Journal of Mathematical Biology

, Volume 50, Issue 3, pp 257–292 | Cite as

State-dependent impulsive models of integrated pest management (IPM) strategies and their dynamic consequences

  • Sanyi TangEmail author
  • Robert A. Cheke


A state-dependent impulsive model is proposed for integrated pest management (IPM). IPM involves combining biological, mechanical, and chemical tactics to reduce pest numbers to tolerable levels after a pest population has reached its economic threshold (ET). The complete expression of an orbitally asymptotically stable periodic solution to the model with a maximum value no larger than the given ET is presented, the existence of which implies that pests can be controlled at or below their ET levels. We also prove that there is no periodic solution with order larger than or equal to three, except for one special case, by using the properties of the LambertW function and Poincaré map. Moreover, we show that the existence of an order two periodic solution implies the existence of an order one periodic solution. Various positive invariant sets and attractors of this impulsive semi-dynamical system are described and discussed. In particular, several horseshoe-like attractors, whose interiors can simultaneously contain stable order 1 periodic solutions and order 2 periodic solutions, are found and the interior structure of the horseshoe-like attractors is discussed. Finally, the largest invariant set and the sufficient conditions which guarantee the global orbital and asymptotic stability of the order 1 periodic solution in the meaningful domain for the system are given using the Lyapunov function. Our results show that, in theory, a pest can be controlled such that its population size is no larger than its ET by applying effects impulsively once, twice, or at most, a finite number of times, or according to a periodic regime. Moreover, our theoretical work suggests how IPM strategies could be used to alter the levels of the ET in the farmers’ favour.

Key words or phrases:

