种群数量模型的脉冲效应研究
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摘要
种群动力学模型的研究因其在生态与生物领域的实际意义,近年来受到了广泛的关注.目前,对于种群动力学的研究,许多学者针对以前一直采用连续或者离散的模型研究的基础上,考虑到种群生态管理上的一些实际问题以及脉冲输入在这些实际问题上的意义,开始建立起具有脉冲效应的种群动力学模型,并取得了一系列的成果.
害虫控制管理(IPM)是进行蔬菜、水果等植物类的种植时,利用物种的生物特性,配合辅助物理以及化学药品,来更好地控制害虫的数量,尽量将对环境,甚至生态系统的影响降至最低,以一种更加健康的方式来达到维持物种种群数量平衡或是取得产品最大产量或是在一定的时间内取得最大经济效益的目的.对现在提倡的绿色产品和环境保护方面都起着积极的作用,适应目前可持续发展的战略,有非常重大的实际应用意义.
本文主要针对害虫控制管理(1PM)中捕食者定期输入的影响,运用脉冲微分方程的相关理论,研究了相应的带脉冲效应的种群动力学模型,包括平衡点及其稳定性,周期解的存在性,以及系统的灭绝性与持久性等的研究.主要内容包括以下几个部分：
论文第三章讨论了一类带HollingⅡ功能反应函数的一食饵、一捕食者的种群捕食模型的相关动力学行为.对现有捕食模型进行害虫控制管理,对捕食者数量进行脉冲式投放,对已有的种群模型进行了优化改进,利用常微分方程的相关理论讨论了系统的平衡点,给出并证明了平衡点局部渐近稳定的充分条件.利用Floquet理论、比较定理等相关理论,讨论了系统的灭绝性与持久性,给出并证明了系统的害虫灭绝周期解的全局渐近稳定与系统持久性生存的充分性条件.最后,利用仿真实例验证了条件的有效性.
论文第四章研究的是一类两个捕食者、一个食饵的HollingⅡ型捕食系统,讨论了对两种捕食者脉冲迁入来抑制害虫数量的脉冲系统的动力学性质.利用Floquet理论与小幅干扰等相关的理论,建立并证明了系统害虫根除周期解的全局渐近稳定的充分条件,然后用分析的方法讨论了系统的持久性条件.通过数值模型证明了条件的正确性.
论文第五章研究了一类带HollingⅡ.功能反应函数的两食饵,一捕食者的捕食系统模型,利用释放两种害虫的共同的天敌的方法,构造新的带有脉冲效应的种群系统,以达到控制种群数量的目的,利用脉冲微分方程的相关理论,建立并证明了系统的害虫根除周期解的局部渐近稳定与持久性的充分条件.
论文第六章,对目前工作做了总结,并对后续的工作做出了展望.其中,提出了对已讨论的有脉冲输入的种群系统施加切换效应的想法,并对其进行了数值模拟,验证了此方法的可行性.
The study on population dynamic model has been widely concerned because of the practical significance in the biological field. In recent years, bas-ed on using continuous or discrete equations to study population models, many scholars estabilished population dynamic model with impulsive effects in order to describe the practical problems more appropriate.
Integrated pest managenment (IPM) is a very suitable method to managing pests by combing biological, cultural, physical and chemical tools in a way that minimize economic, health and protecting environmental. It needs to study further in modeling and analyzing the population dynamic by impulsive system. Consequently. the population models with impulsive effects need to be optimiz-ed. In this dissertation, population dynamic models are studied to consider population controls by means of the theory and methods of impulsive differe-ntial equations.
Capter 1 surverys the history and development of population dynamic models. The main methods and results of impulsive systems are introduced as well. The brief contents about this paper are given as follow.
In Capter 2. the basic theory of impulsive systems systems are introduced. The definition and theory be used for later are given in detail.
Capter 3 studies the dynamic behaviors of the predator-prey of Holling typeⅡfunction response system with impulsive effects. On the theory of controlling pest, the releases of predators are impulsive. We consider changing some coefficients to make the existing model better. By using the theory of the differential eqution, it is shown that it has a positive equilibrium. The sufficient conditions that the equilibrium point is locally asymptotically stable are given. The extinction and permanence of the system is studied via the Floquet theory and comparison theory. It is shown that the pest-eradication periodic solution is globally asymptotically stable and system is permanent.
