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The effects of impulsive releasing methods of natural enemies on pest control and dynamical complexity. (English) Zbl 1238.93044

Summary: In the paper, different releasing methods including constant releasing and proportional to the predator population are considered and analyzed. The effects of these releasing methods of natural enemies on dynamical behavior are investigated. We firstly take into account the model with an impulsive effect at fixed moments, and the results imply that under some conditions the pest may serve to extinction. Several types of attractors can coexist, with switch-like transitions among their attractors showing that varying dosages and frequencies of insecticide applications and the numbers of natural enemies released are crucial. Secondly, the model with unfixed moments is further investigated. Different periodic solutions also exist and the maximum amplitude of the host is always less than the economic threshold. Comparing the results obtained for the two models concludes that the proportional releasing predator has strong effects on the dynamical behavior.

MSC:

93C15 Control/observation systems governed by ordinary differential equations
92D30 Epidemiology
93C95 Application models in control theory
93A15 Large-scale systems
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References:

[1] Van Lenteren, J. C., Measures of success in biological control of anthropoids by augmentation of natural enemies, (Wratten, S.; Gurr, G., Measures of Success in Biological Control (2000), Kluwer Academic Publishers: Kluwer Academic Publishers Dordrecht), 77
[2] Van Lenteren, J. C., Integrated pest management in protected crops, (Dent, D., Integrated Pest Management (1995), Chapman & Hall: Chapman & Hall London), 311
[3] Van Lenteren, J. C.; Woets, J., Biological and integrated pest control in greenhouses, Ann. Rev. Ent., 33, 239 (1988)
[4] Tang, S. Y.; Cheke, R. A., Models for integrated pest control and their biological implications, Math. Biosci., 215, 115 (2008) · Zbl 1156.92046
[5] Tang, S. Y.; Cheke, R. A., State-dependent impulsive models of integrated pest management(IPM) strategies and their dynamic consequences, J. Math. Biol., 50, 257 (2005) · Zbl 1080.92067
[6] Tang, S. Y.; Chen, L. S., Modelling and analysis of integrated pest management strategy, Dis. Cont. Dyn. Sys. B, 4, 761 (2004)
[7] Tang, S. Y.; Xiao, Y. N.; Cheke, R. A., Multiple attractors of host-parasitoid models with integrated pest management strategies: Eradication, persistence and outbreak, Theor. Popul. Biol., 73, 181 (2008) · Zbl 1208.92093
[8] M.P. Hoffmann, A.C. Frodsham, Natural enemies of vegetable insect pests, Cooperative Extension, Cornell University, Ithaca, NY, 1993.; M.P. Hoffmann, A.C. Frodsham, Natural enemies of vegetable insect pests, Cooperative Extension, Cornell University, Ithaca, NY, 1993.
[9] Stinner, R. E., Efficacy of inundative releases, A. Res. Ent., 22, 515 (1977)
[10] Summers, T. E.; King, E. G.; Martin, D. F.; Jackson, R. D., Biological control of Diatraea sacchanalis in Florida by periodic releases of Lixophaga diatraeae, Entomophaga, 4, 359 (1976)
[11] Lotka, A. J., Undamped oscillations derived from the law of mass action, J. Amer. Chem. Soc., 42, 1595 (1920)
[12] Volterra, V., Variations and fluctuations of a number of individuals in animal species living together, (Chapman, R. N., Animal Ecology (1931), McGraw Hill: McGraw Hill New York), 409, Translation
[13] Grebogi, C.; Ott, E.; Yorke, J. A., Crises, sudden changes in chaotic attractors and chaotic transients, Physica D, 7, 181 (1983)
[14] Corless, R. M.; Gonnet, G. H.; Hare, D. E.G.; Jeffrey, D. J.; Knuth, D. E., On the Lambert W function, Adv. Comput. Math., 5, 329 (1996) · Zbl 0863.65008
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