Abstract and Applied Analysis
- Abstr. Appl. Anal.
- Volume 2014, Special Issue (2013), Article ID 963072, 18 pages.
Geometric Analysis of an Integrated Pest Management Model Including Two State Impulses
Wencai Zhao, Yulin Liu, Tongqian Zhang, and Xinzhu Meng
Full-text: Open access
Abstract
According to integrated pest management strategies, we construct and investigate the dynamics of a Holling-Tanner predator-prey system with state dependent impulsive effects by releasing natural enemies and spraying pesticide at different thresholds. Applying the Dulacs criterion, the global stability of the positive equilibrium in the system without impulsive effect is discussed. By using impulsive differential equation geometry theory and the method of successor functions, we prove the existence of periodic solution of the system with state dependent impulsive effects. Furthermore, the stability conditions of periodic solutions are obtained. Some simulations are exerted to illustrate the feasibility of our main results.
Article information
Source
Abstr. Appl. Anal., Volume 2014, Special Issue (2013), Article ID 963072, 18 pages.
Dates
First available in Project Euclid: 6 October 2014
Permanent link to this document
https://projecteuclid.org/euclid.aaa/1412607549
Digital Object Identifier
doi:10.1155/2014/963072
Mathematical Reviews number (MathSciNet)
MR3186991
Zentralblatt MATH identifier
07023408
Citation
Zhao, Wencai; Liu, Yulin; Zhang, Tongqian; Meng, Xinzhu. Geometric Analysis of an Integrated Pest Management Model Including Two State Impulses. Abstr. Appl. Anal. 2014, Special Issue (2013), Article ID 963072, 18 pages. doi:10.1155/2014/963072. https://projecteuclid.org/euclid.aaa/1412607549
References
- Agricultural Information Network of China, http://www.agri.gov.c/.URL: Link to item
- C. C. Yang, C. S. Wang, Y. N. Zheng et al., “Sustained effects of Trichogramma dendrolimi on Ostrinia furnacali,” Journal of Maize Sciences, vol. 19, no. 1, pp. 139–142, 2011 (Chinese).
- D. D. Bainov and P. S. Simeonov, Impulsive Differential Equations: Periodic Solutions and Applications, CRC Press, Boca Raton, Fla, USA, 1993.
- V. Lakshmikantham, D. D. Baĭnov, and P. S. Simeonov, Theory of Impulsive Differential Equations, vol. 6 of Series in Modern Applied Mathematics, World Scientific, Singapore, 1989.Mathematical Reviews (MathSciNet): MR1082551
- P. S. Simeonov and D. D. Baĭnov, “Orbital stability of periodic solutions of autonomous systems with impulse effect,” International Journal of Systems Science, vol. 19, no. 12, pp. 2561–2585, 1988.Mathematical Reviews (MathSciNet): MR975426
Zentralblatt MATH: 0669.34044
Digital Object Identifier: doi:10.1080/00207728808547133 - E. M. Bonotto, “LaSalles theorems in impulsive semidynamical systems,” Cadernos de Matematica, vol. 9, pp. 157–168, 2008.
- E. M. Bonotto and M. Federson, “Topological conjugation and asymptotic stability in impulsive semidynamical systems,” Journal of Mathematical Analysis and Applications, vol. 326, no. 2, pp. 869–881, 2007.Mathematical Reviews (MathSciNet): MR2280949
Zentralblatt MATH: 1162.37008
Digital Object Identifier: doi:10.1016/j.jmaa.2006.03.042 - J. J. Nieto and D. O'Regan, “Variational approach to impulsive differential equations,” Nonlinear Analysis: Real World Applications, vol. 10, no. 2, pp. 680–690, 2009.Mathematical Reviews (MathSciNet): MR2474254
Zentralblatt MATH: 1167.34318
Digital Object Identifier: doi:10.1016/j.nonrwa.2007.10.022 - J. J. Nieto, “Periodic boundary value problems for first-order impulsive ordinary differential equations,” Nonlinear Analysis: Theory, Methods & Applications, vol. 51, no. 7, pp. 1223–1232, 2002.
- L. S. Chen, “Pest control and geometric theory of semi-continuous dynamical system,” Journal of Beihua University (Natural Science), vol. 12, no. 1, pp. 1–9, 2011 (Chinese).
