Advertisement

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
  • 128 Downloads

Abstract

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.

Keywords

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

Notes

Acknowledgements

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.

References

  1. 1.
    Volterra, V.: Variations and fluctuations of the number of individuals in animal species living together. ICES J. Mar. Sci. 3(1), 3–51 (1928)CrossRefGoogle Scholar
  2. 2.
    Sabelis, M.W., Diekmann, O., Jansen, V.A.A.: Metapopulation persistence despite local extinction: predator-prey patch models of the Lotka-Volterra type. Biol. J. Linn. Soc. 42, 267–283 (1991)CrossRefGoogle Scholar
  3. 3.
    Boukal, D.S., Křivan, V.: Lyapunov functions for Lotka-Volterra predator-prey models with optimal foraging behavior. J. Math. Biol. 39, 493–517 (1999)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Seo, G., DeAngelis, D.L.: A predator-prey model with a Holling type I functional response including a predator mutual interference. J. Nonlinear. Sci. 21, 811–833 (2011)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Pang, P.Y.H., Wang, M.: Strategy and stationary pattern in a three-species predator-prey model. J. Differ. Equ. 200, 245–273 (2004)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Aziz-Alaoui, M.A., Daher, M.: Okiye: boundedness and global stability for a predator-prey model with modified Leslie-Gower and Holling-type II schemes. Appl. Math. Lett. 16, 1069–1075 (2003)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Zhang, S., Meng, X., Zhang, T.: Dynamics analysis and numerical simulations of a stochastic non-autonomous predator-prey system with impulsive effects. Nonlinear Anal. Hybrid Syst. 26, 19–37 (2017)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Roy, A.B., Solimano, F.: Global stability and oscillations in classical Lotka-Volterra loops. J. Math. Biol. 24, 603–617 (1987)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Choo, S.: Global stability in n-dimensional discrete Lotka-Volterra predator-prey models. Adv. Differ. Equ. NY. 11, 1–17 (2014)MathSciNetGoogle Scholar
  10. 10.
    Beretta, E., Capasso, V., Rinaldi, F.: Global stability results for a generalized Lotka-Volterra system with distributed delays: applications to predator-prey and to epidemic systems. J. Math. Biol. 26, 661–688 (1988)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Kuang, Y., Smith, H.L.: Global stability for infinite delay Lotka-Volterra type systems. J. Differ. Equ. 103, 221–246 (1993)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Li, Y., Kuang, Y.: Periodic solutions of periodic delay Lotka-Volterra equations and systems. J. Math. Anal. Appl. 255, 260–280 (2001)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Zhu, G., Meng, X., Chen, L.: The dynamics of a mutual interference age structured predator-prey model with time delay and impulsive perturbations on predators. Appl. Math. Comput. 216(1), 308–316 (2010)MathSciNetzbMATHGoogle Scholar
  14. 14.
    Wang, B., Yan, J., Cheng, J., Zhong, S.: New criteria of stability analysis for generalized neural networks subject to time-varying delayed signals. Appl. Math. Comput. 314, 322–333 (2017)MathSciNetGoogle Scholar
  15. 15.
    Zeng, G., Chen, L., Chen, J.: Persistence and periodic orbits for two-species nonautonomous diffusion Lotka-Volterra models. Math. Comput. Model. 20, 69–80 (1994)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Cao, F., Chen, L.: Asymptotic behavior of nonautonomous diffusive Lotka-Volterra model. System Sci. Math. Sci. 11, 107–111 (1998)MathSciNetzbMATHGoogle Scholar
  17. 17.
    Cui, J., Chen, L.: Permanence and extinction in logistic and Lotka-Volterra systems with diffusion. J. Math. Anal. Appl. 258, 512–535 (2001)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Hastings, A.: Global stability in Lotka-Volterra systems with diffusion. J. Math. Biol. 6, 163–168 (1978)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Flint, M.L., van den Bosch, R.: Introduction to integrated pest management. Plenum press, New York (1981)CrossRefGoogle Scholar
