【作者】 张玉娟；

【导师】 陈兰荪； 孙丽华；

【作者基本信息】 大连理工大学 ， 计算数学， 2004， 博士

【摘要】 经过近三十年的研究,脉冲微分方程的理论已经得到深入的发展,但是这些理论在实际中很难应用,全局稳定性等几乎没有什么结果.因此,讨论脉冲微分系统在各领域、各学科的具体应用仍具有较高的理论价值和实际意义.对于种群动力学模型的研究,学者们一直采用连续的或离散的模型来进行研究,而忽略了外界的干扰,可是现实世界中有许多生物现象以及人们对某些生命现象的优化、控制是脉冲的.本文针对种群生态管理上的一些实际问题以及脉冲在这些实际问题上的意义建立具有脉冲效应的种群动力学模型,并以脉冲微分方程的理论为基础,同时结合离散的、连续的动力系统和算子理论的相关理论和方法,并借助于计算机模拟讨论所提出模型的各种动力学行为,包括周期解的存在性与全局稳定性,一致持久性与灭绝性、系统的动力复杂性.我们的研究具有很强的生物背景,所得结论能为生产实际提供可靠的决策依据.本文的主要结果可以概括如下:第二章基于害虫控制问题建立了固定时刻的具有脉冲效应的种群动力学模型并研究了这些模型的动力学性质.基于天敌助增,我们研究了具有脉冲效应的两个食饵一个捕食者系统,利用脉冲微分方程的Floquet乘子理论、比较定理和分析的方法,证明了两个害虫都被根除的周期解的全局渐近稳定性,给出了系统持续生存的条件,和主要害虫灭绝,次要害虫(或者不是害虫)和天敌持续生存的条件,此外还把连续释放天敌与脉冲释放天敌进行比较.基于综合害虫管理,讨论了具有脉冲效应和相互干扰的捕食者-食饵系统,给出了灭绝与持续生存的条件,利用分支理论证明了正周期解的存在性,此外还讨论了综合策略的有效性,并通过数值模拟讨论了当正周期解的稳定性失去时,系统会出现哪些复杂现象.基于综合害虫管理,还讨论了具有脉冲效应和相互干扰的Holling Ⅰ捕食者-食饵系统,给出了灭绝与持续生存的条件,并把受迫系统与非受迫系统作以比较,讨论脉冲扰动如何影响非受迫系统的动力学性质.第三章考虑了周期环境中具有脉冲效应和相互干扰的捕食者-食饵系统.利用Floquet乘子理论讨论了边界周期解的稳定性,给出了系统持续生存的条件,利用分支理论,算子理论及Lyapunov-Schmidt小参数法讨论正周期解的存在性.第四章建立了种子季节性传播的数学模型并讨论了种子季节性传播对种群生长产生的影响.利用脉冲微分方程所对应的差分方程及Lyapunov直接方法,证明系统存在一个全局渐近稳定的周期解.第五章研究了可更新生物资源生物经济管理的脉冲捕获模型.分别讨论了满足Gompertz增长规律的单种群和阶段结构种群的脉冲捕获问题.对于单种群脉冲捕获间题,分别讨论了捕获量为常值和捕获量与种群的存量水平成比例两种情况.当捕获里为常值时,得到了最大承受生产.如果捕获量小于最大承受生产,系统存在两个正周期解,讨论了这两个正周期解的稳定性与吸引性.捕获量等于最大承受生产时,系统存在一个正周期解并且是半稳定的.当捕获量与种群的存量水平成比例时,我们证明了系统存在唯一一个正周期解,并且是全局渐近稳定的.因此,比例捕获优于常值捕获.对于比例捕获我们给出了以单位时间最大持续捕获量为管理目标的最优脉冲捕获策略,得到了单位时间最大持续捕获量,最优的捕获努力量和相应的最优种群水平.对于阶段结构种群的脉冲控制问题,也讨论了常值捕获和比例捕获两种情况.当捕获量为常值时,讨论了周期解的存在性与稳定性,得到了最大承受生产.对于比例捕获我们证明了正周期解的存在性与全局渐近稳定性,讨论了以年持续经济收益为管理目标,捕获努力量为控制变量的最优脉冲控制策略,给出了最优的捕获努力量,最优的幼年和成年种群水平,最大的年持续经济收益.此外我们还讨论了以捕获时刻为控制变量的最优脉冲控制策略.关镇词:脉冲微分方程,种群动力学模型,持续生存,灭绝,全局稳定性,复杂性.更多还原

【Abstract】 Fairly rich results have been made for the theories of impulsive differential equations for almost thirty years. However, these theories are hard to be applied and there are almost no developments in the study of global stability. Therefore, it is of value in theory and of significant in practice to investigate the application of impulsive differential equations in each field. Scholars have been using continuous or discrete equations to study the population dynamic models, while ignoring the external disturbance. However, in the natural world, many biological phenomena and the optimization and control of some biological phenomena are impulsive. In this dissertation, population dynamic models with impulsive effect are established concerning some actual problems of population ecology management and the meaning of impulse in these problems. Mathematically, a combination of approaches to discrete dynamics, continuous dynamics, impulsive dynamics, operator theory and numerical simulations are used to investigate dynamical behaviors including the existence and global stability of periodic solutions, permanence and extinction and all kinds of complexities. The mathematical models are presented based on biological meanings and the results we obtained can be used to provide reliable decisive basis for the production. The main results obtained in this dissertation may be summarized as the following:In Chapter 2, population dynamic models with impulsive effect at fixed moment concerning pest management are established and analyzed. Two-prey one-predator system with impulsive effect is investigated concerning augmentation of natural enemies. By using Floquet theory, two-pest eradication periodic solution is shown to be globally asymptotically stable. Further the conditions for the permanence of the system are obtained by comparison theorem and analytic method, and meanwhile the conditions for the extinction of one of the two prey and permanence of the remaining two species are also given. In addition, continuous release natural enemies is compared with pulse release natural enemies. A predator-prey model with mutual interference and impulsive effect concerning integrated pest management is studied. The conditions for the permanence or extinction are given. Moreover, the existence of positive periodic solution which arises from the pest-eradication periodic solution is proved by bifurcation theory, and the validity of integrated pest management is discussed. When the unique positive periodic solution loses its stability, numerical simulation shows there is a sequence of bifurcation, leading to a chaotic dynam-ics. A Holling I predator-prey model with mutual interference and impulsive effect concerning integrated pest management is also studied. The conditions for the permanence or extinction are given. Finally, the forced system is compared with the corresponding continuous system and to be found the forced system has different dynamical behaviors with different range of initial values which are inside or outside the unstable limit cycle of the unforced continuous system.In Chapter 3, a periodic predator-prey system with mutual interference and impulsive effect is considered. The linear stability of semi-trivial periodic solution is shown by applying Floquet theory. Some conditions for the permanence of the system are obtained. Further the bifurcation theory, operator theory and Lyapunov-Schmidt series expansion are used to show the existence of the positive periodic solution.In Chapter 4, a seasonal seed transfer mathematical model is presented and the effect of seasonal seed transfer on the growth of the species is discussed. Using the corresponding difference equations of impulsive differential equations and the Lyapunov method, a globally asymptotically stable positive periodic solution is shown.In Chapter 5, impulsive harvest models for bioeconomic management of renewable biological resources are investigated. A single species which obeys Gompertz growth law and a stage-structured population are discussed respectively. For t更多还原