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Periodic solution of a pest management Gompertz model with impulsive state feedback control

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Abstract

In this paper, a new model with two state impulses is proposed for pest management. According to different thresholds, an integrated strategy of pest management is considered, that is to say if the density of the pest population reaches the lower threshold \(h_1\) at which pests cause slight damage to the forest, biological control (releasing natural enemy) will be taken to control pests; while if the density of the pest population reaches the higher threshold \(h_2\) at which pests cause serious damage to the forest, both chemical control (spraying pesticide) and biological control (releasing natural enemy) will be taken at the same time. For the model, firstly, we qualitatively analyse its singularity. Then, we investigate the existence of periodic solution by successor functions and Poincaré-Bendixson theorem and the stability of periodic solution by the stability theorem for periodic solutions of impulsive differential equations. Lastly, we use numerical simulations to illustrate our theoretical results.

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Acknowledgments

We would like to thank the anonymous referees for their valuable comments and suggestions. The research has been supported by National Natural Science Foundation of China (Grant No. 11371230), Natural Sciences Fund of Shandong Province (Grant No. ZR2012AM012) and A Project for Higher Educational Science and Technology Program of Shandong Province (Grant No. J13LI05).

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Authors and Affiliations

  1. College of Mathematics and Systems Science, Shandong University of Science and Technology, Qingdao, 266590, People’s Republic of China

    Tongqian Zhang, Xinzhu Meng & Rui Liu

  2. Department of Mathematics, Swinburne University of Technology, Melbourne, VIC , 3122, Australia

    Tonghua Zhang

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  1. Tongqian Zhang
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  2. Xinzhu Meng
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Correspondence to Tongqian Zhang.

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Zhang, T., Meng, X., Liu, R. et al. Periodic solution of a pest management Gompertz model with impulsive state feedback control. Nonlinear Dyn 78, 921–938 (2014). https://doi.org/10.1007/s11071-014-1486-y

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