具有脉冲作用的捕食系统的研究
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摘要
本文利用脉冲微分方程理论对几个捕食系统进行了研究,讨论了种群模型的持久性及其稳定性.全文共分为三章.
第一章绪论,我们介绍了本文的研究背景和主要工作以及预备知识.
第二章主要研究具有脉冲作用的捕食系统.首先讨论了一类具有Beddington-DeAngelis型功能反应的捕食系统.在此系统中,通过对捕食者(天敌)施行周期投放,对食饵(害虫)施行周期捕获的策略来控制害虫,利用脉冲微分方程的Floquet理论和比较定理,得到了食饵灭绝周期解的存在性和局部渐近稳定性的充分条件,以及种群持续生存的条件.其次,讨论了食饵具有脉冲捕获,捕食者具有阶段结构的Leslis-Gower HollingⅡ功能反应的Gomportz捕食模型,通过利用由频闪映射决定的离散动力系统,以及脉冲方程的比较定理,得到了相应的临界条件以确保捕食者灭绝周期解的全局吸引和所讨论系统的持久性.最后,基于综合害虫管理,提出并研究了一类具有脉冲效应和Ivlev类功能反应的两个捕食者一个食饵系统,利用脉冲微分方程的理论,得到了种群灭绝和持续生存的充分条件.
第三章讨论了一类具有性别偏食现象和阶段结构的HollingⅡ类功能反应食饵捕食者模型,利用对不等式放缩得到了种群一致持续生存的条件,通过构造liapunov函数证明了系统周期解的全局渐近稳定性.
In this paper, by use of the theory on impulsive differential equation, we study some predator-prey systems and discuss the permanence and stablity of population ecological models. The article is divided into three chapters.
In Chapter 1, We introduce some knowledge of biology mathematics, the main works and some prelimineries.
In Chapter 2, we study predator-prey system with impulsive effects. At first, the article dis-cussed a predator-prey system with the Beddington-DeAnglis functional response, controlled the pest by using the stratege which is periodically releasing predator (natural enemy) and periodi-cally capturing prey (pest). By using the Floquet theory of impulsive equation and comparison theorem, we obtain sufficient conditions of the locally asymptotical stability of prey-extinction periodic solution and the permanence of the system. Next, we consider a predator-prey model with Leslie-Gower Holling II type schemes and Gomportz which has periodic harvesting for the prey and stage-structured for the predator. By use of the discrete dynamical system deter-mimed by the stroboscopic map and the comparison theory of impulsive equation, we obtain some corresponding threshold conditions which guarantee the globally asymptotical stability of predator-extinction periodic solution and the permanence of this system. At last, based on the integrated pest management program, a predator-prey system with impulsive effect and Ivlev functional response is studied. By using the Floquet theory of impulsive equation and comparison theorem, sufficient conditions for the population to be extinct and permanence are given.
In Chapter 3, a predator-prey model with sexual favoritism is considered which has stage-structure and Holling II functional reponse. conditions of permanence for the system were obtained by constructing inequality. the global stability of the periodic solution were approved by Liapunov method.
第一章绪论,我们介绍了本文的研究背景和主要工作以及预备知识.
第二章主要研究具有脉冲作用的捕食系统.首先讨论了一类具有Beddington-DeAngelis型功能反应的捕食系统.在此系统中,通过对捕食者(天敌)施行周期投放,对食饵(害虫)施行周期捕获的策略来控制害虫,利用脉冲微分方程的Floquet理论和比较定理,得到了食饵灭绝周期解的存在性和局部渐近稳定性的充分条件,以及种群持续生存的条件.其次,讨论了食饵具有脉冲捕获,捕食者具有阶段结构的Leslis-Gower HollingⅡ功能反应的Gomportz捕食模型,通过利用由频闪映射决定的离散动力系统,以及脉冲方程的比较定理,得到了相应的临界条件以确保捕食者灭绝周期解的全局吸引和所讨论系统的持久性.最后,基于综合害虫管理,提出并研究了一类具有脉冲效应和Ivlev类功能反应的两个捕食者一个食饵系统,利用脉冲微分方程的理论,得到了种群灭绝和持续生存的充分条件.
第三章讨论了一类具有性别偏食现象和阶段结构的HollingⅡ类功能反应食饵捕食者模型,利用对不等式放缩得到了种群一致持续生存的条件,通过构造liapunov函数证明了系统周期解的全局渐近稳定性.
In this paper, by use of the theory on impulsive differential equation, we study some predator-prey systems and discuss the permanence and stablity of population ecological models. The article is divided into three chapters.
In Chapter 1, We introduce some knowledge of biology mathematics, the main works and some prelimineries.
