Journal of Mathematical Biology

, Volume 66, Issue 1–2, pp 1–35 | Cite as

Threshold conditions for integrated pest management models with pesticides that have residual effects

  • Sanyi Tang
  • Juhua Liang
  • Yuanshun Tan
  • Robert A. Cheke


Impulsive differential equations (hybrid dynamical systems) can provide a natural description of pulse-like actions such as when a pesticide kills a pest instantly. However, pesticides may have long-term residual effects, with some remaining active against pests for several weeks, months or years. Therefore, a more realistic method for modelling chemical control in such cases is to use continuous or piecewise-continuous periodic functions which affect growth rates. How to evaluate the effects of the duration of the pesticide residual effectiveness on successful pest control is key to the implementation of integrated pest management (IPM) in practice. To address these questions in detail, we have modelled IPM including residual effects of pesticides in terms of fixed pulse-type actions. The stability threshold conditions for pest eradication are given. Moreover, effects of the killing efficiency rate and the decay rate of the pesticide on the pest and on its natural enemies, the duration of residual effectiveness, the number of pesticide applications and the number of natural enemy releases on the threshold conditions are investigated with regard to the extent of depression or resurgence resulting from pulses of pesticide applications and predator releases. Latin Hypercube Sampling/Partial Rank Correlation uncertainty and sensitivity analysis techniques are employed to investigate the key control parameters which are most significantly related to threshold values. The findings combined with Volterra’s principle confirm that when the pesticide has a strong effect on the natural enemies, repeated use of the same pesticide can result in target pest resurgence. The results also indicate that there exists an optimal number of pesticide applications which can suppress the pest most effectively, and this may help in the design of an optimal control strategy.


Residual effects of pesticides Pest control IPM Volterra’s principle Pest-natural enemy system 

Mathematics Subject Classification (2000)

