HIV病毒感染与微生物絮凝相关问题的全局动力学

发布时间：2018-01-19 11:04

本文关键词： 时滞微分方程 Lyapunov-LaSalle定理稳定性持久性　出处：《北京科技大学》2017年博士论文　论文类型：学位论文

【摘要】：时滞生物动力系统是生物数学中重要的研究领域,具有理论意义及广泛的实际背景和应用价值.时滞生物动力系统的稳定性与持久性是研究其动力学性质的重要内容.本文主要应用Lyapunov泛函法,并结合Lyapunov-LaSalle定理或其变体以及持久性理论方法,来研究所建的HIV病毒感染和微生物絮凝相关的时滞模型的全局动力学.基于上述模型的研究方法,给出了 Lyapunov-LaSalle定理的一个推广及其一些应用.具体研究内容如下:在第3章中，我们建立了一类具有(由感染细胞诱导引起的)细胞凋亡效应和一般非线性发生率的时滞微分方程HIV病毒动力学模型,并考虑了该模型系统的全局性质.通过利用Lyapunov第二方法或构造适当的Lyapunov泛函,证明了当基本再生数R01时,该系统的未感染平衡点E0是全局渐近稳定的;当R0 = 1,E0是全局吸引的.证明了当R01时,该系统的感染平衡点E*是局部渐近稳定的且该系统是持久的,并提出了寻求这类系统正解的具体最终下界的方法.在第4章中,我们考虑了一类具有时滞的微生物絮凝模型的全局动力学.该模型系统在一定条件下存在前向分支或后向分支.这类系统的动力学不易用上述病毒系统的动力学方法来分析,故此,我们采用了一种新的思想.也就是,通过考虑过一个给定的初始数据(?)且在某时刻T =(?)后的轨道上的Lyapunov泛函L来确定(?)的ω-极限集ω(?).故而,利用此思想,研究了该系统在适当条件下平衡点的全局稳定性.进一步地,研究了该系统的持久性并给出了微生物浓度的一个具体的最终下界公式.在第5章中,我们建立了一个具有饱和功能反应和时滞的微生物絮凝模型.我们首先分析了模型系统在参数条件下的局部动力学,然后证明了当阈值参数R01时,微生物的收集是可持续的,并提出了在大相空间中寻求这类微生物浓度的具体最终下界的方法.在一定的条件下,若ωR01,则该系统存在一个后向分支,这意味着无微生物平衡点与微生物平衡点共存.在这些情况下,引入了Lyapunov-LaSalle定理的变体思想,换句话说,就是通过考虑初始数据(?)的ω-极限集ω((?))上的Lyapunov泛函L(这蕴含着某时刻T=(?)后轨道上的Lyapunov泛函L)来确定ω(?).由此，建立了无微生物平衡点与微生物平衡点在相应条件下全局稳定性的一些充分条件.在第6章中,基于前文系统全局动力学的研究方法,我们给出了 Lyapunov-LaSalle定理及其现有改进版的一个推广,并建立了一般自治时滞微分系统全局稳定性的一些判别法.
[Abstract]:The time-delay biodynamic system is an important research field in biological mathematics. The stability and persistence of time-delay biodynamic systems are important to study its dynamic properties. Lyapunov functional method is mainly used in this paper. Combined with the Lyapunov-LaSalle theorem or its variants, as well as the persistence of the theoretical method. To study the global dynamics of time-delay models related to HIV virus infection and microbial flocculation. A generalization of Lyapunov-LaSalle theorem and some applications are given. The main contents are as follows: in Chapter 3. A class of time-delay differential equation (HIV) viral dynamics models with apoptosis and general nonlinear incidence were established. The global properties of the model system are considered. By using the second method of Lyapunov or the construction of appropriate Lyapunov functional, it is proved that the basic reproduction number R01 is obtained. The uninfected equilibrium E0 of the system is globally asymptotically stable. It is proved that the infection equilibrium point E * of the system is locally asymptotically stable and that the system is persistent when R0 = 1N E0 is globally attractive. The method of finding the concrete final lower bound of the positive solution of this kind of system is also presented in chapter 4. In this paper, we consider the global dynamics of a class of microbial flocculation models with time delay. Under certain conditions, the model system has forward-branching or backward branching. The dynamics of this kind of system is not easy to use the dynamics of the virus system mentioned above. Methods to analyze. So we adopt a new idea, that is, by considering a given initial data? ) and at some point? ) to determine the Lyapunov functional L in the orbit after? 蠅 -limit set 蠅? Therefore, by using this idea, the global stability of the equilibrium point of the system under suitable conditions is studied. The persistence of the system is studied and a specific final lower bound formula for microbial concentration is given in chapter 5. We establish a microbial flocculation model with saturation response and time delay. We first analyze the local dynamics of the model system under the condition of parameters, and then prove that the threshold parameter R01 is used. The collection of microorganism is sustainable, and a method to find the specific lower bound of microorganism concentration in large phase space is proposed. Under certain conditions, if 蠅 R01, the system has a backward branch. This means that the non-microbial equilibrium point coexists with the microbial equilibrium point. In these cases, the idea of a variant of the Lyapunov-LaSalle theorem is introduced, in other words. By considering the initial data? Of 蠅 -limit set? The Lyapunov functional L (which implies a certain time? ) the Lyapunov functional L) on the posterior orbit to determine 蠅? Therefore, some sufficient conditions for the global stability of the non-microbial equilibrium point and the microbial equilibrium point under the corresponding conditions are established. In Chapter 6, based on the previous research methods of global dynamics of the system, some sufficient conditions for the global stability of the non-microbial equilibrium point and the microbial equilibrium point under the corresponding conditions are established. In this paper, we give a generalization of Lyapunov-LaSalle theorem and its improved version, and establish some criteria for the global stability of general autonomous delay differential systems.
【学位授予单位】：北京科技大学
【学位级别】：博士
【学位授予年份】：2017
【分类号】：O175

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