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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
Article

Abstract

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.

Keywords

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

Mathematics Subject Classification (2000)

92D05 92D25 92D40 

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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

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