A predator-prey model of one-predator two-prey concering impulsive effects is analyzed in Capter 4. After optimizing the existing model, the dynamic behaviors of this impulsive system are studied. Using Floquet theory and small amplitude perturbation method, we give the sufficient conditions that the pest-eradication periodic solution is globally asymptotically stable. Then we prove it. Finally, we analyze the permanence of the system.
In Capter 5, a predator-prey model of two-predator one-prey concering impulsive effects is estabilished and discussed. Using biological control, we control the population of pest. We show that there exist the sufficient conditions of a locally asymptotically stable pest-eradication periodic solution with the impulsive effects. The sufficient condition for the permanence is also given.
Finally, a conclusion of the dissertation and some future research topics in the field are given in Capter 6. The idea of population dynamic with impulsive and switching effects is considered, and the numerical simulations are given. The simulation results show the feasibility of this idea.
害虫控制管理(IPM)是进行蔬菜、水果等植物类的种植时,利用物种的生物特性,配合辅助物理以及化学药品,来更好地控制害虫的数量,尽量将对环境,甚至生态系统的影响降至最低,以一种更加健康的方式来达到维持物种种群数量平衡或是取得产品最大产量或是在一定的时间内取得最大经济效益的目的.对现在提倡的绿色产品和环境保护方面都起着积极的作用,适应目前可持续发展的战略,有非常重大的实际应用意义.
本文主要针对害虫控制管理(1PM)中捕食者定期输入的影响,运用脉冲微分方程的相关理论,研究了相应的带脉冲效应的种群动力学模型,包括平衡点及其稳定性,周期解的存在性,以及系统的灭绝性与持久性等的研究.主要内容包括以下几个部分：
论文第三章讨论了一类带HollingⅡ功能反应函数的一食饵、一捕食者的种群捕食模型的相关动力学行为.对现有捕食模型进行害虫控制管理,对捕食者数量进行脉冲式投放,对已有的种群模型进行了优化改进,利用常微分方程的相关理论讨论了系统的平衡点,给出并证明了平衡点局部渐近稳定的充分条件.利用Floquet理论、比较定理等相关理论,讨论了系统的灭绝性与持久性,给出并证明了系统的害虫灭绝周期解的全局渐近稳定与系统持久性生存的充分性条件.最后,利用仿真实例验证了条件的有效性.
论文第四章研究的是一类两个捕食者、一个食饵的HollingⅡ型捕食系统,讨论了对两种捕食者脉冲迁入来抑制害虫数量的脉冲系统的动力学性质.利用Floquet理论与小幅干扰等相关的理论,建立并证明了系统害虫根除周期解的全局渐近稳定的充分条件,然后用分析的方法讨论了系统的持久性条件.通过数值模型证明了条件的正确性.
论文第五章研究了一类带HollingⅡ.功能反应函数的两食饵,一捕食者的捕食系统模型,利用释放两种害虫的共同的天敌的方法,构造新的带有脉冲效应的种群系统,以达到控制种群数量的目的,利用脉冲微分方程的相关理论,建立并证明了系统的害虫根除周期解的局部渐近稳定与持久性的充分条件.
论文第六章,对目前工作做了总结,并对后续的工作做出了展望.其中,提出了对已讨论的有脉冲输入的种群系统施加切换效应的想法,并对其进行了数值模拟,验证了此方法的可行性.
The study on population dynamic model has been widely concerned because of the practical significance in the biological field. In recent years, bas-ed on using continuous or discrete equations to study population models, many scholars estabilished population dynamic model with impulsive effects in order to describe the practical problems more appropriate.
Integrated pest managenment (IPM) is a very suitable method to managing pests by combing biological, cultural, physical and chemical tools in a way that minimize economic, health and protecting environmental. It needs to study further in modeling and analyzing the population dynamic by impulsive system. Consequently. the population models with impulsive effects need to be optimiz-ed. In this dissertation, population dynamic models are studied to consider population controls by means of the theory and methods of impulsive differe-ntial equations.
