Bulletin of Mathematical Biology

, Volume 67, Issue 1, pp 115–135

# 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.

## Keywords

Periodic Solution Natural Enemy Mathematical Biology Integrate Pest Management Pest Population
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

## 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.
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.
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.
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.
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.
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.
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.
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
18. Nåsell, I., 2001. Extinction and quasi-stationarity in the Verhulst logistic model. J. Theor. Biol. 211, 11–27.
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.
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.
22. Tang, S.Y., Chen, L.S., 2003. Multiple attractors in stage-structured population models with birth pulses. Bull. Math. Biol. 65, 479–495.
23. Tang, S.Y., Chen, L.S., 2004. The effect of seasonal harvesting on stage-structured population models. J. Math. Biol. 48, 357–374.
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.
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.
32. Waldvogel, J., 1986. The period in the Volterra-Lotka system is monotonic. J. Math. Anal. Appl. 114, 178–184.
33. Xiao, Y.N., Van Den Bosch, F., 2003. The dynamics of an eco-epidemic model with biological control. Ecol. Modelling 168, 203–214.

© Society for Mathematical Biology 2005

## Authors and Affiliations

• Sanyi Tang
• 1
Email author
• 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