Bulletin of Mathematical Biology

, Volume 67, Issue 1, pp 115–135 | Cite as

Integrated pest management models and their dynamical behaviour

  • Sanyi Tang
  • Yanni Xiao
  • Lansun Chen
  • Robert A. Cheke
Article

Abstract

Two impulsive models of integrated pest management (IPM) strategies are proposed, one with fixed intervention times and the other with these unfixed. The first model allows natural enemies to survive but under some conditions may lead to extinction of the pest. We use a simple prey-dependent consumption model with fixed impulsive effects and show that there exists a globally stable pesteradication periodic solution when the impulsive period is less than certain critical values. The effects of pest resistance to pesticides are also studied. The second model is constructed in the light of IPM practice such that when the pest population reaches the economic injury level (EIL), a combination of biological, cultural, and chemical tactics that reduce pests to tolerable levels is invoked. Using analytical methods, we show that there exists an orbitally asymptotically stable periodic solution with a maximum value no larger than the given Economic Threshold (ET). The complete expression for this periodic solution is given and the ET is evaluated for given parameters.We also show that in some cases control costs can be reduced by replacing IPM interventions at unfixed times with periodic interventions. Further, we show that small perturbations of the system do not affect the existence and stability of the periodic solution. Thus, we provide the first demonstration using mathematical models that an IPM strategy is more effective than classical control methods.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Arlian, L.G., Neal, J.S., Vyszenki-Moher, D.L., 1999. Reducing relative humidity to control the house dust mite Dermatophagoides farinae. J. Allergy Clin. Immunol. 104, 852–856.CrossRefGoogle Scholar
  2. Bainov, D.D., Simeonov, P.S., 1989. Systems with Impulse Effect, Theory and Applications. Ellis Harwood Series in Mathematics and its Applications, Ellis Harwood, Chichester.Google Scholar
  3. Bainov, D.D., Simeonov, P.S., 1993. Impulsive Differential Equations: Periodic Solutions and Applications. Pitman Monographs and Surveys in Pure and Applied Mathematics, vol. 66.Google Scholar
  4. Barclay, H.J., 1982. Models for pest control using predator release, habitat management and pesticide release in combination. J. Appl. Ecol. 19, 337–348.Google Scholar
  5. Chellaboina, V.S., Bhat, S.P., Haddad, W.M., 2003. An invariance principle for nonlinear hybrid and impulsive dynamical systems. Nonlinear Anal. TMA 53, 527–550.MathSciNetCrossRefGoogle Scholar
  6. Corless, R.M., Gonnet, G.H., Hare, D.E.G., Jeffrey, D.J., Knuth, D.E., 1996. On the Lambert W function. Adv. Comput. Math. 5, 329–359.MathSciNetCrossRefGoogle Scholar
  7. Flint, M.L. (Ed.), 1987. Integrated pest management for walnuts. University of California Statewide Integrated Pest Management Project, Division of Agriculture and Natural Resources, 2nd edn, publication 3270. University of California, Oakland, CA, p. 3641.Google Scholar
  8. Grasman, J., Van Herwaarden, O.A., Hemerik, L., Van Lenteren, J.C., 2001. 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.MathSciNetCrossRefGoogle Scholar
  9. Greathead, D.J., 1992. Natural enemies of tropical locusts and grasshoppers: their impact and potential as biological control agents. In: Lomer, C.J., Prior, C. (Eds.), Biological Control of Locusts and Grasshoppers. C.A.B. International, Wallingford, UK, pp. 105–121.Google Scholar
  10. Grebogi, C., Ott, E., Yorke, J.A., 1983. Fractal basin boundaries, long-lived chaotic transients and unstable pair bifurcation. Phys. Rev. Lett. 50, 935–943.MathSciNetCrossRefGoogle Scholar
  11. Hastings, A., Higgins, K., 1994. Persistence of transitions in spatially structured ecological models. Science 263, 1133–1137.Google Scholar
