兰州大学机构库 >数学与统计学院
一类三物种捕食模型平衡解的全局分歧及稳定性
Alternative TitleGlobal bifurcation and stability of steady-state solutions of the three-species prey-predator model
王芳
Thesis Advisor钟承奎
2008-05-26
Degree Grantor兰州大学
Place of Conferral兰州
Degree Name硕士
Keyword生态系统 椭圆型方程组 主特征值 局部分歧 全局分歧
Abstract在这篇硕士学位论文中, 我们主要考虑种群生态学中的共存问题. 种群生态学是生态学中的一个重要的分支,也是迄今数学在生态学中应用得最为广泛和深入、发展得最为系统和成熟的分支. 它包括对给定种群本身的动力学特性和结构的研究,以及给定种群与相关种群相互作用下演变规律的研究.在近几十年里,许多生物学家和数学家都致力于研究种群的持续生存问题, 由于这类问题具有一定的现实意义, 因此备受关注. 大量的文献研究了系统的稳态解, 该稳态解问题就是对应的椭圆型方程组的边值问题, 而椭圆型方程组正解的存在性其实就是种群生态学中的共存问题. 本文主要利用分歧定理得到了生态系统非平凡正稳态解的充分性条件并研究了这个共存态的稳定性.第一章是综述, 简要介绍了相关工作的背景、历史以及目前进展,然后介绍了本文的主要工作. 第二章先介绍了模型的相关背景以及建模思想, 然后介绍了相关的基础知识, 包括算子的特征值、分歧理论、椭圆方程组的解存在性和唯一性. 第三章的第一部分中分别以c 和a 作分歧参数,讨论关于半平凡解处的分歧解 , 我们证明了如果一定的条件成立,则三物种能共存,并且当以c 为参数时, 捕食者的出生率是有范围的,即新生捕食者不能太多也不能太少; 第二部分中以a 为分歧参数, 先得到了当u =0 时, 半平凡解的唯一性, 利用局部分歧得到了半平凡解邻域内正解存在的充分条件, 并且利用全局分歧定理知道了正解的全局分支. 第三部分中讨论了半平凡解(0,0,w); 得到了在不同参数情况下非平凡正解的存在性. 最后,利用Candall-Rabinowitz 关于分歧解稳定性的理论,得到了不同共存态的稳定性或不稳定性的充分条件.
Other AbstractIn this master’s dissertation, we mainly consider the co-existence states of population ecology. Population ecology is an important branch of ecology. Mathematics is wildly applied and developed systematically in this branch of ecology. This subject involved in the study of population dynamical property and structure, and also involved in the study of the interaction of given population and related population ecology. The mathematical model and methods based on this subject is not only improving the development of ecology, but also make effect on other field in mathematical biology. In recent years, many scholars have devoted themselves to the research of co-existence states, It’s result is a guide of real life. Therefore, coexistence states of ecological models have attracted considerable attention. A number of studies have been made on the existence of steady-state solutions of the system of ecology. In fact, the problem on the existence of steady-state solutions of the system of ecology is the problem on the existence of positive solutions of elliptical system. This paper deals with the research about one type of population dynamic system—having functional reaction on a predator and two preys by means of locally and global bifurcation theory. We have proved the existence of non¬trivial positive solutions and also given the local stability results for the coexistence states. This presentation is divided in four chapter: In Chapter 1, the background and history of current research situation about the related work and major work of this presentation are introduced. In Chapter 2, we introduce the background of mathematical eco1ogy mode and some preliminaries. In Section 3.1 of Chapter 3 we regard c and a as a bifurcation parameter, respectively, and discuss the bifurcation solutions of (A2) which are relative to the unique semi-trivial solution of the form(u ,v, 0). We prove that, if the condition is satisfied, then two prey, a predator can co-exist. when c is bifurcation parameter, then the birth-rate of predator has the limitations. In Section 3.2, we assume conditions under which the predator–prey subsystem(u=0) has a unique positive solution , and using a as a bifurcation parameter, we obtain a continuum of positive solution of (A2). In Section 3.3, we regard a and b as a bifurcation parameter, respectively, and discuss the bifurcation solutions of (A2) which are relative to the unique semi-trivial solution of the form (...
URL查看原文
Language中文
Document Type学位论文
Identifierhttps://ir.lzu.edu.cn/handle/262010/224459
Collection数学与统计学院
Recommended Citation
GB/T 7714
王芳. 一类三物种捕食模型平衡解的全局分歧及稳定性[D]. 兰州. 兰州大学,2008.
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