In Chapter 2, we study predator-prey system with impulsive effects. At first, the article dis-cussed a predator-prey system with the Beddington-DeAnglis functional response, controlled the pest by using the stratege which is periodically releasing predator (natural enemy) and periodi-cally capturing prey (pest). By using the Floquet theory of impulsive equation and comparison theorem, we obtain sufficient conditions of the locally asymptotical stability of prey-extinction periodic solution and the permanence of the system. Next, we consider a predator-prey model with Leslie-Gower Holling II type schemes and Gomportz which has periodic harvesting for the prey and stage-structured for the predator. By use of the discrete dynamical system deter-mimed by the stroboscopic map and the comparison theory of impulsive equation, we obtain some corresponding threshold conditions which guarantee the globally asymptotical stability of predator-extinction periodic solution and the permanence of this system. At last, based on the integrated pest management program, a predator-prey system with impulsive effect and Ivlev functional response is studied. By using the Floquet theory of impulsive equation and comparison theorem, sufficient conditions for the population to be extinct and permanence are given.
In Chapter 3, a predator-prey model with sexual favoritism is considered which has stage-structure and Holling II functional reponse. conditions of permanence for the system were obtained by constructing inequality. the global stability of the periodic solution were approved by Liapunov method.
引文
[1]Agur Z., Cojocaru L., Anderson R., Pulse mass measles vaccination across age cohorts, Proceedings of the National Academy of Sciences of the United States of America,1993,90:11698-11702.
[2]Zhou Y., Liu H., Stability of perioic solution for an SIS model with pulse vaccination, Mathematical and Computer Modelling,2003,38:299-308.
[3]Liu X.N., Chen L., Complex dymamics of Holling type II Lotke-Volterra predator-prey system with impulsive perturbation on the predator, Chaos,Solitions and Fractals,2003,16:311-320.
[4]Zhang X.Y., Shuai Z.S.,Wang K., Optimal impulsive havesting policy for single population, Nonlin-ear Analysis:Rear World Applicition,2005,44:185-199.
[5]Hastings A., Age-dependent predation is not a simple process.I.Continuous time models, Theoretical Population Biology,1983,23(3):47-62.
[6]Hastings A., Delays in recruitment at different trophic levels:Effects on stability, Journal of Mathematical Biology,1984,21(1):35-44.
[7]Wang W., Chen L., A predator-prey system with stage-structure for predator, Applied Mathematical and Computation,1997,33:83-91.
[8]Fisher M.E., Goh B.S., Stiblity results for delay-recuitment models in population dynamics, Bulletin of Mathematical Biology,1986,48:485.
[9]Liu S., Chen L., Extinction and permanence in competitive stage-structrured system with time-delay, Nonliear Analysis,TMA,2002,51:1347-1361.
[10]Yan J., Stablity for impulsive delay differential equations, Nonlinear Analysis, TMA,2005,63:66-80.
[11]Liu X., Ballinger G., Boundeness for impulsive delay differential equations and applications to population growth models, Nonlinear Analysis,TMA,2003,53:1041-1062.
[12]Sun S.L., Chen L., Existence of positive periodic solution of an impulsive delay Logistic model, Applied Mathematical and Computation,2007,184:617-623.
[13]刘汉武,王荣欣,刘建欣,具性别结构的食饵捕食者模型,生物数学学报,2005,20(2)：179-182.
[14]何朝晖,崔景安,具有年龄结构捕食系统的永久持续生存,南京师大学报,2005,28(1)：24-29.
[15]祝惠娇,胡志兴,具性别偏食的Beddington-DeAngelis反应的三种群系统的研究,北京工商大学学报,2006,24(6)：64-67.
[16]杜学武,原三领,关于三种群第三类功能反应周期系数模型的研究,工程数学学报,2000,17(3)：139-142.
[17]徐长永,王美娟,周艳丽,具阶段结构和HollingⅡ类功能反应的捕食系统研究,上海理工大学学报,2006,28(5)：49-54.
[18]徐长永,王美娟,刘妍,一类食饵具有性别结构的非自治两种群捕食系统研究,上海理工大学学报,2008,30(5)：449-455.
[19]孙凯玲,王美娟,朱春娟,具有性别偏食和Holling III类功能反应的食饵捕食者模型,上海理工大学学报,2009,31(1)：6-10.
[20]Zhang S.W., Chen L., A study of predator-prey models with the Beddington-DeAnglis functional response and impulsive effect, Chaos,Solitons and Fractals,2006,27:237-248.
[21]liu B., Zhang Y.j., Chen L., Dynamic complexities of a Holling I Predator-prey model concerning periodic biological and chemical control. Chaos,Solitons and Fractals,2004,22:123-134.
