【作者】 杨静；

【导师】 林支桂；

【作者基本信息】 扬州大学 ， 基础数学， 2014， 硕士

【摘要】 数学作为一门基础学科已经广泛渗透到自然科学的各个领域。如天文上很多小行星的发现,包括轨道的计算都有赖于数学；物理学更是如此,量子论和相对论的提出都深深打下了数学的印记。生物数学是目前应用数学研究的热点方向之一,它是生命科学与数学交叉形成的一门边缘学科,它是应用数学理论与计算机技术研究生命科学中数量性质、空间结构形式,分析复杂的生物系统的内在特性,揭示在大量生物实验数据中所隐含的生物信息。而传染病动力学是生物数学的一个重要的组成部分。它根据种群生长的特性,病毒传播的规律,建立适当的数学模型。通过对模型的定量定性分析以及数值模拟来研究传染病的发病机制、传染途径和流行规律,这为人们预防和控制疾病提供了理论依据。自1927年Kermack和McKendrick建立数学模型研究传染病,传染病模型便深受国内外专家、学者的关注。在标准传染病学模型中,常常只考虑单个种群现象,然而真实情况并非如此。在自然环境中,种群在传播疾病时不可能单独存在,它们还会为生存空间、食物等原因与其他种群竞争,同时还会遭受被其他种群捕食的危险。因此在研究传染病学模型的动态行为时,考虑种群的相互作用更具生物意义。同时,在已研究的工作中,大部分工作都是研究自治系统。即假设系统中的参数是与时间和空间无关的常数。然而非自治现象在生活中无处不在的,甚至更加符合现实生活。如许多传染病都具有季节性特征或地域性特征。所以这里我们有必要研究非自治生态传染病模型。基于对以上两方面的考虑,本文研究一类具齐次Neumann边界条件非自治两种群捕食反应扩散问题,其中病毒在被捕食种群中传播。第一章,我们简要地介绍了相关背景知识及问题的来源,并简单阐述本文研究的主要内容。第二章,我们给出一些定义、符号及一维非自治微分方程的相关引理的证明,这些将在后面的章节中应用。第三章,首先我们通过比较原理和常微分方程中相关的结论得到捕食及被捕食种群持续的充分条件,同时也给出了病毒的蔓延和消退的充分条件,最后通过构造Lyapunov函数,讨论问题的全局一致吸引性。第四章,我们将用Matlab软件对所得到的结果进行数值模拟,并画出图像来验证已经得到的理论结果。第五章,我们对整篇文章作了一个总结,并提出一些今后可以继续考虑的问题。更多还原

【Abstract】 Mathematics as a basic discipline has been widely infiltrated all areas of the natural sciences such as the discovering of many asteroids astronomical and the calculation of the track, they all relies heavily on mathematics; especially physics, quantum theory and relativity which are deeply proposed to lay a mark of mathematics.Mathematical Biology which has been one of the most well recognized subjects in modern applied mathematics is a frontier discipline overlapping life sciences and mathematics. It uses mathematical theory and computer technology to study quantities and types of spatial structure in nature in the life sciences, and to analyze inherent characteristics of complex biological systems, to find bio-informatics concealed in the data from a large number of biological experiments. Epidemic dynamics is an important component of biological mathematics. Based on the characteristics of population growth and the process of disease spreading, it uses to establish the appropriate mathematical models. Then through quantitative and qualitative analysis and numerical simulation, it uses to study the pathogenesis of infectious diseases, infectious pathways and characteristic of prevalence, which provides a theoretical basis for the people to prevent and control the disease.It is well known that Kermack and McKendrick constructed a mathematical model to study epidemiology in1927. From then on, more and more attention has been focused on epidemic models. But in the early model of infectious diseases, they just consider a single population. As we know, the actual situation is not so. In the natural environment, the population can not survive alone, they not only compete from other populations for the living space, food, but also suffer from the risk of being prey to other populations. So it is more biological significance to consider the interaction of populations when we study the dynamical behavior of infectious diseases model.Most previous work has considered the autonomous system, that is, the parameters were assumed to be constants, independent of time and position. However, non-autonomous phenomenon is so prevalent in the real life that many epidemiological problems can be modeled by non-autonomous systems of nonlinear differential equations. For example, many diseases show seasonal or regional behavior. So it is necessary to consider the non-autonomous eco-epidemiology model.Based on the above factors, in this paper we deal with the behavior of positive solutions to a nonautonomous reaction-diffusion system with homogeneous Neumann boundary conditions the model describes a two-species predator-prey system in which there is an infectious disease in prey.The first chapter, we briefly introduce the background and present the reaction-diffusion problem, and the main contents of this paper are also given.The second chapter is devoted to some notations, and some useful lemmas for the non-autonomous differential equation which will be useful in the sequel.The third chapter, we establish the sufficient conditions on the permanence of prey and the predator by using the comparison, and give the sufficient conditions on the spreading and vanishing of the disease. Also, by constructing a Lyapunov functional, we obtain the global attractivity of the corresponding autonomous model. Numerical simulations are also included in the final part to verify our analytical results.The fourth chapter deals with numerical simulations by using Matlab. The numerical results help one to better understand the theoretical conclusions obtained in former chapters.The fifth chapter, we first give a short summary, and then present some work to be considered.更多还原