脉冲控制理论在害虫综合治理中的应用研究
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摘要
害虫控制在农业生产中具有重要作用,而微分方程数学模型在描述害虫种群动力学行为中起到了非常重要的作用,特别是用脉冲微分方程来描述害虫种群动力学模型能够更合理、更精确地反映各种变化规律,因为现实世界中害虫的繁殖和危害、以及人类的控制行为几乎都是阶段性的。
本文基于害虫的综合管理策略,利用脉冲微分方程的相关理论和方法建立并研究了三类具有脉冲效应(即分别在不同的固定时刻分别喷洒杀虫剂和释放染病害虫(或天敌))的生态-流行病模型。它们分别是一类具有脉冲效应和非线性发生率的生态-流行病模型,一类具有脉冲效应和Ivlev功能性反应的生态-流行病模型和一类具有脉冲效应的食饵依赖生态-流行病模型。
我们分别证明了各系统的所有解是一致最终有界的。并利用Floquet乘子理论及脉冲比较定理,证明了当脉冲周期小于某个临界值时,系统存在一个全局渐近稳定的易感者害虫根除周期解,否则,系统是持续生存的。最后,为了证实我们的分析结果,借助计算机数值模拟讨论所提出模型的动力学行为,包括系统周期解的存在性、持续生存性、绝灭性和稳定性。所得到的结果,从数学的角度给出在农业中利用害虫综合管理策略控制害虫增长的一个理论依据。
Pest control plays an important role in agriculture. Mathematical models of di?erentialequations play an important role in describing pest population dynamic behavior. Especially,impulsive di?erential equations describe pest population dynamic models, which is morereasonable and precise on re?ecting all kinds of change orderliness, since the reproductionand damager of pests, human control behavior are almost periodical in the natural world.
In this paper, we investigate and study the dynamic behaviors of three classes ofeco-epidemiological models by means of the theory and method of impulsive di?erentialequations concerning integrated pest management strategy, i.e., periodic releasing infec-tive pests (or natural enemy) and spraying pesticide at di?erent fixed time. They arean eco-epidemiological model with impulsive e?ect and nonlinear incidence rate, an eco-epidemiological model with impulsive e?ect and Ivlev functional response and a class ofprey-dependent eco-epidemiological model with impulsive e?ect, respectively.
We prove that all solutions of the investigated system are uniformly ultimately boundedrespectively. By applying the Floquet theorem and comparison theorem, we prove that thereexists a globally asymptotically stable susceptible pest-eradication periodic solution when theimpulsive period is less than some critical value. Otherwise, the system can be permanent.Finally, to substantiate our analytical results, numerical simulations are used to investigatedynamical behaviors including the existence of periodic solution, permanence, extinctionand stability. Mathematically, the theoretical evidence of the controlling pests using theintegrated pest management strategy in agriculture is given by the obtained results.
本文基于害虫的综合管理策略,利用脉冲微分方程的相关理论和方法建立并研究了三类具有脉冲效应(即分别在不同的固定时刻分别喷洒杀虫剂和释放染病害虫(或天敌))的生态-流行病模型。它们分别是一类具有脉冲效应和非线性发生率的生态-流行病模型,一类具有脉冲效应和Ivlev功能性反应的生态-流行病模型和一类具有脉冲效应的食饵依赖生态-流行病模型。
我们分别证明了各系统的所有解是一致最终有界的。并利用Floquet乘子理论及脉冲比较定理,证明了当脉冲周期小于某个临界值时,系统存在一个全局渐近稳定的易感者害虫根除周期解,否则,系统是持续生存的。最后,为了证实我们的分析结果,借助计算机数值模拟讨论所提出模型的动力学行为,包括系统周期解的存在性、持续生存性、绝灭性和稳定性。所得到的结果,从数学的角度给出在农业中利用害虫综合管理策略控制害虫增长的一个理论依据。
Pest control plays an important role in agriculture. Mathematical models of di?erentialequations play an important role in describing pest population dynamic behavior. Especially,impulsive di?erential equations describe pest population dynamic models, which is morereasonable and precise on re?ecting all kinds of change orderliness, since the reproductionand damager of pests, human control behavior are almost periodical in the natural world.
In this paper, we investigate and study the dynamic behaviors of three classes ofeco-epidemiological models by means of the theory and method of impulsive di?erentialequations concerning integrated pest management strategy, i.e., periodic releasing infec-tive pests (or natural enemy) and spraying pesticide at di?erent fixed time. They arean eco-epidemiological model with impulsive e?ect and nonlinear incidence rate, an eco-epidemiological model with impulsive e?ect and Ivlev functional response and a class ofprey-dependent eco-epidemiological model with impulsive e?ect, respectively.