IPM strategy Economic threshold Impulsive semi-dynamical system Positive invariant set Horseshoe-like attractor Limit set 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Andronov, A.A., Leontovich, E.A., Gordan, L.L., Maier, A.G.: Qualitative theory of second-order dynamic systems. Translated from Russian by D. Louvish, John Wiley & Sons, New York, 1973Google Scholar
  2. 2.
    Bainov, D.D., Simeonov, P.S.: Impulsive differential equations: periodic solutions and applications, Pitman Monographs and Surveys in Pure and Appl. Math. 66, (1993)Google Scholar
  3. 3.
    Bainov, D.D., Simeonov, P.S.: Systems with impulse effect, theory and applications, Ellis Hardwood series in Mathematics and its Applications, Ellis Hardwood, Chichester, 1989Google Scholar
  4. 4.
    Barclay, H.J.: Models for pest control using predator release, habitat management and pesticide release in combination. J. Appl. Ecol. 19, 337–348 (1982)Google Scholar
  5. 5.
    Chellaboina, V.S., Bhat, S.P., Haddad, W.M.: An invariance principle for nonlinear hybrid and impulsive dynamical systems. Nonlinear Anal. TMA 53, 527–550 (2003)CrossRefGoogle Scholar
  6. 6.
    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)zbMATHGoogle Scholar
  7. 7.
    Flint, M.L., ed.: Integrated Pest Management for Walnuts, University of California Statewide Integrated Pest Management Project, Division of Agriculture and Natural Resources, Second Edition, University of California, Oakland, CA, publication 3270, 1987, pp. 3641Google Scholar
  8. 8.
    Grasman, J., Van Herwaarden, O.A., Hemerik, L., Van Lenteren, J.C.: A two-component model of host-parasitoid interactions: determination of the size of inundative releases of parasitoids in biological pest control. Math. Biosci. 169, 207–216 (2001)CrossRefzbMATHGoogle Scholar
  9. 9.
    Greathead, D.J.: Natural enemies of tropical locusts and grasshoppers: their impact and potential as biological control agents. In: C.J. Lomer, C. Prior (eds.), Biological control of locusts and grasshoppers. Wallingford, UK: C.A.B. International, 1992, pp. 105–121Google Scholar
  10. 10.
    Guckenheimer, J., Holmes, P.: Nonlinear oscillations, dynamical systems, and bifurcations of vector fields. Appl. Math. Sci. 42, (1983)Google Scholar
  11. 11.
    Kaul, S.: On impulsive semidynamical systems. J. Math. Anal. Appl. 150, 120–128 (1990)zbMATHGoogle Scholar
  12. 12.
    Lakshmikantham, V., Bainov, D.D., Simeonov, P.S.: Theory of impulsive differential equations. World Scientific series in Modern Mathematics, Vol. 6, Singapore, 1989Google Scholar
  13. 13.
    Matveev, A.S., Savkin, A.V.: Qualitative theory of hybrid dynamical systems. Birkhäuser, 2000Google Scholar
  14. 14.
    Pedigo, L.P., Higley, L.G.: A new perspective of the economic injury level concept and environmental quality. Am. Entomologist 38, 12–20 (1992)Google Scholar
  15. 15.
    Pedigo, L.P.: Entomology and Pest Management. Second Edition. Prentice-Hall Pub., Englewood Cliffs, NJ, 1996, p. 679Google Scholar
  16. 16.
    Qi, J.G., Fu, X.L.: Existence of limit cycles of impulsive differential equations with impulses as variable times. Nonl. Anal. TMA 44, 345–353 (2001)CrossRefGoogle Scholar
  17. 17.
    Simeonov, P.S., Bainov, D.D.: Orbital stability of periodic solutions of autonomous systems with impulsive effect. INT. J. Syst. SCI 19, 2561–2585 (1988)zbMATHGoogle Scholar
  18. 18.
    Stern, V.M., Smith, R.F., van den Bosch, R., Hagen, K.S.: The integrated control concept. Hilgardia 29, 81–93 (1959)Google Scholar
  19. 19.
    Stern, V.M.: Economic Thresholds. Ann. Rev. Entomol., 259–280 (1973)Google Scholar
  20. 20.
    Tang, S.Y., Chen, L.S.: Density-dependent birth rate, birth pulses and their population dynamic consequences. J. Math. Biol. 44, 185–199 (2002)CrossRefzbMATHGoogle Scholar
  21. 21.
    Tang, S.Y., Chen, L.S.: Multiple attractors in stage-structured population models with birth pulses. Bull. Math. Biol. 65, 479–495 (2003)Google Scholar
  22. 22.
    Tang, S.Y., Chen, L.S.: The effect of seasonal harvesting on stage-structured population models. J. Math. Biol. 48, 357–374 (2004a)Google Scholar
  23. 23.
    Tang, S.Y., Chen, L.S.: Modelling and analysis of integrated pest management strategy. Disc. Continuous Dyn. Syst. B 4, 759–768 (2004b)Google Scholar
  24. 24.
    Tang, S.Y., Chen, L.S.: Global attractivity in a “food limited” population model with impulsive effects. J. Math. Anal. Appl. 292, 211–221 (2004c)Google Scholar
  25. 25.
    Van Lenteren, J.C.: Integrated pest management in protected crops. In: Dent, D., (ed.), Integrated pest management, Chapman & Hall, London, 1995, pp. 311–320Google Scholar
  26. 26.
    Van Lenteren, J.C.: Measures of success in biological control of arthropods by augmentation of natural enemies. In: Wratten, S., Gurr G. (eds.), Measures of Success in Biological Control, Kluwer Academic Publishers, Dordrecht, 2000, pp. 77–89Google Scholar
  27. 27.
    Van Lenteren, J.C., Woets, J.: Biological and integrated pest control in greenhouses. Ann. Rev. Ent. 33, 239–250 (1988)CrossRefGoogle Scholar
  28. 28.
    Volterra, V.: Variations and fluctuations of a number of individuals in animal species living together. Translation In: R.N. Chapman: Animal Ecology, New York: McGraw Hill, 1931, pp. 409–448Google Scholar
  29. 29.
    Xiao, Y.N., Van Den Bosch, F.: The dynamics of an eco-epidemic model with biological control. Ecol. Modelling 168, 203–214 (2003)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2004

Authors and Affiliations

  1. 1.Mathematics InstituteUniversity of WarwickUK
  2. 2.Natural Resources InstituteUniversity of GreenwichChathamUK

Personalised recommendations