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The dynamics of a Beddington-type system with impulsive control strategy

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  • Li, Zhenqing
  • Wang, Weiming
  • Wang, Hailing

Abstract

In this paper, by using the theories and methods of ecology and ordinary differential equation, a prey–predator system with Beddington-type functional response and impulsive control strategy is established. Conditions for the system to be extinct are given by using the theories of impulsive equation and small amplitude perturbation skills. It is proved that the system is permanent via the method of comparison involving multiple Liapunov functions. Furthermore, by using the method of numerical simulation, the influence of the impulsive control strategy on the inherent oscillation are investigated, which shows rich dynamics, such as period doubling bifurcation, crises, symmetry-breaking pitchfork bifurcations, chaotic bands, quasi-periodic oscillation, narrow periodic window, wide periodic window, period-halving bifurcation, etc. That will be useful for study of the dynamic complexity of ecosystems.

Suggested Citation

  • Li, Zhenqing & Wang, Weiming & Wang, Hailing, 2006. "The dynamics of a Beddington-type system with impulsive control strategy," Chaos, Solitons & Fractals, Elsevier, vol. 29(5), pages 1229-1239.
    • Handle: RePEc:eee:chsofr:v:29:y:2006:i:5:p:1229-1239
    DOI: 10.1016/j.chaos.2005.08.195
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    References listed on IDEAS

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    1. Zhang, Yujuan & Xiu, Zhilong & Chen, Lansun, 2005. "Dynamic complexity of a two-prey one-predator system with impulsive effect," Chaos, Solitons & Fractals, Elsevier, vol. 26(1), pages 131-139.
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    Cited by:

    1. Lv, Songjuan & Zhao, Min, 2008. "The dynamic complexity of a host–parasitoid model with a lower bound for the host," Chaos, Solitons & Fractals, Elsevier, vol. 36(4), pages 911-919.
    2. Wang, Hailing & Wang, Weiming, 2008. "The dynamical complexity of a Ivlev-type prey–predator system with impulsive effect," Chaos, Solitons & Fractals, Elsevier, vol. 38(4), pages 1168-1176.
    3. Wang, Weiming & Wang, Xiaoqin & Lin, Yezhi, 2008. "Complicated dynamics of a predator–prey system with Watt-type functional response and impulsive control strategy," Chaos, Solitons & Fractals, Elsevier, vol. 37(5), pages 1427-1441.
    4. Wang, Weiming & Wang, Hailing & Li, Zhenqing, 2008. "Chaotic behavior of a three-species Beddington-type system with impulsive perturbations," Chaos, Solitons & Fractals, Elsevier, vol. 37(2), pages 438-443.
    5. Zhao, Min & Wang, Xitao & Yu, Hengguo & Zhu, Jun, 2012. "Dynamics of an ecological model with impulsive control strategy and distributed time delay," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 82(8), pages 1432-1444.
    6. Wang, Xiaoqin & Wang, Weiming & Lin, Xiaolin, 2009. "Dynamics of a two-prey one-predator system with Watt-type functional response and impulsive control strategy," Chaos, Solitons & Fractals, Elsevier, vol. 40(5), pages 2392-2404.
    7. Lv, Songjuan & Fang, Zhongmiao, 2009. "The dynamic complexity of a host–parasitoid model with a Beddington–DeAngelis functional response," Chaos, Solitons & Fractals, Elsevier, vol. 41(5), pages 2617-2623.
    8. 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.
    9. Yu, Hengguo & Zhong, Shouming & Ye, Mao, 2009. "Dynamic analysis of an ecological model with impulsive control strategy and distributed time delay," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 80(3), pages 619-632.
    10. Yu, Hengguo & Zhao, Min & Lv, Songjuan & Zhu, Lili, 2009. "Dynamic complexities in a parasitoid-host-parasitoid ecological model," Chaos, Solitons & Fractals, Elsevier, vol. 39(1), pages 39-48.
    11. Zhao, Jiantao & Wei, Junjie, 2009. "Stability and bifurcation in a two harmful phytoplankton–zooplankton system," Chaos, Solitons & Fractals, Elsevier, vol. 39(3), pages 1395-1409.
    12. Wang, Xiaoqin & Wang, Weiming & Lin, Xiaolin, 2008. "Chaotic behavior of a Watt-type predator–prey system with impulsive control strategy," Chaos, Solitons & Fractals, Elsevier, vol. 37(3), pages 706-718.
    13. Kang, Li & Tang, Sanyi, 2016. "The reverse effects of random perturbation on discrete systems for single and multiple population models," Chaos, Solitons & Fractals, Elsevier, vol. 91(C), pages 198-209.
    14. 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.
    15. Wang, Xiaoqin & Wang, Weiming & Lin, Yezhi & Lin, Xiaolin, 2009. "The dynamical complexity of an impulsive Watt-type prey–predator system," Chaos, Solitons & Fractals, Elsevier, vol. 40(2), pages 731-744.
    16. Wang, Xiaoqin & Wang, Weiming & Lin, Xiaolin, 2009. "Dynamics of a periodic Watt-type predator–prey system with impulsive effect," Chaos, Solitons & Fractals, Elsevier, vol. 39(3), pages 1270-1282.

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