【作者】 刘巍；

【导师】 李勇； 常小军；

【作者基本信息】 吉林大学 ， 应用数学， 2013， 硕士

【摘要】 生物数学根本的价值在于它来源于现实又应用于现实.微分方程是从实际间题中抽象出来的数学模型,它刻画的是事物的变化规律与其状态之间的关系.许多现实问题又可归结为微分方程的概周期解问题,概周期现象能更全面的反映事物的变化规律,比如机械振动、天体力学、电力系统、经济学、生态学等领域,研究它们的概周期现象要比周期现象更符合实际,更能解释现实生活.在生态系统不断的发展过程中,周期解、概周期解的存在性问题更吸引了很多学者的注意,成为数学生态学研究的重要课题.在种群动力学、生态学的研究中,提出了很多数学模型,种群动力学中最经典的模型之一就是Lotka-Volterra系统,由于它的理论与现实意义,Lotka-Volterra系统得到了快速的发展,Lotka-Volterra系统及其推广的模型的动力学性质是数十年来人们经久不息深入研究的课题,由此建立起来的理论方法是生物种群模型研究成果中最基本的内容.本文是一篇综述类文章,主要对Lotka-Volterra竞争系统及其推广模型就概周期解存在且全局吸引性的问题进行概述.在第一章中介绍了生物数学,概周期解及Lotka-Volterra竞争系统模型的发展历程.第二章中主要介绍了非自治n种群Lotka-Volterra竞争系统模型利用构造适当的Lyapunov函数方法得到了存在唯一概周期解是全局吸引的一些充分条件.定理2.2.1如果这个系统满足下列假设成立：(H2)存在一个正数α,使得则系统存在唯一的正概周期解G(t)={u1(t),...,un(t)},t∈R而且G(t)的模包含于这里第三章主要是介绍了Lotka-Volterra竞争系统模型的推广模型具有反馈控制的多种群竞争系统模型：利用Lyapunov函数法研究竞争系统概周期解,给出具有反馈控制的多种群竞争系统概周期解全局吸引性的充分条件.定理3.2.1对于此系统,若假设：(H1*)bi(t),ri(t),aij(t),cij(t),di(t),ei(t),fi(t)都是定义在t∈(-∞,+∞)上的非负概周期函数,成立,则系统在R上存在唯一正有界解若下列假设也成立,即存在常数ωi,si,εi,δi,使得则Y(t)是唯一正概周期解是全局吸引的.更多还原

【Abstract】 The fundamental value of biological mathematics embodied in that its originates from the reality and application in reality. The differential equation is a mathematical model from practical problems, it is a portrait of the relationship between the changes of things and the state. Many practical problems can be attributed to the almost periodic solutions of differential equations. Almost periodic phenomenon can more comprehensively reflect the changes of things, such as mechanical vibration, celestial mechanics, electric power system, economics, ecology etc. And the studying about the almost periodic phenomenon of is more practical than the periodic phenomenon, sometimes the former can explain the reality of life. In the process of development of ecological system, periodic solutions and almost periodic solutions attracted a lot of attention, and they have become an important topic in mathematical ecology.In the study of population dynamics and ecology, people had put forward many mathematical models, one of the most classical models of population dynamics is Lotka-Volterra system. Because of its theoretical and practical significance, the Lotka-Volterra system has been development rapidly. And the dynamic properties of Lotka-Volterra system and the extension of the model are the prolonged study for people dozens of years. This established theoretical method is the most basic research results of biological population model content.This paper is mainly on the Lotka-Volterra competition system and its extension model for the existence and global stability of almost periodic solutions of the problem are summarized. The first chapter mainly introduces the mathematical biology, history of almost periodic solutions and almost periodic solutions of Lotka-Volterra competitive system model.The second chapter mainly introduces the nonautonomous Lotka-Volterra competitive n-dimensional system model. the only sufficient condition for almost periodic solution is globally attractive. By constructing suitable Lyapunov function proved thatTheorem2.2.1If this system satisfies (H2)There is a positive constant α>0, so that then the system has a strictly pusitive almost periodic solution G(t)={u1(t),...,un(t)}, t∈R, whose module is contained in that of WhereThe third chapter mainly introduces the generalized model of Lotka-Volterra competition system model, it has a variety of group competition system feedback control.The almost periodic solutions are given using the Lyapunov function method to study the competition system. Given the multi-species competition system with feedback control of the sufficient conditions for global stability of the almost periodic solution.Theorem3.2.1For this system, if set up:(H1*)bi(t),ri(t), aij(t), cij(t), di(t), ei(t), fi(t) are almost periodic functions defined on(-∞,+∞),(H2*)aii1>0,ri1>0.m(bi(t))>0,m(ei(t))>0, (H3*)m(bi(t)-∑j=1,i≠j(?)aij(t)xj*(t)-di(t)ui*(t)). the system on R contain the only positive bounded solution If the following assumptions are also established,there are constants ωi,si,εi,δi,and so that then the unique existence almost periodic solution Y(t)of the system is global attracted.更多还原