  20. 20.
    Van Lenteren, J.C.: Integrated pest management in protected crops. In: Dent D (ed) Integrated Pest Management, pp. 311–320. Chapman Hall, London (1995)Google Scholar
  21. 21.
    Tang, S.Y., Chen, L.S.: Modelling and analysis of integrated pest management strategy. Discrete Cont. Dyn. B 4, 759–768 (2004)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Tang, S.Y., Cheke, R.A.: State-dependent impulsive models of integrated pest management (IPM) strategies and their dynamic consequences. J. Math. Biol. 50, 257–292 (2005)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Tang, S.Y., Xiao, Y.N., Chen, L.S., Cheke, R.A.: Integrated pest management models and their dynamical behaviour. B. Math. Biol. 67, 115–135 (2005)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Liu, B., Zhang, Y.J., Chen, L.S., Sun, L.H.: The dynamics of a prey-dependent consumption model concerning integrated pest management. Acta Math. Sin. 21(3), 541–554 (2005)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Liu, X.N., Chen, L.S.: Complex dynamics of Holling type II Lotka-Volterra predator-prey system with impulsive perturbations on the predator. Chaos Solitons Fract. 16, 311–320 (2003)MathSciNetCrossRefGoogle Scholar
  26. 26.
    Tang, S.Y., Tang, B., Wang, A.L., Xiao, Y.N.: Holling II predator-prey impulsive semi-dynamic model with complex Poincare map. Nonlinear Dynam. 81, 1579–1596 (2015)MathSciNetzbMATHGoogle Scholar
  27. 27.
    Yang, J., Tang, S.: Holling type II predator-prey model with nonlinear pulse as state-dependent feedback control. J. Comput. Appl. Math. 291, 225–241 (2016)MathSciNetCrossRefGoogle Scholar
  28. 28.
    Feng, L., Liu, Z.: An impulsive periodic predator-prey Lotka-Volterra type dispersal system with mixed functional responses. J. Appl. Math. Comput. 45, 235–257 (2014)MathSciNetCrossRefGoogle Scholar
  29. 29.
    Tang, S.Y., Pang, W.H., Cheke, R.A., Wu, J.H.: Global dynamics of a state-dependent feedback control system. Adv. Differ. Equ. 2015(1), 322 (2015)MathSciNetCrossRefGoogle Scholar
  30. 30.
    Wang, X., Tian, Y., Tang, S.: A holling type II pest and natural enemy model with density dependent IPM strategy. Math. Probl. Eng. 2017, 1–12 (2017)MathSciNetGoogle Scholar
  31. 31.
    Corless, R.M., Gonnet, G.H., Hare, D.E.G., Jeffrey, D.J., Knuth, D.E.: On The Lambert W function. Adv. Comput. Math. 5, 329–359 (1996)MathSciNetCrossRefGoogle Scholar
  32. 32.
    Ciesielski, K.: On stability in impulsive dynamical systems. Bull. Pol. Acad. Sci. Math. 52, 81–91 (2004)MathSciNetCrossRefGoogle Scholar
  33. 33.
    Kaul, S.: On impulsive semidynamical systems. J. Math. Anal. Appl. 150, 120–128 (1990)MathSciNetCrossRefGoogle Scholar
  34. 34.
    Kaul, S.: On impulsive semidynamical systems III: Lyapunov stability. Recent Trends Differ. Equ. Ser. Appl. Anal. 1, 335–345 (1992)Google Scholar
  35. 35.
    Bainov, D.D., Simeonov, P.S.: Systems with Impulse Effect: Stability. Theory and Applications. Ellis Hordwood limited, Chichester (1989)zbMATHGoogle Scholar
  36. 36.
    Ciesielski, K.: On semicontinuity in impulsive dynamical systems. Bull. Pol. Acad. Sci. Math. 52, 71–80 (2004)MathSciNetCrossRefGoogle Scholar
  37. 37.
    Gao, W., Tang, S.Y.: The effects of impulsive releasing methods of natural enemies on pest control and dynamical complexity. Nonlinear Anal. Hybri. 5, 540–553 (2011)MathSciNetCrossRefGoogle Scholar
  38. 38.
    Jiang, G., Lu, Q., Qian, L.: Complex dynamics of a Holling type II prey-predator system with state feedback control. Chaos Solitons Fract. 31, 448–461 (2007)MathSciNetCrossRefGoogle Scholar
  39. 39.
    Li, S., Liu, W.: A delayed Holling type III functional response predator-prey system with impulsive perturbation on the prey. Adv. Differ. Equ. 2016, 42 (2016)MathSciNetCrossRefGoogle Scholar
  40. 40.
    Huang, M.Z., Li, J.X., Song, X.Y., Guo, H.J.: Modeling impulsive injections of insulin: towards artificial pancreas. SIAM J. Appl. Math. 72, 1524–1548 (2012)MathSciNetCrossRefGoogle Scholar

Copyright information

© 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

Personalised recommendations