiang-Ping, 2009. "Hopf bifurcations in a predator–prey system with multiple delays," Chaos, Solitons & Fractals, Elsevier, vol. 42(2), pages 1273-1285.
    5. Andrei D. Polyanin & Vsevolod G. Sorokin, 2023. "Exact Solutions of Reaction–Diffusion PDEs with Anisotropic Time Delay," Mathematics, MDPI, vol. 11(14), pages 1-19, July.
    6. Han, Renji & Dai, Binxiang, 2017. "Hopf bifurcation in a reaction-diffusive two-species model with nonlocal delay effect and general functional response," Chaos, Solitons & Fractals, Elsevier, vol. 96(C), pages 90-109.
    7. Andrei D. Polyanin & Alexei I. Zhurov, 2022. "Multi-Parameter Reaction–Diffusion Systems with Quadratic Nonlinearity and Delays: New Exact Solutions in Elementary Functions," Mathematics, MDPI, vol. 10(9), pages 1-28, May.
    8. Andrei D. Polyanin & Vsevolod G. Sorokin, 2023. "Reductions and Exact Solutions of Nonlinear Wave-Type PDEs with Proportional and More Complex Delays," Mathematics, MDPI, vol. 11(3), pages 1-25, January.

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