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Dynamic Analysis of a Delayed Reaction‐Diffusion Predator‐Prey System with Modified Holling‐Tanner Functional Response

Author

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

Abstract

A predator‐prey model with modified Holling‐Tanner functional response and time delays is considered. By regarding the delays as bifurcation parameters, the local and global asymptotic stability of the positive equilibrium are investigated. The system has been found to undergo a Hopf bifurcation at the positive equilibrium when the delays cross through a sequence of critical values. In addition, the direction of the Hopf bifurcation and the stability of bifurcated periodic solutions are also studied, and an explicit algorithm is obtained by applying normal form theory and the center manifold theorem. The main results are illustrated by numerical simulations.

Suggested Citation

  • Xinhong Pan & Min Zhao & Chuanjun Dai & Yapei Wang, 2015. "Dynamic Analysis of a Delayed Reaction‐Diffusion Predator‐Prey System with Modified Holling‐Tanner Functional Response," Abstract and Applied Analysis, John Wiley & Sons, vol. 2015(1).
    • Handle: RePEc:wly:jnlaaa:v:2015:y:2015:i:1:n:620891
    DOI: 10.1155/2015/620891
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    References listed on IDEAS

    as
    1. Zhao, Min & Lv, Songjuan, 2009. "Chaos in a three-species food chain model with a Beddington–DeAngelis functional response," Chaos, Solitons & Fractals, Elsevier, vol. 40(5), pages 2305-2316.
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    3. Ruiqing Shi & Junmei Qi & Sanyi Tang, 2013. "Stability and Bifurcation Analysis for a Predator‐Prey Model with Discrete and Distributed Delay," Abstract and Applied Analysis, John Wiley & Sons, vol. 2013(1).
    4. Yanhui Zhai & Haiyun Bai & Ying Xiong & Xiaona Ma, 2013. "Hopf Bifurcation Analysis for the Modified Rayleigh Price Model with Time Delay," Abstract and Applied Analysis, John Wiley & Sons, vol. 2013(1).
    5. Lv, Songjuan & Zhao, Min, 2008. "The dynamic complexity of a three species food chain model," Chaos, Solitons & Fractals, Elsevier, vol. 37(5), pages 1469-1480.
    6. 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.
    7. Fengying Wei & Yuhua Cai, 2013. "Global Asymptotic Stability of Stochastic Nonautonomous Lotka‐Volterra Models with Infinite Delay," Abstract and Applied Analysis, John Wiley & Sons, vol. 2013(1).
    8. 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).
    9. Ruiqing Shi & Junmei Qi & Sanyi Tang, 2013. "Stability and Bifurcation Analysis for a Predator-Prey Model with Discrete and Distributed Delay," Abstract and Applied Analysis, Hindawi, vol. 2013, pages 1-12, June.
    10. Yanhui Zhai & Haiyun Bai & Ying Xiong & Xiaona Ma, 2013. "Hopf Bifurcation Analysis for the Modified Rayleigh Price Model with Time Delay," Abstract and Applied Analysis, Hindawi, vol. 2013, pages 1-6, August.
    11. Zhang, Limin & Zhao, Min, 2009. "Dynamic complexities in a hyperparasitic system with prolonged diapause for host," Chaos, Solitons & Fractals, Elsevier, vol. 42(2), pages 1136-1142.
    12. 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.
    13. 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).
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