Optimal Control Policies of Pests for Hybrid Dynamical Systems
Baolin Kang
School of Mathematical Sciences, Dalian University of Technology, Dalian, Liaoning 116024, China dlut.edu.cn
Department of Mathematics, Anshan Normal University, Anshan, Liaoning 114007, China asnc.edu.cn
Search for more papers by this authorCorresponding Author
Mingfeng He
School of Mathematical Sciences, Dalian University of Technology, Dalian, Liaoning 116024, China dlut.edu.cn
Search for more papers by this authorBing Liu
Department of Mathematics, Anshan Normal University, Anshan, Liaoning 114007, China asnc.edu.cn
Search for more papers by this authorBaolin Kang
School of Mathematical Sciences, Dalian University of Technology, Dalian, Liaoning 116024, China dlut.edu.cn
Department of Mathematics, Anshan Normal University, Anshan, Liaoning 114007, China asnc.edu.cn
Search for more papers by this authorCorresponding Author
Mingfeng He
School of Mathematical Sciences, Dalian University of Technology, Dalian, Liaoning 116024, China dlut.edu.cn
Search for more papers by this authorBing Liu
Department of Mathematics, Anshan Normal University, Anshan, Liaoning 114007, China asnc.edu.cn
Search for more papers by this authorAbstract
We improve the traditional integrated pest management (IPM) control strategies and formulate three specific management strategies, which can be described by hybrid dynamical systems. These strategies can not only effectively control pests but also reduce the abuse of pesticides and protect the natural enemies. The aim of this work is to study how the factors, such as natural enemies optimum choice in the two kinds of different pests, timings of natural enemy releases, dosages and timings of insecticide applications, and instantaneous killing rates of pesticides on both pests and natural enemies, can affect the success of IPM control programmes. The results indicate that the pests outbreak period or frequency largely depends on the optimal selective feeding of the natural enemy between one of the pests and the control tactics. Ultimately, we obtain the only pest x2 needs to be controlled below a certain threshold while not supervising pest x1.
References
- 1 Finch S. and Collier R. H., Integrated pest management in field vegetable crops in northern Europe—with focus on two key pests, Crop Protection. (2000) 19, no. 8-10, 817–824, 2-s2.0-0033729303, https://doi.org/10.1016/S0261-2194(00)00109-5.
- 2 Bange M. P., Deutscher S. A., Larsen D., Linsley D., and Whiteside S., A handheld decision support system to facilitate improved insect pest management in Australian cotton systems, Computers and Electronics in Agriculture. (2004) 43, no. 2, 131–147, 2-s2.0-1842844482, https://doi.org/10.1016/j.compag.2003.12.003.
- 3 Bisignanesi V. and Borgas M. S., Models for integrated pest management with chemicals in atmospheric surface layers, Ecological Modelling. (2007) 201, no. 1, 2–10, 2-s2.0-33845994403, https://doi.org/10.1016/j.ecolmodel.2006.07.040.
- 4 Hashemi S. M., Hosseini S. M., and Damalas C. A., Farmers′ competence and training needs on pest management practices: participation in extension workshops, Crop Protection. (2009) 28, no. 11, 934–939, 2-s2.0-70049114170, https://doi.org/10.1016/j.cropro.2009.07.007.
- 5 Strand J. F., Some agrometeorological aspects of pest and disease management for the 21st century, Agricultural and Forest Meteorology. (2000) 103, no. 1-2, 73–82, 2-s2.0-0034212461, https://doi.org/10.1016/S0168-1923(00)00119-2.
- 6 Yorobe J. M., Rejesus R. M., and Hammig M. D., Insecticide use impacts of Integrated Pest Management (IPM) Farmer Field Schools: evidence from onion farmers in the Philippines, Agricultural Systems. (2011) 104, no. 7, 580–587, 2-s2.0-79960432486, https://doi.org/10.1016/j.agsy.2011.05.001.
- 7 Liu B., Zhang Y. J., and Chen L. S., The dynamical behaviors of a Lotka-Volterra predator-prey model concerning integrated pest management, Nonlinear Analysis: Real World Applications. (2005) 6, no. 2, 227–243, 2-s2.0-10244240509, https://doi.org/10.1016/j.nonrwa.2004.08.001.
- 8 Zhang Y. J., Liu B., and Chen L. S., Dynamical behavior of Volterra model with mutual interference concerning IPM, Mathematical Modelling and Numerical Analysis. (2004) 38, no. 1, 143–155, 2-s2.0-1342326243, https://doi.org/10.1051/m2an:2004007.
- 9 Liu B., Zhang Y. J., and Chen L., The dynamical behaviors of a Lotka-Volterra predator-prey model concerning integrated pest management, Nonlinear Analysis: Real World Applications. (2005) 6, no. 2, 227–243, 2-s2.0-10244240509, https://doi.org/10.1016/j.nonrwa.2004.08.001.
