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Extinction and permanence of a two-prey two-predator system with impulsive on the predator

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  • Xiang, Zhongyi
  • Song, Xinyu

Abstract

In this paper, the dynamic behaviors of a two-prey two-predator system with impulsive effect on the predator of fixed moment are investigated. By applying the Floquet theory of liner periodic impulsive equation, we show that there exists a globally asymptotically stable two-prey eradication periodic solution when the impulsive period is less than some critical value. Further, we prove that the system is permanent if the impulsive period is large than some critical value, and meanwhile the conditions for the extinction of one of the two prey and permanence of the remaining three species are given. Finally, numerical simulation shows that there exists a stable positive periodic solution with a maximum value no larger than a given level. Thus, we can use the stability of the positive periodic solution and its period to control insect pests at acceptably low levels.

Suggested Citation

  • Xiang, Zhongyi & Song, Xinyu, 2006. "Extinction and permanence of a two-prey two-predator system with impulsive on the predator," Chaos, Solitons & Fractals, Elsevier, vol. 29(5), pages 1121-1136.
    • Handle: RePEc:eee:chsofr:v:29:y:2006:i:5:p:1121-1136
    DOI: 10.1016/j.chaos.2005.08.076
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    References listed on IDEAS

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    1. Zhang, Shuwen & Chen, Lansun, 2005. "Chaos in three species food chain system with impulsive perturbations," Chaos, Solitons & Fractals, Elsevier, vol. 24(1), pages 73-83.
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    1. Xiang, Zhongyi & Song, Xinyu & Zhang, Fengqin, 2008. "Bifurcation and complex dynamics of a two-prey two-predator system concerning periodic biological and chemical control," Chaos, Solitons & Fractals, Elsevier, vol. 37(2), pages 424-437.
    2. 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.
    3. 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.
    4. Xiang, Zhongyi & Song, Xinyu, 2009. "The dynamical behaviors of a food chain model with impulsive effect and Ivlev functional response," Chaos, Solitons & Fractals, Elsevier, vol. 39(5), pages 2282-2293.

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