- L. S. Chen, “Theory and application of semi-continuous dynamical system,” Journal of Yulin Normal University (Natural Science), vol. 34, no. 2, pp. 1–9, 2013 (Chinese).
- X. Song, M. Hao, and X. Meng, “A stage-structured predator-prey model with disturbing pulse and time delays,” Applied Mathematical Modelling, vol. 33, no. 1, pp. 211–223, 2009.Mathematical Reviews (MathSciNet): MR2458507
Zentralblatt MATH: 1167.34372
Digital Object Identifier: doi:10.1016/j.apm.2007.10.020 - S. Sun and L. Chen, “Mathematical modelling to control a pest population by infected pests,” Applied Mathematical Modelling, vol. 33, no. 6, pp. 2864–2873, 2009.Mathematical Reviews (MathSciNet): MR2502149
Zentralblatt MATH: 1205.34065
Digital Object Identifier: doi:10.1016/j.apm.2008.08.018 - S. Tang and R. A. Cheke, “Models for integrated pest control and their biological implications,” Mathematical Biosciences, vol. 215, no. 1, pp. 115–125, 2008.Mathematical Reviews (MathSciNet): MR2459534
Zentralblatt MATH: 1156.92046
Digital Object Identifier: doi:10.1016/j.mbs.2008.06.008 - S. Tang, Y. Xiao, and R. A. Cheke, “Multiple attractors of host-parasitoid models with integrated pest management strategies: eradication, persistence and outbreak,” Theoretical Population Biology, vol. 73, no. 2, pp. 181–197, 2008.
- B. Liu, Y. Zhang, and L. Chen, “The dynamical behaviors of a Lotka-Volterra predator-prey model concerning integrated pest management,” Nonlinear Analysis: Real World Applications, vol. 6, no. 2, pp. 227–243, 2005.Mathematical Reviews (MathSciNet): MR2111652
Digital Object Identifier: doi:10.1016/j.nonrwa.2004.08.001 - L. Mailleret and F. Grognard, “Global stability and optimisation of a general impulsive biological control model,” Mathematical Biosciences, vol. 221, no. 2, pp. 91–100, 2009.Mathematical Reviews (MathSciNet): MR2561136
Zentralblatt MATH: 1175.92070
Digital Object Identifier: doi:10.1016/j.mbs.2009.07.002 - R. Shi and L. Chen, “The study of a ratio-dependent predator-prey model with stage structure in the prey,” Nonlinear Dynamics, vol. 58, no. 1-2, pp. 443–451, 2009.Mathematical Reviews (MathSciNet): MR2550826
Zentralblatt MATH: 1183.92083
Digital Object Identifier: doi:10.1007/s11071-009-9491-2 - X. Meng, J. Jiao, and L. Chen, “The dynamics of an age structured predator-prey model with disturbing pulse and time delays,” Nonlinear Analysis: Real World Applications, vol. 9, no. 2, pp. 547–561, 2008.Mathematical Reviews (MathSciNet): MR2382398
Zentralblatt MATH: 1142.34054
Digital Object Identifier: doi:10.1016/j.nonrwa.2006.12.001 - X. Meng, Z. Li, and J. J. Nieto, “Dynamic analysis of Michaelis-Menten chemostat-type competition models with time delay and pulse in a polluted environment,” Journal of Mathematical Chemistry, vol. 47, no. 1, pp. 123–144, 2010.Mathematical Reviews (MathSciNet): MR2576641
Zentralblatt MATH: 1194.92075
Digital Object Identifier: doi:10.1007/s10910-009-9536-2 - J. Jiao, W. Long, and L. Chen, “A single stage-structured population model with mature individuals in a polluted environment and pulse input of environmental toxin,” Nonlinear Analysis: Real World Applications, vol. 10, no. 5, pp. 3073–3081, 2009.Mathematical Reviews (MathSciNet): MR2523269
Zentralblatt MATH: 1162.92330
Digital Object Identifier: doi:10.1016/j.nonrwa.2008.10.007 - T. Zhang, X. Meng, and Y. Song, “The dynamics of a high-dimensional delayed pest management model with impulsive pesticide input and harvesting prey at different fixed moments,” Nonlinear Dynamics, vol. 64, no. 1-2, pp. 1–12, 2011.Mathematical Reviews (MathSciNet): MR2782953
Zentralblatt MATH: 06251524
Digital Object Identifier: doi:10.1007/s11071-010-9840-1 - T. Q. Zhang, X. Z. Meng, Y. Song, and T. H. Zhang, “A stage-structured predator-prey SI model with disease in the prey and impulsive effects,” Mathematical Modelling and Analysis, vol. 18, no. 4, pp. 505–528, 2013.Mathematical Reviews (MathSciNet): MR3175661
Digital Object Identifier: doi:10.3846/13926292.2013.840866 - H. Zhang, J. Jiao, and L. Chen, “Pest management through continuous and impulsive control strategies,” BioSystems, vol. 90, no. 2, pp. 350–361, 2007.
