Nonlinear Dynamics

, Volume 94, Issue 3, pp 2243–2263 | Cite as

Nonlinear state-dependent feedback control of a pest-natural enemy system

  • Yuan Tian
  • Sanyi TangEmail author
  • Robert A. Cheke
Original Paper


The numbers of pests and of natural enemies released to control them as part of integrated pest management strategies are density dependent. Therefore, the numbers of natural enemies to be released and the rate at which they kill pests should depend on their densities when the number of the pest population has reached the economic threshold. Bearing this in mind, a classic Lotka–Volterra system but with nonlinear state-dependent feedback control tactics is proposed and analysed in this paper. Furthermore, the definition and properties of the Poincaré map which is defined in the phase set were investigated for various cases, allowing us to address the existence and global stability of an order-1 periodic solution of the model with nonlinear feedback control. Moreover, the existence and nonexistence of periodic solutions with an order larger than 2 or 3 are also discussed. The modelling methods and analytical techniques developed could be widely used and applied in other systems with threshold control such as the glucose insulin regulatory system.


Nonlinear control Poincaré map Order-1 periodic solution Global stability Pest control 



This work was supported by the National Natural Science Foundation of China (NSFCs, 11471201, 11631012, 61772017), and by the Fundamental Research Funds for the Central Universities GK201701001, and by the Youth Foundation of Hubei University For Nationalities MY2017Q007.

Compliance with ethical standards

Conflicts of interest

All the authors declare that they have no conflict of interest.


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© Springer Nature B.V. 2018

Authors and Affiliations

  1. 1.School of Mathematics and Information ScienceShaanxi Normal UniversityXi’anPeople’s Republic of China
  2. 2.School of ScienceHubei University For NationalitiesEnshiPeople’s Republic of China
  3. 3.Natural Resources InstituteUniversity of Greenwich at MedwayChathamUK

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