Capter 1 surverys the history and development of population dynamic models. The main methods and results of impulsive systems are introduced as well. The brief contents about this paper are given as follow.
In Capter 2. the basic theory of impulsive systems systems are introduced. The definition and theory be used for later are given in detail.
Capter 3 studies the dynamic behaviors of the predator-prey of Holling typeⅡfunction response system with impulsive effects. On the theory of controlling pest, the releases of predators are impulsive. We consider changing some coefficients to make the existing model better. By using the theory of the differential eqution, it is shown that it has a positive equilibrium. The sufficient conditions that the equilibrium point is locally asymptotically stable are given. The extinction and permanence of the system is studied via the Floquet theory and comparison theory. It is shown that the pest-eradication periodic solution is globally asymptotically stable and system is permanent.
A predator-prey model of one-predator two-prey concering impulsive effects is analyzed in Capter 4. After optimizing the existing model, the dynamic behaviors of this impulsive system are studied. Using Floquet theory and small amplitude perturbation method, we give the sufficient conditions that the pest-eradication periodic solution is globally asymptotically stable. Then we prove it. Finally, we analyze the permanence of the system.
In Capter 5, a predator-prey model of two-predator one-prey concering impulsive effects is estabilished and discussed. Using biological control, we control the population of pest. We show that there exist the sufficient conditions of a locally asymptotically stable pest-eradication periodic solution with the impulsive effects. The sufficient condition for the permanence is also given.
Finally, a conclusion of the dissertation and some future research topics in the field are given in Capter 6. The idea of population dynamic with impulsive and switching effects is considered, and the numerical simulations are given. The simulation results show the feasibility of this idea.
引文
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[2]V. Lakshmikantham, D.D. Bainov, P.S. Simconov. Theory of impulsive differential equations [M]. Singapore,World Scientific,1989.
[3]徐克学.生物数学[M].北京：科技出版社,1999,1-11.
[4]D.D. Bainov, P.S. Simeonov. Impulsive differential equations:periodic solutions and applications [M]. New York, Longman Scientific and Teschnical,1993.
[5]M.S. Alwan, X.Z. Liu. On stability of linear and weakly nonlinear switched systems with time delay [J]. Math. Comput. Modelling.48,1150-1157, 2008.
[6]荆海英,赵立纯.微生物连续培养模型的溢流量控制[J].生物数学学报,1998,13(5)：599-603.
[7]赵立纯,荆海英.微生物连续培养模型的最优溢流量控制[J].生物数学学报,1999,14(1)：32—35.
[8]W. Perk, P. Saruds. Input—output analysis and for total input rate and total traced mass of body cholesterol in man [J]. Circ Res,1969,25:191-199.
[9]JG. Wapber, JG. Sipynar, JJ. Feryr. Michaelis-mebten elimination kinetics: Areas under curves, steady-state concentrations, and clearances for compantment models with different types of input [J]. Bioharm Drug Disp, 1985,6:177-186.
[10]S. Ruan, D. Xiao. Global analysis in a predator—prey system with nonmonotonic functional response [J]. SIAM J Appl. Math.,2001,61: 1445-472.
[11]程荣福,蔡淑云.一类具功能反应的食饵-捕食者两种群模型的定性分析[J].生物数学学报,2002,17(4)：406-410.
[12]颜向平,张存华.一类具功能反应的食饵-捕食者两种群模型的定性分析[J].生物数学学报,2004,19(3)：323-327.
[13]张平光,赵申琪.一类三次系统极限环的唯一性[J].高校应用数学学报A辑,2003,18(1)：27-32.
[14]Y. Kuang. Rich dynamics of Gause-type ratio-dependent predator-prey system [J]. Fields Institute Communications,1999,21,325-337.
[15]W.D. Wang, L.S. Chen, Z.Y. Lu. Global stability of a population dispersal in a two-Patch environment [J]. Dynamic Systems and Applications,1997, 6,207-216.
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