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Dynamics of an ecological model with impulsive control strategy and distributed time delay

Author

Listed:
  • Zhao, Min
  • Wang, Xitao
  • Yu, Hengguo
  • Zhu, Jun

Abstract

In this paper, using the theories and methods of ecology and ordinary differential equations, an ecological model with an impulsive control strategy and a distributed time delay is defined. Using the theory of the impulsive equation, small-amplitude perturbations, and comparative techniques, a condition is identified which guarantees the global asymptotic stability of the prey-(x) and predator-(y) eradication periodic solution. It is proved that the system is permanent. Furthermore, the influences of impulsive perturbations on the inherent oscillation are studied numerically, an oscillation which exhibits rich dynamics including period-halving bifurcation, chaotic narrow or wide windows, and chaotic crises. Computation of the largest Lyapunov exponent confirms the chaotic dynamic behavior of the model. All these results may be useful for study of the dynamic complexity of ecosystems.

Suggested Citation

  • 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.
    • Handle: RePEc:eee:matcom:v:82:y:2012:i:8:p:1432-1444
    DOI: 10.1016/j.matcom.2011.08.009
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    References listed on IDEAS

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    1. Chen, C.W. & Chiang, W.L. & Hsiao, F.H., 2004. "Stability analysis of T–S fuzzy models for nonlinear multiple time-delay interconnected systems," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 66(6), pages 523-537.
    2. 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.
    3. 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.
    4. Song, Xinyu & Li, Yongfeng, 2007. "Dynamic complexities of a Holling II two-prey one-predator system with impulsive effect," Chaos, Solitons & Fractals, Elsevier, vol. 33(2), pages 463-478.
    5. Chen, Cheng-Wu, 2009. "The stability of an oceanic structure with T–S fuzzy models," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 80(2), pages 402-426.
    6. 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.
    7. 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.
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    Cited by:

    1. Juneja, Nishant & Agnihotri, Kulbhushan & Kaur, Harpreet, 2018. "Effect of delay on globally stable prey–predator system," Chaos, Solitons & Fractals, Elsevier, vol. 111(C), pages 146-156.
    2. Dai, Chuanjun & Zhao, Min & Chen, Lansun, 2012. "Complex dynamic behavior of three-species ecological model with impulse perturbations and seasonal disturbances," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 84(C), pages 83-97.
    3. 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.
    4. Kim, Hye Kyung & Baek, Hunki, 2013. "The dynamical complexity of a predator–prey system with Hassell–Varley functional response and impulsive effect," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 94(C), pages 1-14.
    5. Tiancai Liao & Hengguo Yu & Chuanjun Dai & Min Zhao, 2019. "Impact of Cell Size Effect on Nutrient-Phytoplankton Dynamics," Complexity, Hindawi, vol. 2019, pages 1-23, November.

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