- H. Zhang, L. Chen, and J. J. Nieto, “A delayed epidemic model with stage-structure and pulses for pest management strategy,” Nonlinear Analysis: Real World Applications, vol. 9, no. 4, pp. 1714–1726, 2008.Mathematical Reviews (MathSciNet): MR2422575
Zentralblatt MATH: 1154.34394
Digital Object Identifier: doi:10.1016/j.nonrwa.2007.05.004 - J.-J. Jiao, L.-S. Chen, J. J. Nieto, and A. Torres, “Permanence and global attractivity of stage-structured predator-prey model with continuous harvesting on predator and impulsive stocking on prey,” Applied Mathematics and Mechanics, vol. 29, no. 5, pp. 653–663, 2008.Mathematical Reviews (MathSciNet): MR2414687
Zentralblatt MATH: 1231.34021
Digital Object Identifier: doi:10.1007/s10483-008-0509-x - S. Tang, Y. Xiao, L. Chen, and R. A. Cheke, “Integrated pest management models and their dynamical behaviour,” Bulletin of Mathematical Biology, vol. 67, no. 1, pp. 115–135, 2005.Mathematical Reviews (MathSciNet): MR2204249
Digital Object Identifier: doi:10.1016/j.bulm.2004.06.005 - S. Tang and R. A. Cheke, “State-dependent impulsive models of integrated pest management (IPM) strategies and their dynamic consequences,” Journal of Mathematical Biology, vol. 50, no. 3, pp. 257–292, 2005.Mathematical Reviews (MathSciNet): MR2135823
Zentralblatt MATH: 1080.92067
Digital Object Identifier: doi:10.1007/s00285-004-0290-6 - G. Zeng, L. Chen, and L. Sun, “Existence of periodic solution of order one of planar impulsive autonomous system,” Journal of Computational and Applied Mathematics, vol. 186, no. 2, pp. 466–481, 2006.Mathematical Reviews (MathSciNet): MR2176360
Zentralblatt MATH: 1088.34040
Digital Object Identifier: doi:10.1016/j.cam.2005.03.003 - L. Zhao, L. Chen, and Q. Zhang, “The geometrical analysis of a predator-prey model with two state impulses,” Mathematical Biosciences, vol. 238, no. 2, pp. 55–64, 2012.Mathematical Reviews (MathSciNet): MR2947084
Zentralblatt MATH: 1250.92047
Digital Object Identifier: doi:10.1016/j.mbs.2012.03.011 - L. Nie, Z. Teng, L. Hu, and J. Peng, “Existence and stability of periodic solution of a predator-prey model with state-dependent impulsive effects,” Mathematics and Computers in Simulation, vol. 79, no. 7, pp. 2122–2134, 2009.Mathematical Reviews (MathSciNet): MR2504088
Zentralblatt MATH: 1185.34123
Digital Object Identifier: doi:10.1016/j.matcom.2008.11.015 - L. Nie, Z. Teng, L. Hu, and J. Peng, “Qualitative analysis of a modified Leslie-Gower and Holling-type II predator-prey model with state dependent impulsive effects,” Nonlinear Analysis: Real World Applications, vol. 11, no. 3, pp. 1364–1373, 2010.Mathematical Reviews (MathSciNet): MR2646552
Zentralblatt MATH: 1228.37058
Digital Object Identifier: doi:10.1016/j.nonrwa.2009.02.026 - B. Liu, Y. Tian, and B. Kang, “Dynamics on a Holling II predator-prey model with state-dependent impulsive control,” International Journal of Biomathematics, vol. 5, no. 3, Article ID 1260006, 2012.
