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Stability and Hopf Bifurcation Analysis of a Nutrient‐Phytoplankton Model with Delay Effect

Author

Listed:
  • Xinhong Pan
  • Min Zhao
  • Chuanjun Dai
  • Yapei Wang

Abstract

A delay differential system is investigated based on a previously proposed nutrient‐phytoplankton model. The time delay is regarded as a bifurcation parameter. Our aim is to determine how the time delay affects the system. First, we study the existence and local stability of two equilibria using the characteristic equation and identify the condition where a Hopf bifurcation can occur. Second, the formulae that determine the direction of the Hopf bifurcation and the stability of periodic solutions are obtained using the normal form and the center manifold theory. Furthermore, our main results are illustrated using numerical simulations.

Suggested Citation

  • Xinhong Pan & Min Zhao & Chuanjun Dai & Yapei Wang, 2014. "Stability and Hopf Bifurcation Analysis of a Nutrient‐Phytoplankton Model with Delay Effect," Abstract and Applied Analysis, John Wiley & Sons, vol. 2014(1).
    • Handle: RePEc:wly:jnlaaa:v:2014:y:2014:i:1:n:471507
    DOI: 10.1155/2014/471507
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    References listed on IDEAS

    as
    1. Weiwei Zhang & Min Zhao, 2013. "Dynamical Complexity of a Spatial Phytoplankton‐Zooplankton Model with an Alternative Prey and Refuge Effect," Journal of Applied Mathematics, John Wiley & Sons, vol. 2013(1).
    2. Yanuo Zhu & Yongli Cai & Shuling Yan & Weiming Wang, 2012. "Dynamical Analysis of a Delayed Reaction-Diffusion Predator-Prey System," Abstract and Applied Analysis, Hindawi, vol. 2012, pages 1-23, October.
    3. Weiwei Zhang & Min Zhao, 2013. "Dynamical Complexity of a Spatial Phytoplankton-Zooplankton Model with an Alternative Prey and Refuge Effect," Journal of Applied Mathematics, Hindawi, vol. 2013, pages 1-10, May.
    4. Ming Zhao, 2013. "Hopf Bifurcation Analysis for a Semiratio-Dependent Predator-Prey System with Two Delays," Abstract and Applied Analysis, Hindawi, vol. 2013, pages 1-13, September.
    5. Xinze Lian & Shuling Yan & Hailing Wang, 2013. "Pattern Formation in Predator‐Prey Model with Delay and Cross Diffusion," Abstract and Applied Analysis, John Wiley & Sons, vol. 2013(1).
    6. Xinze Lian & Shuling Yan & Hailing Wang, 2013. "Pattern Formation in Predator-Prey Model with Delay and Cross Diffusion," Abstract and Applied Analysis, Hindawi, vol. 2013, pages 1-10, November.
    7. Yanuo Zhu & Yongli Cai & Shuling Yan & Weiming Wang, 2012. "Dynamical Analysis of a Delayed Reaction‐Diffusion Predator‐Prey System," Abstract and Applied Analysis, John Wiley & Sons, vol. 2012(1).
    8. Ming Zhao, 2013. "Hopf Bifurcation Analysis for a Semiratio‐Dependent Predator‐Prey System with Two Delays," Abstract and Applied Analysis, John Wiley & Sons, vol. 2013(1).
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    Citations

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    Cited by:

    1. Beibei Wang & Min Zhao & Chuanjun Dai & Hengguo Yu & Nan Wang & Pengfei Wang, 2016. "Dynamics Analysis of a Nutrient‐Plankton Model with a Time Delay," Discrete Dynamics in Nature and Society, John Wiley & Sons, vol. 2016(1).
    2. Du, Yanfei & Zhang, Cong, 2025. "The effects of delays on stability and spatiotemporal dynamics in a diffusive nutrient-phytoplankton model," Chaos, Solitons & Fractals, Elsevier, vol. 199(P3).

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