  12. Kaul, S., 1990. On impulsive semidynamical systems. J. Math. Anal. Appl. 150, 120–128.MATHMathSciNetCrossRefGoogle Scholar
  13. Lakshmikantham, V., Bainov, D.D., Simeonov, P.S., 1989. Theory of Impulsive Differential Equations. World Scientific Series in Modern Mathematics, vol. 6. World Scientific, Singapore.Google Scholar
  14. Lotka, A.J., 1920. Undamped oscillations derived from the law of mass action. J. Am. Chem. Soc. 42, 1595–1599.CrossRefGoogle Scholar
  15. Løvik, R.M.M., Gardner, P.I., 1998. Allergy. (Suppl.), Munksgaard, Copenhagen.Google Scholar
  16. Matveev, A.S., Savkin, A.V., 2000. Qualitative Theory of Hybrid Dynamical Systems. Birkhäuser, Basel.Google Scholar
  17. Nevada Agricultural Statistics Service, 1997. Nevada Agricultural Statistics 1996–1997. p. 20.Google Scholar
  18. Nåsell, I., 2001. Extinction and quasi-stationarity in the Verhulst logistic model. J. Theor. Biol. 211, 11–27.CrossRefGoogle Scholar
  19. Parker, F.D., 1971. Management of pest populations by manipulating densities of both host and parasites through periodic releases. In: Huffaker, C.B. (Ed.), Biological Control. Plenum Press, New York.Google Scholar
  20. Roberts, M.G., Kao, R.R., 1998. The dynamics of an infectious disease in a population with birth pulses. Math. Biosci. 149, 23–36.CrossRefGoogle Scholar
  21. Tang, S.Y., Chen, L.S., 2002. Density-dependent birth rate, birth pulses and their population dynamic consequences. J. Math. Biol. 64, 169–184.MathSciNetGoogle Scholar
  22. Tang, S.Y., Chen, L.S., 2003. Multiple attractors in stage-structured population models with birth pulses. Bull. Math. Biol. 65, 479–495.CrossRefGoogle Scholar
  23. Tang, S.Y., Chen, L.S., 2004. The effect of seasonal harvesting on stage-structured population models. J. Math. Biol. 48, 357–374.MathSciNetCrossRefGoogle Scholar
  24. Tang, S.Y., Cheke, R.A., 2004. State-dependent impulsive models of integrated pest management (IPM) strategies and their dynamic consequences. J. Math. Biol. (in press).Google Scholar
  25. Van den Bosch, R., 1978. The Pesticide Conspiracy. Doubleday & Co, Garden City, NY.Google Scholar
  26. Van Lenteren, J.C., 1987. Environmental manipulation advantageous to natural enemies of pests. In: Delucchi, V. (Ed.), Integrated Pest Management. Parasitis, Geneva, pp. 123–166.Google Scholar
  27. Van Lenteren, J.C., 1995. Integrated pest management in protected crops. In: Dent, D. (Ed.), Integrated Pest Management. Chapman & Hall, London, pp. 311–320.Google Scholar
  28. Van Lenteren, J.C., 2000. 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, pp. 77–89.Google Scholar
  29. Van Lenteren, J.C., Woets, J., 1988. Biological and integrated pest control in greenhouses. Ann. Rev. Ent. 33, 239–250.CrossRefGoogle Scholar
  30. Volterra, V., 1931. Variations and fluctuations of a number of individuals in animal species living together, translation. In: Chapman, R.N. (Ed.), Animal Ecology. McGraw-Hill, New York, pp. 409–448.Google Scholar
  31. Waldvogel, J., 1983. The period in the Volterra-Lotka predator-prey model. SIAM J. Numer. Anal. 20, 1264–1272.MATHMathSciNetCrossRefGoogle Scholar
  32. Waldvogel, J., 1986. The period in the Volterra-Lotka system is monotonic. J. Math. Anal. Appl. 114, 178–184.MATHMathSciNetCrossRefGoogle Scholar
  33. Xiao, Y.N., Van Den Bosch, F., 2003. The dynamics of an eco-epidemic model with biological control. Ecol. Modelling 168, 203–214.CrossRefGoogle Scholar

Copyright information

© Society for Mathematical Biology 2005

Authors and Affiliations

  • Sanyi Tang
    • 1
  • Yanni Xiao
    • 2
  • Lansun Chen
    • 3
  • Robert A. Cheke
    • 4
  1. 1.Mathematics InstituteUniversity of WarwickCoventryUK
  2. 2.Department of Mathematical SciencesThe University of LiverpoolLiverpoolUK
  3. 3.Academy of Mathematics and System Sciences, Chinese Academy of SciencesBeijingPR China
  4. 4.Natural Resources InstituteUniversity of GreenwichKentUK

Personalised recommendations