[22]贾建文,乔梅红,一类具有脉冲效应和Holling III类功能反应的捕食系统分析,生物数学学报,2008,23(2)：279-288.
[23]张玉娟,陈兰荪,孙丽华,一类具有脉冲效应的捕食者-食饵系统的灭绝与持续生存,大连理工大学学报,2004,40(2)：769-77.
[24]Song X., Chen L., Optimal harvesting and stability for a two-species competitive system with stage structure, Mathematical Biosciences,2001,170:173-186.
[25]Song X.Y., Hao M.Y., Meng X.Z., A stage-structured predator-prey model with disturbing pulse and time delays, Applied Mathematical Modelling,2009,33:211-223.
[26]Liu K.Y., Meng X.Z., Chen L., A new stage structured predator-prey Gomportz model with time de-lay and impulsive perturbations on the prey, Applied Mathematics and Computation,2008,196:705-719.
[27]Jiang X.W., Hao M.Y., Song X.Y., An Impulsive predator-prey system with Stage-Structured and Modified Leslie-Gower Holling II type Schemes,生物数学学报,2008,23(2)：202-208.
[28]Onofrio A.D., Pulse vaccination strategy in the SIR epidemic model:global asymptotic stable erad-ication in presence of vaccine failures, Mathematical and Computer Modelling,2002,36:473-489.
[29]Liu X., Ballinger G., Boundedness for impulsive delay differential equations and applications to population growth, Nonlinear Analysis,2003,53:1041-1062.
[30]Zhang S., Tan D., Chen L., Chaos in periodically forced Holling type Ⅱ predator-prey system with impulsive perturbations, Chaos,Solitons and Fractals,2006,28:367-376.
[31]张树文,陈兰荪,具有脉冲效应和综合害虫控制的捕食系统,系统科学与数学,2005,25(3)：264-275.
[32]Pang G.P., Chen L., Complexity of an Ivlev's predator-prey model with pulse, Advances in Complex Systems,2007,10(2):217-231.
[33]陈以平,一类具脉冲效应的三种群捕食系统的动力学性质,数学的实践与认识,2009,39(8)：135-141.
[34]Lakshmikantham V., Bainov D., Simeonov P., Theroy of Impulsive Differential Equations, World Scientific,Singapore,1989.
[35]Bainov D. D., Simeonov P. S., Impulsive Differential Equations:Periodic Solutions and Applications, New York:John Wiley Sons,1993.
[36]Kuang Y., Delay differential equation with application in population dynamics, New York:Aca-demic press,1993, p:67-70.
[37]王虎,二维Brouwer不动点定理的改进,数学的实践与认识,1986(2)：59.
[38]陈兰荪,宋新宇,非自治阶段结构捕食系统,平顶山师专学报,1999,5：1-4.
[39]刘秀湘,冯佑和,具性别偏食的二种群捕食者食饵系统模型,生物数学学报,2002,17(1)：5-11.
[40]张银萍,两种群竞争系统的持久性,生物数学学报,1998,13(5)：661-664.
[41]马知恩,周义仓,常微分方程定性与稳定性方法,北京,科学出版社,2001.
[2]Zhou Y., Liu H., Stability of perioic solution for an SIS model with pulse vaccination, Mathematical and Computer Modelling,2003,38:299-308.
[3]Liu X.N., Chen L., Complex dymamics of Holling type II Lotke-Volterra predator-prey system with impulsive perturbation on the predator, Chaos,Solitions and Fractals,2003,16:311-320.
[4]Zhang X.Y., Shuai Z.S.,Wang K., Optimal impulsive havesting policy for single population, Nonlin-ear Analysis:Rear World Applicition,2005,44:185-199.
[5]Hastings A., Age-dependent predation is not a simple process.I.Continuous time models, Theoretical Population Biology,1983,23(3):47-62.
[6]Hastings A., Delays in recruitment at different trophic levels:Effects on stability, Journal of Mathematical Biology,1984,21(1):35-44.
[7]Wang W., Chen L., A predator-prey system with stage-structure for predator, Applied Mathematical and Computation,1997,33:83-91.
[8]Fisher M.E., Goh B.S., Stiblity results for delay-recuitment models in population dynamics, Bulletin of Mathematical Biology,1986,48:485.
[9]Liu S., Chen L., Extinction and permanence in competitive stage-structrured system with time-delay, Nonliear Analysis,TMA,2002,51:1347-1361.
[10]Yan J., Stablity for impulsive delay differential equations, Nonlinear Analysis, TMA,2005,63:66-80.
[11]Liu X., Ballinger G., Boundeness for impulsive delay differential equations and applications to population growth models, Nonlinear Analysis,TMA,2003,53:1041-1062.
