Skip to main content
SpringerLink
Log in

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

  • Published:
Journal of Mathematical Biology Aims and scope Submit manuscript

Abstract.

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  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, 1973

  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)

  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, 1989

  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. 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)

    Article  Google Scholar 

  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)

    MATH  Google Scholar 

  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. 3641

  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)

    Article  MATH  Google Scholar 

  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. 105–121

  10. Guckenheimer, J., Holmes, P.: Nonlinear oscillations, dynamical systems, and bifurcations of vector fields. Appl. Math. Sci. 42, (1983)

  11. Kaul, S.: On impulsive semidynamical systems. J. Math. Anal. Appl. 150, 120–128 (1990)

    MATH  Google Scholar 

  12. Lakshmikantham, V., Bainov, D.D., Simeonov, P.S.: Theory of impulsive differential equations. World Scientific series in Modern Mathematics, Vol. 6, Singapore, 1989

  13. Matveev, A.S., Savkin, A.V.: Qualitative theory of hybrid dynamical systems. Birkhäuser, 2000

  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. Pedigo, L.P.: Entomology and Pest Management. Second Edition. Prentice-Hall Pub., Englewood Cliffs, NJ, 1996, p. 679

  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)

    Article  Google Scholar 

  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)

    MATH  Google Scholar 

  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. Stern, V.M.: Economic Thresholds. Ann. Rev. Entomol., 259–280 (1973)

  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)

    Article  MATH  Google Scholar 

  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. 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. Tang, S.Y., Chen, L.S.: Modelling and analysis of integrated pest management strategy. Disc. Continuous Dyn. Syst. B 4, 759–768 (2004b)

  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. Van Lenteren, J.C.: Integrated pest management in protected crops. In: Dent, D., (ed.), Integrated pest management, Chapman & Hall, London, 1995, pp. 311–320

  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. 77–89

  27. Van Lenteren, J.C., Woets, J.: Biological and integrated pest control in greenhouses. Ann. Rev. Ent. 33, 239–250 (1988)

    Article  Google Scholar 

  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. 409–448

  29. Xiao, Y.N., Van Den Bosch, F.: The dynamics of an eco-epidemic model with biological control. Ecol. Modelling 168, 203–214 (2003)

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Mathematics Institute, University of Warwick, CV4 7AL, UK

    Sanyi Tang

  2. Natural Resources Institute, University of Greenwich, Central Avenue, Chatham Maritime, Chatham, Kent, ME4 4TB, UK

    Robert A. Cheke

Authors
  1. Sanyi Tang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Robert A. Cheke
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Sanyi Tang.

Additional information

Mathematics Subject Classifications (2000): 34A36, 34D45, 92B05, 34C25

Acknowledgement The authors thank Professor L.S. Chen for support and encouragement in this research, as well as Y.N. Xiao for fruitful discussions. We would also like to thank the referees for their careful reading of the original manuscript and many valuable comments and suggestions, including the new method of proving proposition 2.1 attached in Appendix, that greatly improved the presentation of this paper.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Tang, S., Cheke, R. State-dependent impulsive models of integrated pest management (IPM) strategies and their dynamic consequences. J. Math. Biol. 50, 257–292 (2005). https://doi.org/10.1007/s00285-004-0290-6

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00285-004-0290-6

Key words or phrases:

Search

Navigation