Nonlinear Dynamics

, Volume 78, Issue 2, pp 921–938 | Cite as

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

  • Tongqian Zhang
  • Xinzhu Meng
  • Rui Liu
  • Tonghua Zhang
Original Paper

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.

Keywords

Gompertz growth rate Integrated pest management (IPM) State impulse State feedback control Periodic solution Stability 

Notes

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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Copyright information

© Springer Science+Business Media Dordrecht 2014

Authors and Affiliations

  • Tongqian Zhang
    • 1
  • Xinzhu Meng
    • 1
  • Rui Liu
    • 1
  • Tonghua Zhang
    • 2
  1. 1.College of Mathematics and Systems ScienceShandong University of Science and TechnologyQingdaoPeople’s Republic of China
  2. 2.Department of MathematicsSwinburne University of TechnologyMelbourneAustralia

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