[12]Sun S.L., Chen L., Existence of positive periodic solution of an impulsive delay Logistic model, Applied Mathematical and Computation,2007,184:617-623.
[13]刘汉武,王荣欣,刘建欣,具性别结构的食饵捕食者模型,生物数学学报,2005,20(2)：179-182.
[14]何朝晖,崔景安,具有年龄结构捕食系统的永久持续生存,南京师大学报,2005,28(1)：24-29.
[15]祝惠娇,胡志兴,具性别偏食的Beddington-DeAngelis反应的三种群系统的研究,北京工商大学学报,2006,24(6)：64-67.
[16]杜学武,原三领,关于三种群第三类功能反应周期系数模型的研究,工程数学学报,2000,17(3)：139-142.
[17]徐长永,王美娟,周艳丽,具阶段结构和HollingⅡ类功能反应的捕食系统研究,上海理工大学学报,2006,28(5)：49-54.
[18]徐长永,王美娟,刘妍,一类食饵具有性别结构的非自治两种群捕食系统研究,上海理工大学学报,2008,30(5)：449-455.
[19]孙凯玲,王美娟,朱春娟,具有性别偏食和Holling III类功能反应的食饵捕食者模型,上海理工大学学报,2009,31(1)：6-10.
[20]Zhang S.W., Chen L., A study of predator-prey models with the Beddington-DeAnglis functional response and impulsive effect, Chaos,Solitons and Fractals,2006,27:237-248.
[21]liu B., Zhang Y.j., Chen L., Dynamic complexities of a Holling I Predator-prey model concerning periodic biological and chemical control. Chaos,Solitons and Fractals,2004,22:123-134.
[22]贾建文,乔梅红,一类具有脉冲效应和Holling III类功能反应的捕食系统分析,生物数学学报,2008,23(2)：279-288.
[23]张玉娟,陈兰荪,孙丽华,一类具有脉冲效应的捕食者-食饵系统的灭绝与持续生存,大连理工大学学报,2004,40(2)：769-77.
[24]Song X., Chen L., Optimal harvesting and stability for a two-species competitive system with stage structure, Mathematical Biosciences,2001,170:173-186.
[25]Song X.Y., Hao M.Y., Meng X.Z., A stage-structured predator-prey model with disturbing pulse and time delays, Applied Mathematical Modelling,2009,33:211-223.
[26]Liu K.Y., Meng X.Z., Chen L., A new stage structured predator-prey Gomportz model with time de-lay and impulsive perturbations on the prey, Applied Mathematics and Computation,2008,196:705-719.
[27]Jiang X.W., Hao M.Y., Song X.Y., An Impulsive predator-prey system with Stage-Structured and Modified Leslie-Gower Holling II type Schemes,生物数学学报,2008,23(2)：202-208.
[28]Onofrio A.D., Pulse vaccination strategy in the SIR epidemic model:global asymptotic stable erad-ication in presence of vaccine failures, Mathematical and Computer Modelling,2002,36:473-489.
[29]Liu X., Ballinger G., Boundedness for impulsive delay differential equations and applications to population growth, Nonlinear Analysis,2003,53:1041-1062.
[30]Zhang S., Tan D., Chen L., Chaos in periodically forced Holling type Ⅱ predator-prey system with impulsive perturbations, Chaos,Solitons and Fractals,2006,28:367-376.
[31]张树文,陈兰荪,具有脉冲效应和综合害虫控制的捕食系统,系统科学与数学,2005,25(3)：264-275.
[32]Pang G.P., Chen L., Complexity of an Ivlev's predator-prey model with pulse, Advances in Complex Systems,2007,10(2):217-231.
[33]陈以平,一类具脉冲效应的三种群捕食系统的动力学性质,数学的实践与认识,2009,39(8)：135-141.
[34]Lakshmikantham V., Bainov D., Simeonov P., Theroy of Impulsive Differential Equations, World Scientific,Singapore,1989.
[35]Bainov D. D., Simeonov P. S., Impulsive Differential Equations:Periodic Solutions and Applications, New York:John Wiley Sons,1993.
[36]Kuang Y., Delay differential equation with application in population dynamics, New York:Aca-demic press,1993, p:67-70.
[37]王虎,二维Brouwer不动点定理的改进,数学的实践与认识,1986(2)：59.
[38]陈兰荪,宋新宇,非自治阶段结构捕食系统,平顶山师专学报,1999,5：1-4.
[39]刘秀湘,冯佑和,具性别偏食的二种群捕食者食饵系统模型,生物数学学报,2002,17(1)：5-11.
[40]张银萍,两种群竞争系统的持久性,生物数学学报,1998,13(5)：661-664.
[41]马知恩,周义仓,常微分方程定性与稳定性方法,北京,科学出版社,2001.