- C. Dai, M. Zhao, and L. Chen, “Homoclinic bifurcation in semi-continuous dynamic systems,” International Journal of Biomathematics, vol. 5, no. 6, Article ID 1250059, 2012.Mathematical Reviews (MathSciNet): MR2969658
Digital Object Identifier: doi:10.1142/S1793524512500593 - J. B. Fu and Y. Z. Wang, “The mathematical study of pest management strategy,” Discrete Dynamics in Nature and Society, vol. 2012, Article ID 251942, 19 pages, 2012.
- C. Wei and L. Chen, “Periodic solution of prey-predator model with Beddington-DeAngelis functional response and impulsive state feedback control,” Journal of Applied Mathematics, vol. 2012, Article ID 607105, 17 pages, 2012.
- C. Wei and L. Chen, “Heteroclinic bifurcations of a prey-predator fishery model with impulsive harvesting,” International Journal of Biomathematics, vol. 6, no. 5, Article ID 1350031, 2013.Mathematical Reviews (MathSciNet): MR3123176
Digital Object Identifier: doi:10.1142/S1793524513500319 - L. S. Chen, X. Z. Meng, and J. J. Jiao, Biological Dynamics, Science Press, Beijing, China, 2009.
- S. B. Hsu and T. W. Huang, “Global stability for a class of predator-prey systems,” SIAM Journal on Applied Mathematics, vol. 55, no. 3, pp. 763–783, 1995. \endinputMathematical Reviews (MathSciNet): MR1331585
Zentralblatt MATH: 0832.34035
Digital Object Identifier: doi:10.1137/S0036139993253201
- You have access to this content.
- You have partial access to this content.
- You do not have access to this content.
More like this
- Two Generalized Predator-Prey Models for Integrated Pest Management with
Stage Structure and Disease in the Prey Population
Shi, Ruiqing, Tang, Sanyi, and Feng, Wenli, Abstract and Applied Analysis, 2013 - Existence and Attractiveness of Order One Periodic Solution of a Holling I Predator-Prey Model
Cheng, Huidong, Zhang, Tongqian, and Wang, Fang, Abstract and Applied Analysis, 2012 - Optimal Control Policies of Pests for Hybrid Dynamical Systems
Kang, Baolin, He, Mingfeng, and Liu, Bing, Abstract and Applied Analysis, 2013
- Two Generalized Predator-Prey Models for Integrated Pest Management with
Stage Structure and Disease in the Prey Population
Shi, Ruiqing, Tang, Sanyi, and Feng, Wenli, Abstract and Applied Analysis, 2013 - Existence and Attractiveness of Order One Periodic Solution of a Holling I Predator-Prey Model
Cheng, Huidong, Zhang, Tongqian, and Wang, Fang, Abstract and Applied Analysis, 2012 - Optimal Control Policies of Pests for Hybrid Dynamical Systems
Kang, Baolin, He, Mingfeng, and Liu, Bing, Abstract and Applied Analysis, 2013 - Generalized Predator-Prey Model with Nonlinear Impulsive Control Strategy
Qin, Wenjie, Tang, Guangyao, and Tang, Sanyi, Journal of Applied Mathematics, 2014 - Dynamic Analysis of a Predator-Prey (Pest) Model with Disease in Prey and Involving an Impulsive Control Strategy
Zhao, Min, Wang, Yanzhen, and Chen, Lansun, Journal of Applied Mathematics, 2012 - Optimal Application Timing of Pest Control Tactics in Nonautonomous Pest Growth Model
Zhang, Shujuan, Liang, Juhua, and Tang, Sanyi, Abstract and Applied Analysis, 2014 - Time Delayed Stage-Structured Predator-Prey Model with Birth Pulse and Pest Control Tactics
Yan, Mei, Li, Yongfeng, and Xiang, Zhongyi, Abstract and Applied Analysis, 2014 - Stability and Permanence of a Pest Management Model with Impulsive Releasing and Harvesting
Yang, Jiangtao and Yang, Zhichun, Abstract and Applied Analysis, 2013 - A predator-prey model of Holling-type II with
state dependent impulsive effects
Ding, Changming and Zhang, Zhongxin, Topological Methods in Nonlinear Analysis, 2015 - Dynamical Analysis of a Pest Management Model with Saturated Growth Rate and State Dependent Impulsive Effects
Zhao, Wencai, Zhang, Tongqian, Meng, Xinzhu, and Yang, Yang, Abstract and Applied Analysis, 2013