本文關(guān)鍵詞： 時滯微分方程 Lyapunov-LaSalle定理 穩(wěn)定性 持久性　出處：《北京科技大學(xué)》2017年博士論文　論文類型：學(xué)位論文

【摘要】：時滯生物動力系統(tǒng)是生物數(shù)學(xué)中重要的研究領(lǐng)域,具有理論意義及廣泛的實際背景和應(yīng)用價值.時滯生物動力系統(tǒng)的穩(wěn)定性與持久性是研究其動力學(xué)性質(zhì)的重要內(nèi)容.本文主要應(yīng)用Lyapunov泛函法,并結(jié)合Lyapunov-LaSalle定理或其變體以及持久性理論方法,來研究所建的HIV病毒感染和微生物絮凝相關(guān)的時滯模型的全局動力學(xué).基于上述模型的研究方法,給出了 Lyapunov-LaSalle定理的一個推廣及其一些應(yīng)用.具體研究內(nèi)容如下:在第3章中，我們建立了一類具有(由感染細胞誘導(dǎo)引起的)細胞凋亡效應(yīng)和一般非線性發(fā)生率的時滯微分方程HIV病毒動力學(xué)模型,并考慮了該模型系統(tǒng)的全局性質(zhì).通過利用Lyapunov第二方法或構(gòu)造適當(dāng)?shù)腖yapunov泛函,證明了當(dāng)基本再生數(shù)R01時,該系統(tǒng)的未感染平衡點E0是全局漸近穩(wěn)定的;當(dāng)R0 = 1,E0是全局吸引的.證明了當(dāng)R01時,該系統(tǒng)的感染平衡點E*是局部漸近穩(wěn)定的且該系統(tǒng)是持久的,并提出了尋求這類系統(tǒng)正解的具體最終下界的方法.在第4章中,我們考慮了一類具有時滯的微生物絮凝模型的全局動力學(xué).該模型系統(tǒng)在一定條件下存在前向分支或后向分支.這類系統(tǒng)的動力學(xué)不易用上述病毒系統(tǒng)的動力學(xué)方法來分析,故此,我們采用了一種新的思想.也就是,通過考慮過一個給定的初始數(shù)據(jù)(?)且在某時刻T =(?)后的軌道上的Lyapunov泛函L來確定(?)的ω-極限集ω(?).故而,利用此思想,研究了該系統(tǒng)在適當(dāng)條件下平衡點的全局穩(wěn)定性.進一步地,研究了該系統(tǒng)的持久性并給出了微生物濃度的一個具體的最終下界公式.在第5章中,我們建立了一個具有飽和功能反應(yīng)和時滯的微生物絮凝模型.我們首先分析了模型系統(tǒng)在參數(shù)條件下的局部動力學(xué),然后證明了當(dāng)閾值參數(shù)R01時,微生物的收集是可持續(xù)的,并提出了在大相空間中尋求這類微生物濃度的具體最終下界的方法.在一定的條件下,若ωR01,則該系統(tǒng)存在一個后向分支,這意味著無微生物平衡點與微生物平衡點共存.在這些情況下,引入了Lyapunov-LaSalle定理的變體思想,換句話說,就是通過考慮初始數(shù)據(jù)(?)的ω-極限集ω((?))上的Lyapunov泛函L(這蘊含著某時刻T=(?)后軌道上的Lyapunov泛函L)來確定ω(?).由此，建立了無微生物平衡點與微生物平衡點在相應(yīng)條件下全局穩(wěn)定性的一些充分條件.在第6章中,基于前文系統(tǒng)全局動力學(xué)的研究方法,我們給出了 Lyapunov-LaSalle定理及其現(xiàn)有改進版的一個推廣,并建立了一般自治時滯微分系統(tǒng)全局穩(wěn)定性的一些判別法.
[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.
【學(xué)位授予單位】：北京科技大學(xué)
【學(xué)位級別】：博士
【學(xué)位授予年份】：2017
【分類號】：O175

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