We prove that all solutions of the investigated system are uniformly ultimately boundedrespectively. By applying the Floquet theorem and comparison theorem, we prove that thereexists a globally asymptotically stable susceptible pest-eradication periodic solution when theimpulsive period is less than some critical value. Otherwise, the system can be permanent.Finally, to substantiate our analytical results, numerical simulations are used to investigatedynamical behaviors including the existence of periodic solution, permanence, extinctionand stability. Mathematically, the theoretical evidence of the controlling pests using theintegrated pest management strategy in agriculture is given by the obtained results.
引文
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[2] Liu X N, Chen L S. Etinction and permanence of a Lotka-volterra predator-prey system withimpulsive e?ect on the predators[J]. Mathematical Biosciences, to appear
[3] Falcon L A. Use of Bacteria for Microbial Control of Insects[M]. New York, N Y: AcademicPress, 1971.
[4] Burges H D, Hussey N W. Microbial Control of Insects and Mites[M]. New York, N Y: Aca-demic Press, 1971,67-95.
[5] Burges H D, Hussey N W. Microbial Control of Insects and Mites[M]. New York, N Y: Aca-demic Press, 1971,861.
[6] Davis P E, Myers K, Hoy J B. Biological control among vertebrates[A]. In: Hu?aker C B,Messenger PS Eds. Theory and Practice of Biological Control[C]. New York, N Y: PlenumPress,1976,501-519.
[7] Goh B S. The potential utility of control theory to pest management[J]. Proc. Ecol. Soc.1971,6:84-89.
[8] Wickwire K. Mathematical models for the control of pests and infectious diseases: a survey[J].Theoret Population Biol. 1977,11(2):182-238.
[9] Goh B S. Management and analysis of biological populations[M]. Amsterdam-Oxford-NewYork: Elsevier Scientific Publishing Company, 1980.
[10] Z. Liu, R. Tan. Impulsive harvesting and stocking in a Monod-Haldane functional responsepredator-prey system[J]. Chaos, Solitons & Fractals 34(2007)454-464.
[11] B. Liu et al. Analysis of a predator-prey model with Holling II functional response con-cerning impulsive control strategy[J]. Journal of Computational and Applied Mathematics193(2006)347-362.
[12] B. Liu et al. The dynamical behaviors of a Lotka-Volterra predator-prey model concerningintegrated pest management[J]. Nonlinear Analysis: Real World Applications 6(2005)227-243.
[13] Z. Liu et al. Impulsive control strategies in biological control of pesticide[J]. Theoretical Pop-ulation Biology 64(2003)39-47.
[14] B. Liu et al. The dynamics of a prey-dependent consumption model concerning impulsivecontrol strategy[J]. Appl. Math. Comput. 169(2005)305-320.
[15] G. Pang et al. Extinction and permanence in delayed stage-structure predator-prey systemwith impulsive e?ects[J]. Chaos, Solitons & Fractals 39(2009)2216-2224.
[16] H. Guo, L. Chen. Time-limited pest control of a Lotka-Volterra model with impulsive har-vest[J]. Nonlinear Analysis: Real World Applications 10(2009)840-848.
[17] S. Tang et al. Integrated pest management models and their dynamical behaviour[J]. Bulletinof Mathematical Biology 67(2005)115-135.
[18]司成斌,沈伯骞.在杀虫剂作用下的一类具有Allee效应的天敌-害虫模型[J].生物数学学报, 2004,19(1):82-86.
[19] J. Jiao, L. Chen, S. Cai. Impulsive control strategy of a pest management SI model withnonlinear incidence rate[J]. Applied Mathematical Modelling 33(2009)555-563.
[20] Chen Y, Liu Z. Modelling and analysis of an impulsive SI model with Monod-Haldane func-tional response[J]. Chaos, Solutions & Fractals 39(2009)1698-1714.
[21] H. Zhang, J. Jiao, L. Chen. Pest management through continuous and impulsive control strate-gies[J]. BioSystems 90(2007)350-361.
[22] G. Pang, L. Chen. Dynamic analysis of a pest-epidemic model with impulsive control[J].Mathematics and Computers in Simulation 79(2008)72-84.
