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Dynamic behaviors of the periodic Lotka–Volterra competing system with impulsive perturbations

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  • Liu, Bing
  • Teng, Zhidong
  • Liu, Wanbo

Abstract

In this paper, we investigate a classical periodic Lotka–Volterra competing system with impulsive perturbations. The conditions for the linear stability of trivial periodic solution and semi-trivial periodic solutions are given by applying Floquet theory of linear periodic impulsive equation, and we also give the conditions for the global stability of these solutions as a consequence of some abstract monotone iterative schemes introduced in this paper, which will be also used to get some sufficient conditions for persistence. By using the method of coincidence degree, the conditions for the existence of at least one strictly positive (componentwise) periodic solution are derived. The theoretical results are confirmed by a specific example and numerical simulations. It shows that the dynamic behaviors of the system we consider are quite different from the corresponding system without pulses.

Suggested Citation

  • Liu, Bing & Teng, Zhidong & Liu, Wanbo, 2007. "Dynamic behaviors of the periodic Lotka–Volterra competing system with impulsive perturbations," Chaos, Solitons & Fractals, Elsevier, vol. 31(2), pages 356-370.
    • Handle: RePEc:eee:chsofr:v:31:y:2007:i:2:p:356-370
    DOI: 10.1016/j.chaos.2005.09.059
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    References listed on IDEAS

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    1. Zhang, Shuwen & Chen, Lansun, 2005. "A Holling II functional response food chain model with impulsive perturbations," Chaos, Solitons & Fractals, Elsevier, vol. 24(5), pages 1269-1278.
    2. 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.
    3. Gao, Shujing & Chen, Lansun & Sun, Lihua, 2005. "Optimal pulse fishing policy in stage-structured models with birth pulses," Chaos, Solitons & Fractals, Elsevier, vol. 25(5), pages 1209-1219.
    4. Gao, Shujing & Chen, Lansun, 2005. "The effect of seasonal harvesting on a single-species discrete population model with stage structure and birth pulses," Chaos, Solitons & Fractals, Elsevier, vol. 24(4), pages 1013-1023.
    5. Zhang, Shuwen & Wang, Fengyan & Chen, Lansun, 2005. "A food chain model with impulsive perturbations and Holling IV functional response," Chaos, Solitons & Fractals, Elsevier, vol. 26(3), pages 855-866.
    6. 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. Li, Dong & Wang, Shilong & Zhang, Xiaohong & Yang, Dan, 2009. "Impulsive control of uncertain Lotka–Volterra predator–prey system," Chaos, Solitons & Fractals, Elsevier, vol. 41(4), pages 1572-1577.
    2. Jiao, Jianjun & Meng, Xinzhu & Chen, Lansun, 2009. "Harvesting policy for a delayed stage-structured Holling II predator–prey model with impulsive stocking prey," Chaos, Solitons & Fractals, Elsevier, vol. 41(1), pages 103-112.
    3. Wen, Sun & Chen, Shihua & Mei, Huihai, 2009. "Positive periodic solution of a more realistic three-species Lotka-Volterra model with delay and density regulation," Chaos, Solitons & Fractals, Elsevier, vol. 40(5), pages 2340-2348.
    4. Stamov, Gani Tr. & Simeonov, Stanislav & Stamova, Ivanka M., 2018. "Uncertain impulsive Lotka–Volterra competitive systems: Robust stability of almost periodic solutions," Chaos, Solitons & Fractals, Elsevier, vol. 110(C), pages 178-184.
    5. 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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