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


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.


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



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).


  1. 1.
    Bainov, D., Simeonov, P.S.: Systems with impulse effect: stability, theory, and applications. Ellis Horwood, Chichester (1989)MATHGoogle Scholar
  2. 2.
    Bainov, D.D., Hristova, S.G., Hu, S., et al.: Periodic boundary value problems for systems of first order impulsive differential equations. Differ. Integral Equ. 2, 37–43 (1989)MATHMathSciNetGoogle Scholar
  3. 3.
    Bainov, D., Simeonov, P.S.: Impulsive differential equations: periodic solutions and applications. Longman, Harlow (1993)MATHGoogle Scholar
  4. 4.
    Bale, J.S., Van Lenteren, J.C., Bigler, F.: Biological control and sustainable food production. Philos. Trans. R. Soc. B 363(1492), 761–776 (2008)CrossRefGoogle Scholar
  5. 5.
    Bonotto, E.M., Federson, M.: Topological conjugation and asymptotic stability in impulsive semidynamical systems. J. Math. Anal. Appl. 326(2), 869–881 (2007)CrossRefMATHMathSciNetGoogle Scholar
  6. 6.
    Bonotto, E.M., Federson, M.: Limit sets and the Poincaré–Bendixson theorem in impulsive semidynamical systems. J. Differ. Equ. 244(9), 2334–2349 (2008)CrossRefMATHMathSciNetGoogle Scholar
  7. 7.
    Bonotto, E.M.: LaSalle‘s theorems in impulsive semidynamical systems. Nonlinear Anal. 71(5), 2291–2297 (2009)CrossRefMATHMathSciNetGoogle Scholar
  8. 8.
    Erbe, L.H., Liu, X.: Existence of periodic solutions of impulsive differential systems. Int. J. Stoch. Anal. 4(2), 137–146 (1991)CrossRefMATHMathSciNetGoogle Scholar
  9. 9.
    Frigon, M., O’Regan, D.: Existence results for first-order impulsive differential equations. J. Math. Anal. Appl. 193(1), 96–113 (1995)CrossRefMATHMathSciNetGoogle Scholar
  10. 10.
    Canadian Forest Service.
  11. 11.
    Dai, C., Zhao, M., Chen, L.: Homoclinic bifurcation in semi-continuous dynamic systems. Int. J. Biomath. 5(06), 1250059 (2012)CrossRefMathSciNetGoogle Scholar
  12. 12.
    Focus On Forest Health, Alberta Environment and Alberta Sustainable Resource Development (2003)Google Scholar
  13. 13.
    Gompertz, B.: On the nature of the function expressive of the law of human mortality, and on a new mode of determining the value of life contingencies. Philos. Trans. R. Soc. Lond. 115, 513–583 (1825)CrossRefGoogle Scholar
  14. 14.
    Huang, M., Liu, S., Song, X., et al.: Periodic solutions and homoclinic bifurcation of a predator-Cprey system with two types of harvesting. Nonlinear Dyn. 64, 1–12 (2013)MathSciNetGoogle Scholar
  15. 15.
    Hui, J., Zhu, D.: Dynamic complexities for prey-dependent consumption integrated pest management models with impulsive effects. Chaos Solitons Fractals 29(1), 233–251 (2006)CrossRefMATHMathSciNetGoogle Scholar
  16. 16.
    Jiao, J., Chen, L., Cai, S.: Impulsive control strategy of a pest management SI model with nonlinear incidence rate. Appl. Math. Model. 33(1), 555–563 (2009)CrossRefMATHMathSciNetGoogle Scholar
  17. 17.
    Lakshmikantham, V., Bainov, D.D., Simeonov, P.S.: Theory of impulsive differential equations. World Scientific Publishing Company, Singapore (1989)CrossRefMATHGoogle Scholar
  18. 18.
    Li, Z., Chen, L.: Periodic solution of a turbidostat model with impulsive state feedback control. Nonlinear Dyn. 58(3), 525–538 (2009)CrossRefMATHGoogle Scholar
  19. 19.
    Li, Z., Chen, L., Huang, J.: Permanence and periodicity of a delayed ratio-dependent predator-prey model with Holling type functional response and stage structure. J. Comput. Appl. Math. 233(2), 173–187 (2009)CrossRefMATHMathSciNetGoogle Scholar
  20. 20.
    Liu, B., Zhang, Y., Chen, L.: Dynamic complexities of a Holling I predator Cprey model concerning periodic biological and chemical control. Chaos Solitons Fractals 22(1), 123–134 (2004)CrossRefMATHMathSciNetGoogle Scholar
  21. 21.
    Liu, B., Zhang, Y., Chen, L.: The dynamical behaviors of a Lotka-Volterra predator-prey model concerning integrated pest management. Nonlinear Anal. 6(2), 227–243 (2005)CrossRefMATHMathSciNetGoogle Scholar
  22. 22.
    Liu, B., Chen, L., Zhang, Y.: The dynamics of a prey-dependent consumption model concerning impulsive control strategy. Appl. Math. Comput. 169(1), 305–320 (2005)CrossRefMATHMathSciNetGoogle Scholar
  23. 23.
    Mailleret, L., Grognard, F.: Global stability and optimisation of a general impulsive biological control model. Math. Biosci. 221(2), 91–100 (2009)CrossRefMATHMathSciNetGoogle Scholar