- 10 Meng X., Jiao J., and Chen L., The dynamics of an age structured predator-prey model with disturbing pulse and time delays, Nonlinear Analysis: Real World Applications. (2008) 9, no. 2, 547–561, MR2382398, ZBL1142.34054, 2-s2.0-37248999845, https://doi.org/10.1016/j.nonrwa.2006.12.001.
- 11 Meng X., Song Z., and Chen L., A new mathematical model for optimal control strategies of integrated pest management, Journal of Biological Systems. (2007) 15, no. 2, 219–234, 2-s2.0-34547241484, https://doi.org/10.1142/S0218339007002143.
- 12
Jiao J.-J. and
Chen L.-S., Nonlinear incidence rate of a pest management SI model with biological and chemical control concern, Applied Mathematics and Mechanics. (2007) 28, no. 4, 541–551, MR2325166, ZBL1231.34019, 2-s2.0-38049115260, https://doi.org/10.1007/s10483-007-0415-y.
10.1007/s10483-007-0415-y Google Scholar
- 13 Liu B., Teng Z. D., and Chen L., Analysis of a predator-prey model with Holling II functional response concerning impulsive control strategy, Journal of Computational and Applied Mathematics. (2006) 193, no. 1, 347–362, MR2228722, ZBL1089.92060, 2-s2.0-33646197126, https://doi.org/10.1016/j.cam.2005.06.023.
- 14 Tang S., Tang G., and Cheke R. A., Optimum timing for integrated pest management: modelling rates of pesticide application and natural enemy releases, Journal of Theoretical Biology. (2010) 264, no. 2, 623–638, 2-s2.0-77951652585, https://doi.org/10.1016/j.jtbi.2010.02.034.
- 15 Baek H. K., Qualitative analysis of Beddington-DeAngelis type impulsive predator-prey models, Nonlinear Analysis: Real World Applications. (2010) 11, no. 3, 1312–1322, MR2646548, ZBL1204.34062, 2-s2.0-77950917103, https://doi.org/10.1016/j.nonrwa.2009.02.021.
- 16 Dimitrov D. T. and Kojouharov H. V., Complete mathematical analysis of predator-prey models with linear prey growth and Beddington-DeAngelis functional response, Applied Mathematics and Computation. (2005) 162, no. 2, 523–538, 2-s2.0-10444225454, https://doi.org/10.1016/j.amc.2003.12.106.
- 17 Naji R. K. and Balasim A. T., On the dynamical behavior of three species food web model, Chaos, Solitons and Fractals. (2007) 34, no. 5, 1636–1648, MR2335411, ZBL1152.34350, 2-s2.0-34250185002, https://doi.org/10.1016/j.chaos.2006.04.064.
- 18
Ermentrout B., Simulating: Analyzing and Animating Dynamical Systems: A Guide to XPPAUT for Researchers and Students, 2002, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, Pa, USA, MR1929404, https://doi.org/10.1137/1.9780898718195, MR1929404.
10.1137/1.9780898718195 Google Scholar
- 19 Genkai-Kato M. and Yamamura N., Unpalatable prey resolves the paradox of enrichment, Proceedings of the Royal Society B. (1999) 266, no. 1425, 1215–1219, 2-s2.0-0033594935, https://doi.org/10.1098/rspb.1999.0765.
- 20 Baĭnov D. D. and Simeonov P. S., Impulsive Differential Equations: Periodic Solutions and Applications, 1993, 66, Longman Scientific, New York, NY, USA, Pitman Monographs and Surveys in Pure and Applied Mathematics, MR1266625.
- 21 Bainov D. D. and Simeonov P. S., System With Impulsive Effect: Stability, Theory and Applications, 1989, John Wiley & Sons, New York, NY, USA.
- 22
Lakshmikantham V.,
Baĭnov D. D., and
Simeonov P. S., Theory of Impulsive Differential Equations, 1989, World Scientific, London, UK, MR1082551.
10.1142/0906 Google Scholar
- 23 Werner E. E. and Hall D., Optimal foraging and the size selection of prey by the bluegill sunfish (Lepomis macrochirus), Ecology. (1974) 55, 1216–1232.
- 24
Ruberson J. R.,
Nemoto H., and
Hirose Y., P. Barbosa, Pesticides and conservation of natural enemies in pest management, Conservation Biological Control, 1998, Academic Press, New York, NY, USA, 207–220.
10.1016/B978-012078147-8/50057-8 Google Scholar
- 25 Wang H., Spreading speeds and traveling waves for non-cooperative reaction-diffusion systems, Journal of Nonlinear Science. (2011) 21, no. 5, 747–783, MR2841986, ZBL1242.35147, 2-s2.0-80053376525, https://doi.org/10.1007/s00332-011-9099-9.
- 26 Teramoto E., Kawasaki K., and Shigesada N., Switching effect of predation on competitive prey species, Journal of Theoretical Biology. (1979) 79, no. 3, 303–315, MR543086, 2-s2.0-0018667032.
- 27 http://www.chinadaily.com.cn/china/.
- 28 http://www.ecns.cn/cns-wire/2012/08-17/22255.shtml.
