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Periodic solution of a Leslie predator–prey system with ratio-dependent and state impulsive feedback control

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Abstract

In this paper, a Leslie predator–prey system with ratio-dependent and state impulsive feedback control is investigated by applying the geometry theory of differential equation. When the economic threshold level is under the positive equilibrium, the existence, uniqueness and orbital asymptotical stability of order-1 periodic solution for the system can be obtained. When the economic threshold level is above the positive equilibrium, and the positive equilibrium is a focus point, sufficient conditions of the existence, uniqueness and orbital asymptotical stability of order-1 periodic solution for the system are also acquired. Furthermore, when the positive equilibrium is an unstable focus point, the existence of order-1 periodic solution of the impulsive system can be obtained within limit cycle of the continuous system. The mathematical results can be verified by numerical simulations.

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Acknowledgements

This work was supported by the National Natural Science Foundation of PR China (61364020,11361068) and the Major Research Programmes of Yulin Normal University of PR China (2015YJZD02).

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Authors and Affiliations

  1. College of Mathematics and statistics, Guangxi Colleges and Universities Key Lab of Complex System Optimization and Large Data Processing, Yulin Normal University, Yulin, 537000, Guangxi, China

    Zhiqing Liang, Xiaping Zeng, Guoping Pang & Yanhong Liang

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  1. Zhiqing Liang
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  2. Xiaping Zeng
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  3. Guoping Pang
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  4. Yanhong Liang
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Correspondence to Zhiqing Liang.

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Liang, Z., Zeng, X., Pang, G. et al. Periodic solution of a Leslie predator–prey system with ratio-dependent and state impulsive feedback control. Nonlinear Dyn 89, 2941–2955 (2017). https://doi.org/10.1007/s11071-017-3637-4

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  • DOI: https://doi.org/10.1007/s11071-017-3637-4

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