  24. 24.
    Meng, X., Jiao, J., Chen, L.: The dynamics of an age structured predator-prey model with disturbing pulse and time delays. Nonlinear Anal. 9(2), 547–561 (2008)CrossRefMATHMathSciNetGoogle Scholar
  25. 25.
    Meng, X., Chen, L.: Permanence and global stability in an impulsive Lotka-Volterra N-species competitive system with both discrete delays and continuous delays. Int. J. Biomath. 1(02), 179–196 (2008)CrossRefMATHMathSciNetGoogle Scholar
  26. 26.
    Nieto, J.J.: Basic theory for nonresonance impulsive periodic problems of first order. J. Math. Anal. Appl. 205(2), 423–433 (1997)CrossRefMATHMathSciNetGoogle Scholar
  27. 27.
    Nieto, J.J., O’Regan, D.: Variational approach to impulsive differential equations. Nonlinear Anal. 10(2), 680–690 (2009)CrossRefMATHMathSciNetGoogle Scholar
  28. 28.
    Nie, L., Teng, Z., Hu, L., et al.: Existence and stability of periodic solution of a predator-prey model with state-dependent impulsive effects. Math. Comput. Simul. 79(7), 2122–2134 (2009)CrossRefMATHMathSciNetGoogle Scholar
  29. 29.
    Nie, L., Peng, J., Teng, Z., et al.: Existence and stability of periodic solution of a Lotka-Volterra predator-prey model with state dependent impulsive effects. J. Comput. Appl. Math. 224(2), 544–555 (2009)CrossRefMATHMathSciNetGoogle Scholar
  30. 30.
    Nie, L., Teng, Z., Hu, L., et al.: Qualitative analysis of a modified Leslie-Gower and Holling-type II predator-prey model with state dependent impulsive effects. Nonlinear Anal. 11(3), 1364–1373 (2010)CrossRefMATHMathSciNetGoogle Scholar
  31. 31.
    Science Features: Forest Pests: Boring a Hole in Your Wallet,
  32. 32.
    Shi, R., Jiang, X., Chen, L.: A predator-prey model with disease in the prey and two impulses for integrated pest management. Appl. Math. Model. 33(5), 2248–2256 (2009)CrossRefMATHMathSciNetGoogle Scholar
  33. 33.
    Shi, R., Chen, L.: The study of a ratio-dependent predator-prey model with stage structure in the prey. Nonlinear Dyn. 58(1–2), 443–451 (2009)CrossRefMATHMathSciNetGoogle Scholar
  34. 34.
    Song, X., Hao, M., Meng, X.: A stage-structured predator-prey model with disturbing pulse and time delays. Appl. Math. Model. 33(1), 211–223 (2009)CrossRefMATHMathSciNetGoogle Scholar
  35. 35.
    Sun, S., Chen, L.: Mathematical modelling to control a pest population by infected pests. Appl. Math. Model. 33(6), 2864–2873 (2009)CrossRefMATHMathSciNetGoogle Scholar
  36. 36.
    Tang, S., Cheke, R.A.: State-dependent impulsive models of integrated pest management (IPM) strategies and their dynamic consequences. J. Math. Biol. 50(3), 257–292 (2005)CrossRefMATHMathSciNetGoogle Scholar
  37. 37.
    Trzcinski, M.K., Reid, M.L.: Intrinsic and extrinsic determinants of mountain pine beetle population growth. Agric. For. Entomol. 11(2), 185–196 (2009)CrossRefGoogle Scholar
  38. 38.
    Zeng, G., Chen, L., Sun, L.: Existence of periodic solution of order one of planar impulsive autonomous system. J. Comput. Appl. Math. 186(2), 466–481 (2006)CrossRefMATHMathSciNetGoogle Scholar
  39. 39.
    Zhang, T., Meng, X., Song, Y.: The dynamics of a high-dimensional delayed pest management model with impulsive pesticide input and harvesting prey at different fixed moments. Nonlinear Dyn. 64(1), 1–12 (2011)CrossRefMATHMathSciNetGoogle Scholar
  40. 40.
    Zhang, T., Meng, X., Song, Y., et al.: Dynamical analysis of delayed plant disease models with continuous or impulsive cultural control strategies. Abstract and applied analysis. Hindawi Publishing Corporation, New York (2012)Google Scholar
  41. 41.
    Zhang, H., Chen, L., Nieto, J.J.: A delayed epidemic model with stage-structure and pulses for pest management strategy. Nonlinear Anal. 9(4), 1714–1726 (2008)CrossRefMATHMathSciNetGoogle Scholar
  42. 42.
    Zhang, H., Georgescu, P., Chen, L.: An impulsive predator-prey system with Beddington–Deangelis functional response and time delay. Int. J. Biomath. 1(01), 1–17 (2008) Google Scholar
  43. 43.
    Zhao, L., Chen, L., Zhang, Q.: The geometrical analysis of a Predator-prey model with two state impulses. Math. Biosci. 238, 55–64 (2012)CrossRefMATHMathSciNetGoogle Scholar
  44. 44.
    Zhao, W., Zhang, T., Meng, X., Yang, Y.: Dynamical analysis of a pest management model with saturated growth rate and state dependent impulsive effects. Abstract and applied analysis. Hindawi Publishing Corporation, New York (2013)Google Scholar

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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