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Stability and Hopf Bifurcation Analysis for a Stage‐Structured Predator‐Prey Model with Discrete and Distributed Delays

Author

Listed:
  • Ruiqing Shi
  • Junmei Qi
  • Sanyi Tang

Abstract

We propose a three‐dimensional stage‐structured predatory‐prey model with discrete and distributed delays. By use of a new variable, the original three‐dimensional system transforms into an equivalent four‐dimensional system. Firstly, we study the existence and local stability of positive equilibrium of the new system. And, by choosing the time delay τ as a bifurcation parameter, we show that Hopf bifurcation may occur as the time delay τ passes through some critical values. Secondly, by use of normal form theory and central manifold argument, we establish the direction and stability of Hopf bifurcation. At last, some simple discussion is presented.

Suggested Citation

  • Ruiqing Shi & Junmei Qi & Sanyi Tang, 2013. "Stability and Hopf Bifurcation Analysis for a Stage‐Structured Predator‐Prey Model with Discrete and Distributed Delays," Journal of Applied Mathematics, John Wiley & Sons, vol. 2013(1).
    • Handle: RePEc:wly:jnljam:v:2013:y:2013:i:1:n:201936
    DOI: 10.1155/2013/201936
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    References listed on IDEAS

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    1. Sun, Chengjun & Han, Maoan & Lin, Yiping & Chen, Yuanyuan, 2007. "Global qualitative analysis for a predator–prey system with delay," Chaos, Solitons & Fractals, Elsevier, vol. 32(4), pages 1582-1596.
    2. Liao, Xiaofeng & Chen, Guanrong, 2005. "Hopf bifurcation and chaos analysis of Chen’s system with distributed delays," Chaos, Solitons & Fractals, Elsevier, vol. 25(1), pages 197-220.
    3. Wang, Fengyan & Zeng, Guangzhao, 2007. "Chaos in a Lotka–Volterra predator–prey system with periodically impulsive ratio-harvesting the prey and time delays," Chaos, Solitons & Fractals, Elsevier, vol. 32(4), pages 1499-1512.
    4. Liu, Xiaoli & Xiao, Dongmei, 2007. "Complex dynamic behaviors of a discrete-time predator–prey system," Chaos, Solitons & Fractals, Elsevier, vol. 32(1), pages 80-94.
    5. Zhang, Shuwen & Tan, Dejun & Chen, Lansun, 2006. "Chaotic behavior of a chemostat model with Beddington–DeAngelis functional response and periodically impulsive invasion," Chaos, Solitons & Fractals, Elsevier, vol. 29(2), pages 474-482.
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    Cited by:

    1. Bashir Al-Hdaibat & A. Alameer, 2025. "Bifurcation and Chaos in a Nonlinear Population Model with Delayed Nonlinear Interactions," Mathematics, MDPI, vol. 13(13), pages 1-23, June.

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