Optimal application timing of pest control tactics in nonautonomous pest growth model. (English) Zbl 1406.92706

Summary: Considering the effects of the living environment on growth of populations, it is unrealistic to assume that the growth rates of predator and prey are all constants in the models with integrated pest management (IPM) strategies. Therefore, a nonautonomous predator-prey system with impulsive effect is developed and investigated in the present work. In order to determine the optimal application timing of IPM tactics, the threshold value which guarantees the stability of pest-free periodic solution has been obtained firstly. The analytical formula of optimal application timings within a given period for different cases has been obtained such that the threshold value is the smallest, which is the most effective in successful pest control. Moreover, extensively numerical investigations have also been confirmed our main results and the biological implications have been discussed in more detail. The main results can guide the farmer to design the optimal pest control strategies.


92D40 Ecology
34A37 Ordinary differential equations with impulses
Full Text: DOI


[1] Flint, M. L., Integrated Pest Management for Walnuts, seconded. University of California Statewide Integrated Pest Management Project
[2] Van Lenteren, J. C.; Dent, D., Integrated pest management in protected crops, Integrated Pest Management, 311-343 (1995), London, UK: Chapman and Hall, London, UK
[3] Van Lenteren, J. C., Success in biological control of arthropods by augmentation of natural enemies, Biological Control: Measures of Success, 77-103 (2000)
[4] Van Lenteren, J. C.; Woets, J., Biological and integrated pest control in greenhouses, Annual Review of Entomology, 33, 239-269 (1988)
[5] Li, C.; Tang, S., The effects of timing of pulse spraying and releasing periods on dynamics of generalized predator-prey model, International Journal of Biomathematics, 5 (2012) · Zbl 1297.92065
[6] Gao, W.; Tang, S., The effects of impulsive releasing methods of natural enemies on pest control and dynamical complexity, Nonlinear Analysis: Hybrid Systems, 5, 3, 540-553 (2011) · Zbl 1238.93044
[7] Tang, S.; Tang, G.; Cheke, R. A., Optimum timing for integrated pest management: modelling rates of pesticide application and natural enemy releases, Journal of Theoretical Biology, 264, 2, 623-638 (2010) · Zbl 1406.92694
[8] Liang, J.; Tang, S., Optimal dosage and economic threshold of multiple pesticide applications for pest control, Mathematical and Computer Modelling, 51, 5-6, 487-503 (2010) · Zbl 1190.91115
[9] Liu, X.; Chen, L., Complex dynamics of Holling type II Lotka-Volterra predator-prey system with impulsive perturbations on the predator, Chaos, Solitons and Fractals, 16, 2, 311-320 (2003) · Zbl 1085.34529
[10] Tang, S. Y.; Chen, L. S., The periodic predator-prey Lotka-Volterra model with impulsive effect, Journal of Mechanics in Medicine and Biology, 2, 3-4, 267-296 (2002)
[11] Tang, S.; Xiao, Y.; Chen, L.; Cheke, R. A., Integrated pest management models and their dynamical behaviour, Bulletin of Mathematical Biology, 67, 1, 115-135 (2005) · Zbl 1334.91058
[12] Tang, S.; Cheke, R. A., Models for integrated pest control and their biological implications, Mathematical Biosciences, 215, 1, 115-125 (2008) · Zbl 1156.92046
[13] Mailleret, L.; Grognard, F., Global stability and optimisation of a general impulsive biological control model, Mathematical Biosciences, 221, 2, 91-100 (2009) · Zbl 1175.92070
[14] Tang, S.; Chen, L., Modelling and analysis of integrated pest management strategy, Discrete and Continuous Dynamical Systems B: A Journal Bridging Mathematics and Sciences, 4, 3, 759-768 (2004) · Zbl 1114.92074
[15] Tang, S. Y.; Xiao, Y. N., Biological Dynamics of Single Species, Science Press
[16] Safuan, H. M.; Jovanoski, Z.; Towers, I. N.; Sidhu, H. S., Exact solution of a non-autonomous logistic population model, Ecological Modelling, 251, 99-102 (2013)
[17] Meng, X.; Chen, L., Almost periodic solution of non-autonomous Lotka-Volterra predator-prey dispersal system with delays, Journal of Theoretical Biology, 243, 4, 562-574 (2006) · Zbl 1447.92355
[18] Zhang, F. F.; Tang, S. Y., Dynamical behaviour of periodic impulsive chemical control model with periodically switching pesticides, Mathematical Biology, 2, 497-502 (2011)
[19] Lakshmikantham, V.; Bainov, D. D.; Simeonov, P. S., Theory of Impulsive Differential Equations (1989), Singapore: World Scientific, Singapore · Zbl 0719.34002
[20] Bainov, D. D.; Simenov, P. S., Systems with Impulsive Effect: Stability the-Ory and Applications (1989), Chichester, UK: Ellis Horwood, Chichester, UK
[21] Baĭnov, D.; Simeonov, P., Impulsive Differential Equations: Periodic Solutions and Applications. Impulsive Differential Equations: Periodic Solutions and Applications, Pitman Monographs and Surveys in Pure and Applied Mathematics, 66 (1993) · Zbl 0815.34001
[22] Xue, Y.; Tang, S.; Liang, J., Optimal timing of interventions in fishery resource and pest management, Nonlinear Analysis: Real World Applications, 13, 4, 1630-1646 (2012) · Zbl 1401.92170
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