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1、论文题目：论文题目：微分方程在生态系统中的应用微分方程在生态系统中的应用作者姓名：作者姓名： 李聚臣专业名称：专业名称： 应用数学入学时间：入学时间：2006 年 9 月研究方向：研究方向： 微 分 方 程 理 论 及 应 用指导教师：指导教师： 卓相来职职称：称： 教授论文提交日期：论文提交日期：2009 年 5 月论文答辩日期：论文答辩日期：2009 年 6 月 20 日授予学位日期：授予学位日期：The Application of Differential Equationin some Kinds of EcosystemA Dissertation submitted in ful
2、fillment of the requirements of the degree ofMASTER OF PHILOSOPHYfromShandong University of Science and TechnologybyLi JuchenSupervisor: Professor Zhuo XianglaiCollege of Natural Resources and Environmental EngineeringMay 2009声明本人呈交给山东科技大学的这篇硕士学位论文，除了所列参考文献和世所公认的文献外，全部是本人在导师指导下的研究成果。该论文资料尚没有呈交于其它任何学
3、术机关作鉴定。硕士生签名：日期：AFFIRMATIONI declare that this dissertation, submitted in fulfillment of the requirements for the award of Master of Philosophy in Shandong University of Science and Technology, is wholly my own work unless referenced of acknowledge. The document has not been submitted for qualificat
4、ion at any other academic institute.Signature:Date:山东科技大学硕士论文摘要摘要本论文共分四章。第一章是综述部分。介绍了数学生态学的发展状况和相关的一些工作。第二章研究了一类 2n 1次 Kolmogorov 系统。主要讨论了第一象限内系统的平衡点的性态；不存在极限环的条件；存在惟一稳定极限环的条件。第三章对一类具 Holling 类功能反应的捕食者食饵系统进行了讨论。讨论了系统平衡点的性态，给出了系统无闭轨的充分条件以及存在惟一稳定极限环的条件。然后，研究了 PIF（prey influx）效应对于具 Holling 类功能反应的捕食者食饵系
5、统的影响，主要讨论了 PIF 效应对于系统平衡点的稳定性及两种群平衡密度的影响。第四章主要利用 Mawhin 重合度拓展定理和微分不等式研究了两类具偏差变元的微分方程的周期解，得到了其存在周期解的充分条件，扩展了已有文献的一些结论。关键词：捕食者食饵系统，极限环，闭轨，平衡点，稳定性，偏差变元，周期解。山东科技大学硕士论文摘要ABSTRACTThere are four chapters in this thesis.The first chapter is overview, in which we introduce the developing status aboutMathemati
6、cal Ecology and some related works that we have done.In the second chapter, we study a class of 2n 1 degree Kolmogorovs system. Thesystem is discussed mainly about some sufficient conditions to ensure the system has nolimit cycle and the stability of equilibrium , the sufficient conditions of existe
7、nce of a stablelimit cycle of the system in the first quadrant.In the third chapter, we discuss a class of predator-prey system with Holling typefunctional response .The natures of equilibrium points are discussed. The sufficientconditions of nonexistence of the closed orbit and conditions of existe
8、nce of the unique andstable limit cycle are also obtained. Then, we study the effect of prey influx on the predator-prey system with Hollingtype functional response.The impact on the stability andequilibrium density is mainly discussed.In chapter 4, with the help of the continuation theorem of the c
9、oincidence degree anddifferential inequalities, we study two kinds of third order functional differential equationwith deviating argument. Sufficient conditions for the existence of periodic solutions areobtained. Our work generalizes the corresponding work of the past.Keywords: predator-prey system
10、, limit cycles, closed orbit, equilibrium point, stability,deviating argument, periodic solution.山东科技大学硕士学位论文目录目录1 综述.12一类 Kolmogorov 生态模型的定性分析.62.1 平衡点的性态. 6 2.2 闭轨线的不存在性. 10 2.3 极限环的存在性. 113 具 Holling 类功能性反应的两种群的研究. 133.1 一类具 Holling 类功能性反应的两种群的定性分析. 13 3.2 PIF 效应对于具 Holling 类功能反应的捕食者食饵系统的影响. 204
11、具偏差变元微分方程的周期解研究.254.1 一类三阶具偏差变元微分方程的周期解. 25 4.2 一类三阶具多偏差变元微分方程的周期解. 30致谢.39攻读硕士期间主要成果.40参考文献.41山东科技大学硕士学位论文目录Contents1 Overview 12 The Qualitative Analysis of a class of Kolmogorov System62.1 Properties of equilibrim62.2 Noexistence of closed orbit 102.3 Existence of limit cycle113 Research of a cla
12、ss of two Species Predator-prey model with HollingtypeFunctional Response133.1 The qualitative analysis of a class of two species predator-prey model with Hollingtypefunctional response 133.2 Effect of prey influx on a predator-prey system with Holling type functional response 204 Research for a kin
13、d of third order functional differential equation with deviatingargument 254.1 Periodic solutions for a kind of third order functional differential equation with a deviatingargument254.2 Periodic solutions for a kind of third order functional differential equation with some deviatingarguments 30Thanks 39Main Work Achievement of the Author during Working on Master Paper40Main Reference Documents41山东科技大学硕士学位论文综述1 综综 述述生物数学是生物学与数学之间的交叉学科。它以数学方法研究和解决生物学问题,并对与生物学有关的数学方法进行理论研究。数学模型是能够表现和描述真实世界某些现象、特征和状况的数
