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Stability and Hopf bifurcation in a ratio-dependent predator–prey system with stage structure

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  • Xu, Rui
  • Ma, Zhien

Abstract

A ratio-dependent predator–prey model with stage structure for the predator and time delay due to the gestation of the predator is investigated. By analyzing the characteristic equations, the local stability of a positive equilibrium and a boundary equilibrium is discussed, respectively. Further, it is proved that the system undergoes a Hopf bifurcation at the positive equilibrium when τ=τ0. By using an iteration technique, sufficient conditions are derived for the global attractivity of the positive equilibrium. By comparison arguments, sufficient conditions are obtained for the global stability of the boundary equilibrium. Numerical simulations are carried out to illustrate the main results.

Suggested Citation

  • Xu, Rui & Ma, Zhien, 2008. "Stability and Hopf bifurcation in a ratio-dependent predator–prey system with stage structure," Chaos, Solitons & Fractals, Elsevier, vol. 38(3), pages 669-684.
    • Handle: RePEc:eee:chsofr:v:38:y:2008:i:3:p:669-684
    DOI: 10.1016/j.chaos.2007.01.019
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    References listed on IDEAS

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    1. Jing, Zhujun & Yang, Jianping, 2006. "Bifurcation and chaos in discrete-time predator–prey system," Chaos, Solitons & Fractals, Elsevier, vol. 27(1), pages 259-277.
    2. Sun, Chengjun & Han, Maoan & Lin, Yiping, 2007. "Analysis of stability and Hopf bifurcation for a delayed logistic equation," Chaos, Solitons & Fractals, Elsevier, vol. 31(3), pages 672-682.
    3. El-Sheikh, M.M.A. & Mahrouf, S.A.A., 2005. "Stability and bifurcation of a simple food chain in a chemostat with removal rates," Chaos, Solitons & Fractals, Elsevier, vol. 23(4), pages 1475-1489.
    4. Liu, Zhihua & Yuan, Rong, 2006. "Stability and bifurcation in a harvested one-predator–two-prey model with delays," Chaos, Solitons & Fractals, Elsevier, vol. 27(5), pages 1395-1407.
    5. Chen, Yuanyuan & Yu, Jiang & Sun, Chengjun, 2007. "Stability and Hopf bifurcation analysis in a three-level food chain system with delay," Chaos, Solitons & Fractals, Elsevier, vol. 31(3), pages 683-694.
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    Cited by:

    1. Jana, Debaldev & Agrawal, Rashmi & Upadhyay, Ranjit Kumar, 2015. "Dynamics of generalist predator in a stochastic environment: Effect of delayed growth and prey refuge," Applied Mathematics and Computation, Elsevier, vol. 268(C), pages 1072-1094.
    2. Changjin Xu, 2017. "Delay-Induced Oscillations in a Competitor-Competitor-Mutualist Lotka-Volterra Model," Complexity, Hindawi, vol. 2017, pages 1-12, April.
    3. Wang, Xuedi & Peng, Miao & Liu, Xiuyu, 2015. "Stability and Hopf bifurcation analysis of a ratio-dependent predator–prey model with two time delays and Holling type III functional response," Applied Mathematics and Computation, Elsevier, vol. 268(C), pages 496-508.
    4. Jana, Soovoojeet & Chakraborty, Milon & Chakraborty, Kunal & Kar, T.K., 2012. "Global stability and bifurcation of time delayed prey–predator system incorporating prey refuge," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 85(C), pages 57-77.
    5. Xiaohong Tian & Rui Xu, 2011. "Global Stability of a Virus Infection Model with Time Delay and Absorption," Discrete Dynamics in Nature and Society, Hindawi, vol. 2011, pages 1-20, June.

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