种群、传染病及复杂网络微分方程模型动力学行为研究

发布时间：2018-03-01 07:22

本文关键词： 种群模型传染病模型持久性灭绝性复杂网络有限时间同步　出处：《新疆大学》2016年博士论文　论文类型：学位论文

【摘要】：种群模型、传染病模型和复杂网络模型是生物数学模型的几个重要组成部分,近年来受到国内外众多学者的广泛研究.本文研究基于微分方程描述的种群、传染病及复杂网络模型的动力学行为.第一部分研究了具有年龄结构的捕食食饵种群模型的动力学行为.与已有的研究模型相比,作者都要求捕食者为密度制约的条件下进行分析研究,显然这个假设过于理想化,在现实中很难适用于所有生物种群.我们研究了以下两类捕食者非密度制约的捕食食饵模型.(1)研究了一类具有Holling-IV型功能反应周期的捕食食饵模型,其中捕食者是非密度制约的.我们得到关于系统持久性、灭绝性、周期解的存在性积分形式的充分条件.将捕食者密度制约的重要结论推广到了捕食者非密度制约的情形.最后通过数值模拟验证了理论结果的有效性.(2)研究了食饵具有年龄结构捕食者非密度制约的周期时滞捕食食饵模型的全局性质.通过不等式分析技巧、比较原理、动力系统持久性理论,首先得到了所有正解最终有界的充分条件,其次得到了关于系统持久性、灭绝性积分形式的充分条件.充实了具有年龄结构捕食者非密度制约捕食食饵模型的理论研究内容.将捕食者密度制约的重要结论推广到了捕食者非密度制约的情形.最后通过数值模拟验证了结果的有效性,并通过数值模拟给出了一个开问题.第二部分研究了SEIRS传染病模型的动力学行为.传染病模型通常利用常微分方程模型和差分方程模型来刻画.利用常微分方程刻画的就是连续的传染病模型,利用差分方程刻画的就是离散的传染病模型.(1)研究了一类具有免疫的连续非自治SEIRS传染病模型.在非常弱的假设条件下,我们得到了疾病持久(一致持续)和灭绝积分形式的充分条件.并得到了一些积分形式的闽值条件R1,R1*,R2和R2*.这些条件涵盖了自治、周期情形的结果,丰富了非自治SEIRS传染病模型的理论研究结果.最后我们给出数值模拟验证了结论.(2)研究了具有一般非线性发生率的离散SEIRS传染病模型.首先我们得到了满足初始条件的解的正性和有界性,进而得到了如果基本再生数R0≤1那么无病平衡点是全局吸引的.如果R01那么疾病是持久的.当系统退化为SEIR模型的时候,那就意味着恢复者获得了永久的免疫,通过离散Lyapunov函数的方法,我们给出了当R0≤1模型有一个唯一的地方病平衡点是全局吸引的.最后,通过数值模拟提出了一个重要的开问题.第三部分讨论了含有未知参数的复杂网络网络的有限时间同步问题.通过设计滑模控制器研究具有未知参数的复杂网络的有限时间同步问题.基于有限时间稳定性理论,介绍了一种非奇异的滑模面,证明了系统在有限时间内趋向于平衡点.并证明了设计的滑模控制器可以保证系统在有限时间内到达滑模面.我们利用引进多参数方法,构造合适的Lyapunov函数,运用不等式技巧证明了提出的滑模控制策略可以保证系统有限时间到达切换面,以及滑动模运动的有限时间稳定性.
[Abstract]:Population model, infectious disease model and complex network model is an important part of mathematical biology model, in recent years has been widely studied by many scholars at home and abroad. This paper is based on the differential equations describing the dynamic behavior of the population, infectious diseases and complex network model. The first part of the age structure of the predator prey population model the dynamic behavior. Compared with the existing models, the authors are required for the predator density constraint condition analysis, obviously this assumption is too ideal, in reality it is difficult to apply to all populations. We study the following two predator - prey predator model density (1) study. A class of predator-prey cycle with Holling-IV type functional response prey model, which is the predator density independent. We get on the permanence of the system, the extinction cycle. Sufficient conditions for the existence of the integral form of the solution. The predator density dependent important conclusion to the predator density independent. The numerical simulation results validate the theoretical results. (2) studied the prey with age structure predator non global properties of periodic delay predator-prey model density restriction. The principle of comparative analysis techniques, through persistent inequality, dynamical system theory, firstly obtained all positive solutions and sufficient conditions for the ultimate boundedness of the second, is obtained for the permanence of the system, sufficient conditions for the extinction of the integral form. Enriched with age structure theory research content of density dependent predator predator-prey model of predator density. The important conclusion is extended to restrict the predator density independent. The numerical simulation results verify the effectiveness of the results, and through numerical simulation to An open problem. The second part studies the dynamic behavior of the SEIRS infectious disease model. Infectious disease models usually use ordinary differential equation model and differential equation model to describe the use of ordinary differential equations. Characterization is the continuous infectious disease model, the differential equation is portrayed in the discrete epidemic model (1). Study on a class of continuous immune non autonomous SEIRS epidemic model. On the assumption that the condition is very weak, we got the disease persistence and extinction (persistent) integral form and obtain some sufficient conditions. The integral form of the threshold conditions of R1, R1*, R2 and R2*., these conditions include autonomy. Periodic case results, enriches the theoretical results of SEIRS model of infectious disease of non autonomous. Finally we give the numerical simulation to verify the conclusion. (2) studied with general nonlinear incidence rate of discrete SEIRS infectious disease model. First we got to meet the initial conditions of positive solutions and the boundedness, can be obtained if the basic reproduction number R0 = 1 then the disease-free equilibrium is globally attractive. If R01 then the disease is long lasting. When the system degenerates into the SEIR model, which means to obtain a permanent restoration through the method of immunity, discrete Lyapunov function, we give the R0 1 model has a unique endemic equilibrium is globally attractive. Finally, through numerical simulation, put forward an important open problem. The third part discusses the finite time complex network network with unknown parameters of the limited synchronization problem. The time of complex networks by studying the design of sliding mode controller with unknown parameters of the synchronization problem. Based on finite time stability theory, introduces a non singular sliding surface in finite time is proved that the system tends to balance Point. And it is proved that the design of sliding mode controller can guarantee the system to reach the sliding surface in finite time. We use the introduction of multi parameter method, constructing a suitable Lyapunov function, proved that the proposed control strategy can guarantee the system to reach the switching surface in finite time by using inequality technique and finite time sliding mode stability.

【学位授予单位】：新疆大学
【学位级别】：博士
【学位授予年份】：2016
【分类号】：O175

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