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Dynamics in two nonsmooth predator–prey models with threshold harvesting

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Abstract

Considering a good pest control program should reduce the pest to levels acceptable to the public, we investigate the threshold harvesting policy on pests in two predator–prey models. Both models are nonsmooth and the aim of this paper is to provide how threshold harvesting affects the dynamics of the two systems. When the harvesting threshold is larger than some positive level, the harvesting does not affect the ecosystem; when the harvesting threshold is less than the level, the model has complex dynamics with multiple coexistence equilibria, limit cycle, bistability, homoclinic orbit, saddle-node bifurcation, transcritical bifurcation, subcritical and supercritical Hopf bifurcation, Bogdanov–Takens bifurcation, and discontinuous Hopf bifurcation. Firstly, we provide the complete stability analysis and bifurcation analysis for the two models. Furthermore, some numerical simulations are given to illustrate our results. Finally, it is found that harvesting lowers the level of both species for natural enemy–pest system while raises the densities of both species for the pest–crop system. It is seen that the threshold harvesting policy of the enemy system is more effective than the crop system.

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Acknowledgements

We are extremely grateful to the critical comments and invaluable suggestions made by anonymous honorable reviewers. This work is supported by the Natural Science Foundation of China, the Doctoral Program of Higher Education of China and National Natural Science Foundation of China (11101305).

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Authors and Affiliations

  1. School of Mathematical Sciences, Laboratory of Mathematics and Complex Systems, Beijing Normal University, Beijing, 100875, China

    Yunfei Lv & Rong Yuan

  2. School of Science, Tianjin Polytechnic University, Tianjin, 300387, China

    Yongzhen Pei

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  1. Yunfei Lv
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  2. Rong Yuan
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  3. Yongzhen Pei
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Correspondence to Rong Yuan.

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Lv, Y., Yuan, R. & Pei, Y. Dynamics in two nonsmooth predator–prey models with threshold harvesting. Nonlinear Dyn 74, 107–132 (2013). https://doi.org/10.1007/s11071-013-0952-2

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  • DOI: https://doi.org/10.1007/s11071-013-0952-2

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