92D05 92D25 92D40 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Barclay HJ (1982) Models for pest control using predator release, habitat management and pesticide release in combination. J Appl Ecol 19: 337–348CrossRefGoogle Scholar
  2. Barlow ND, Moller H, Beggs JR (1996) A model for the effect of Sphecophaga vesparum as a biological control agent of the common wasp in New Zealand. J Appl Ecol 33: 31–34CrossRefGoogle Scholar
  3. Beddington JR, Free CA, Lawton JH (1978) Characteristics of successful natural enemies in models of biological control of insect pests. Nature 273: 513–519CrossRefGoogle Scholar
  4. Blower SM, Dowlatabadi H (1994) Sensitivity and uncertainty analysis of complex-models of disease transmission? An HIV model, as an example. Int Stat Rev 62: 229–243zbMATHCrossRefGoogle Scholar
  5. Bor Jeffrey YC (1995) Optimal pest management and economic threshold. Agric Syst 49: 113–133CrossRefGoogle Scholar
  6. Debach P (1974) Biological control by natural enemies. Cambridge University Press, LondonGoogle Scholar
  7. Heinrichs EA, Mochida O (1985) From secondary to major pest status: the case of insecticide-induced rice brown planthopper, Nilaparvata lugens, resurgence. Prot Ecol 7: 201–218Google Scholar
  8. Jones WA, Greenberg SM, Legaspi JR (1999) The effect of varying Bemisia argentifolii and Eretmocerus mundus ratios on parasitism. BioControl 44: 13–28CrossRefGoogle Scholar
  9. Liu XN, Chen LS (2004) Global dynamics of the periodic logistic system with periodic impulsive perturbations. J Math Anal Appl 289: 279–291MathSciNetzbMATHCrossRefGoogle Scholar
  10. Liu B, Chen LS (2004) The periodic competing Lotka-Volterra model with impulsive effect. Math Med Biol 38: 1505–1523Google Scholar
  11. Marino S, Ian B, Hogue IB, Ray CJ, Kirschner DE (2008) A methodology for performing global uncertainty and sensitivity analysis in systems biology. J Theor Biol 254: 178–196CrossRefGoogle Scholar
  12. Mckay MD, Beckman RJ, Conover WJ (1979) Comparison of 3 methods for selecting values of input variables in the analysis of output from a computer code. Technometrics 21: 239–245MathSciNetzbMATHGoogle Scholar
  13. Neuenschwander P, Herren HR (1988) Biological control of the Cassava Mealybug, Phenacoccus manihoti, by the exotic parasitoid Epidinocarsis lopezi in Africa. Phil Trans R Soc Lond B 318: 319–333CrossRefGoogle Scholar
  14. Panetta JC (1995) A logistic model of periodic chemotherapy. Appl Math Lett 8: 83–86zbMATHCrossRefGoogle Scholar
  15. Parker FD (1971) Management of pest populations by manipulating densities of both host and parasites through periodic releases. In: Huffaker CB (ed) Biological control. Plenum Press, New YorkGoogle Scholar
  16. Pedigo LP, Higley LG (1992) A new perspective of the economic injury level concept and environmental quality. Am Entomol 38: 12–20Google Scholar
  17. Raupp MJ, Holmes JJ, Sadof C, Shrewsbury P, Davidson JA (2001) Effects of cover sprays and residual pesticides on scale insects and natural enemies in urban forests. J Arboric 27: 203–214Google Scholar
  18. Reed GL, Jensen AS, Riebe J, Head G, Duan JJ (2001) Transgenic Bt potato and conventional insecticides for Colorado Beetle management: comparative efficacy and non-target impacts. Entomologia Experimentalies et Applicata 100: 89–100CrossRefGoogle Scholar
  19. Residual effects of pesticides (2010).
  20. Roughgarden J (1979) Theory of population genetics and evolutionary ecology: an introduction. MacMillan, New YorkGoogle Scholar
  21. Ruberson JR, Nemoto H, Hirose Y (1998) Pesticides and conservation of natural enemies in pest management. In: Barbosa P (ed) Conservation biological control. Academic Press, New York, pp 207–220CrossRefGoogle Scholar
  22. Schaalje GB (1990) Dynamic models of pesticide effectiveness. Environ Entomol 19: 439–447Google Scholar
  23. Schmutterer H (1988) Potential of azadirachtin-containing pesticides for integrated pest control in developing and industrialized countries. J Insect Phys 34: 713–719CrossRefGoogle Scholar
  24. Tang SY, Chen LS (2002) Density-dependent birth rate, birth pulses and their population dynamic consequences. J Math Bio 44: 185–199MathSciNetCrossRefGoogle Scholar
  25. Tang SY, Chen LS (2003) Multiple attractors in stage-structured population models with birth pulses. Bull Math Biol 65: 479–495CrossRefGoogle Scholar
  26. Tang SY, Cheke RA (2005) Stage-dependent impulsive models of integrated pest management (IPM) strategies and their dynamic consequences. J Math Biol 50: 257–292MathSciNetzbMATHCrossRefGoogle Scholar
  27. Tang SY, Xiao YN, Chen LS, Cheke RA (2005) Integrated pest management models and their dynamical behaviour. Bull Math Biol 67: 115–135MathSciNetCrossRefGoogle Scholar
  28. Tang SY, Xiao YN, Cheke RA (2008) Multiple attractors of host-parasitoid models with integrated pest management strategies: eradication, persistence and outbreak. Theor Popul Biol 73: 181–197zbMATHCrossRefGoogle Scholar
  29. Tang SY, Cheke RA (2008) Models for integrated pest control and their biological implications. Math Biosci 215: 115–125MathSciNetzbMATHCrossRefGoogle Scholar
  30. Tang SY, Xiao YN, Cheke RA (2009) Effects of predator and prey dispersal on success or failure of biological control. Bull Math Biol 71: 2025–2047MathSciNetzbMATHCrossRefGoogle Scholar
  31. Tang SY, Tang GY, Cheke RA (2010) Optimum timing for integrated pest management: Modelling rates of pesticide application and natural enemy releases. J Theor Biol 264: 623–638CrossRefGoogle Scholar
  32. Udayagiri S, Norton AP, Welter SC (2000) Integrating pesticide effects with inundative biological control: interpretation of pesticide toxicity curves for Anaphes iole in strawberries. Entomologia Experimentalis et Applicata 95: 87–95CrossRefGoogle Scholar
  33. Van Lenteren JC (1995) Integrated pest management in protected crops. In: Dent D (ed) Integrated pest management. Chapman & Hall, London, pp 311–320Google Scholar
  34. van Lenteren JC (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 77–89CrossRefGoogle Scholar
  35. Van Lenteren JC, Woets J (1988) Biological and integrated pest control in greenhouses. Ann Rev Ent 33: 239–250CrossRefGoogle Scholar
  36. Volterra V (1926) Fluctuation in abundance of a species considered mathematically. Nature 118: 558–560zbMATHCrossRefGoogle Scholar
  37. Waage JK, Hassell MP (1982) Parasitoids as biological control agents-a fundamental approach. Parasitology 84: 241–268CrossRefGoogle Scholar
  38. Waage JK, Hassell MP, Godfray HCJ (1985) The dynamics of pest-parasitoid-insecticide interactions. J Appl Ecol 22: 825–838CrossRefGoogle Scholar
  39. Zacharda M, Hluchy M (1991) Long-term residual efficacy of commercial formulations of 16 pesticides to Typhlodromus pyri Scheuten (Acari: Phytoseiidae) inhabiting commercial vineyards. Exp Appl Acaro 13: 27–40CrossRefGoogle Scholar

Copyright information

© Springer-Verlag 2011

Authors and Affiliations

  • Sanyi Tang
    • 1
  • Juhua Liang
    • 1
  • Yuanshun Tan
    • 2
  • Robert A. Cheke
    • 3
  1. 1.College of Mathematics and Information ScienceShaanxi Normal UniversityXi’anPeople’s Republic of China
  2. 2.Department of Mathematics and PhysicsChongqing Jiaotong UniversityChongqingPeople’s Republic of China
  3. 3.Natural Resources InstituteUniversity of Greenwich at MedwayKentUK

Personalised recommendations