[23] P. Georgescu, G. Morosanu. Pest regulation by means of impulsive controls[J]. Applied Math-ematics and Computation 190(2007)790-803.
[24] J. Jiao, L. Chen. A pest management SI model with periodic biological and chemical controlconcern[J]. Applied Mathematics and Computation 183(2006)1018-1026.
[25] J. Jiao, L. Chen. Nonlinear incidence rate of a pest management SI model with biologi-cal and chemical control concern[J]. Applied Mathematics and Mechanics(English Edition)2007,28(4):541-551.
[26] M. Liu et al. An impulsive predator-prey model with communicable disease in the prey speciesonly[J]. Nonlinear Analysis: Real World Applications 10(2009)3098-3111.
[27] N. Bairagi et al. Harvesting as a disease control measure in an eco-epidemiological system–Atheoretical study[J]. Mathematical Biosciences 217(2009)134-144.
[28] Z. Huang et al. A Predator-Prey system with viral infection and anorexia response[J]. Appl.Math. Comput. 175(2006)1455-1483.
[29] J. Chattopadhyay et al. Pelicans at risk in Salton Sea-an eco-epidemiological model-II[J].Ecological Modelling 167(2003)199-211.
[30] S. Chatterjee et al. Time delay factor can be used as a key factor for preventing the outbreakof a disease-Results drawn from a mathematical study of a one season eco-epidemiologicalmodel[J]. Nonlinear Analysis: Real World Applications 8(2007)1472-1493.
[31]黄友霞,王辉,苏丹丹.食饵有病的生态-流行病模型的稳定性研究[J].生物数学学报,2008,23(1):132-138.
[32]邹娓,谢杰华.食饵种群染病且有垂直传染的生态-流行病模型的分析[J].湖南文理学院学报(自然科学版), 2007,19(4):35-38.
[33] Yanni Xiao, Lansun Chen. Analysis of a Three Species Eco-Epidemiological Model[J]. Math-ematical Analysis and Applications 258(2001),733-754.
[34] R.K. Upadhyay et al. Chaos in eco-epidemiological problem of the Salton Sea and its possiblecontrol[J]. Applied Mathematics and Computation 196(2008)392-401.
[35] A. Kang et al. Dynamic behavior of an eco-epidemic system with impulsive birth[J]. Math.Anal. Appl. 345(2008)783-795.
[36]陆志奇,李爽.具有阶段结构的生态-流行病SI模型的分析[J].河南师范大学学报(自然科学版), 2008,36(1):1-5.
[37]宋新宇,肖燕妮,陈兰荪.具有时滞的生态-流行病模型的稳定性和Hopf分支[J].数学物理学报, 2005,25A(1):57-66.
[38]张靖,宋燕,詹丽.食饵具有流行病的捕食-被捕食(SI)模型的分析[J].渤海大学学报(自然科学版), 2008,29(2):173-176.
[39] P. Georgescu, H. Zhang. An impulsively controlled predator-pest model with disease in thepest[J]. Nonlinear Analysis: Real World Applications. to appear
[40]祝光湖,陈兰荪.一个阶段时滞结构的生态-流行病模型的害虫综合防治策略[J].重庆文理学院学报(自然科学版), 2008,27(6):1-5.
[41] J.F. Zhang et al. Hopf bifurcation and stability of periodic solutions in a delayed eco-epidemiological system[J]. Applied Mathematics and Computation 198(2008)865-876.
[42]孙树林,原存德.捕食者具有流行病的捕食-被捕食模型的分析[J].生物数学学报,2006,21(1):97-104.
[43]孙树林,原存德.捕食者有病的生态-流行病SIS模型的分析[J].工程数学学报,2005,22(1):30-34.
[44]张江山,孙树林.捕食者有病的生态-流行病模型的分析[J].生物数学学报, 2005,20(2):157-164.
[45]詹丽,宋燕,张靖.捕食者有病且具有垂直传染的生态-流行病SIS模型分析[J].沈阳化工学院学报, 2008,22(4):373-376.
[46]张永坡,王辉.一个疾病在捕食者中传播的捕食与被捕食模型的分析[J].北京工商大学学报, 2007,25(4)71-74.
[47] Y. Tan, L. Chen. Modelling approach for biological control of insect pest by releasing infectedpest[J]. Chaos, Solitons & Fractals 39(2009)304-315.
[48] R. Shi, L. Chen. A predator-prey model with disease in the prey and two impulses for integratedpest management[J]. Applied Mathematical Modelling 33(2009)2248-2256.
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