【摘要】：生物數(shù)學(xué)作為一門交叉學(xué)科,近些年已經(jīng)有了飛速的發(fā)展.生物動力學(xué)是生物數(shù)學(xué)的一個分支,數(shù)學(xué)模型在描述生物動力學(xué)行為中起到很大的作用.時滯生物動力系統(tǒng)是一個具有豐富實際背景與廣泛應(yīng)用的領(lǐng)域.時滯動力系統(tǒng)的穩(wěn)定性和分支問題的研究對實際應(yīng)用領(lǐng)域的發(fā)展起著關(guān)鍵作用,其中,穩(wěn)定性體現(xiàn)了結(jié)構(gòu)平衡性,在無窮維空間上對系統(tǒng)進行穩(wěn)定性研究,尤其是對全局穩(wěn)定性的研究會更全面和深入地展示系統(tǒng)的動力學(xué)性質(zhì).所謂分支,是指當(dāng)參數(shù)發(fā)生變化并經(jīng)過某些臨界值時,系統(tǒng)的某些特性發(fā)生突變的現(xiàn)象.Hopf分支是一種常見而重要的分支,它主要研究當(dāng)參數(shù)變化時,平衡點的穩(wěn)定性發(fā)生變化,從而在平衡點附近產(chǎn)生小振幅周期解的現(xiàn)象.本文主要應(yīng)用Lyapunov穩(wěn)定性理論,LaSalle不變性原理,拓?fù)涠壤碚?中心流形定理,規(guī)范型方法以及全局分支定理等理論和方法對幾類生物動力系統(tǒng)的局部和全局穩(wěn)定性、周期解的存在性以及不動點分支、局部和全局Hopf分支,系統(tǒng)的持久性進行研究.具體內(nèi)容如下：首先,討論了一類SIRS模型,選擇時滯為參數(shù),得到了無病平衡點的全局漸近穩(wěn)定性,地方病平衡點的局部漸近穩(wěn)定性和Hopf分支的存在性,以及系統(tǒng)的持久性.之后研究了一類非自治SIR模型,利用重合度理論,得到了系統(tǒng)正周期解全局存在性,唯一性以及全局穩(wěn)定性的充分條件.考慮到疾病潛伏期的影響,進而研究了一類SEIRS模型,得到了無病平衡點的全局漸近穩(wěn)定性,地方病平衡點的局部漸近穩(wěn)定性和全局Hopf分支的存在性,進而也得到了系統(tǒng)持久的充分條件.其次,研究了一類浮游生態(tài)系統(tǒng)的復(fù)雜動力學(xué).首先給出了常微分方程系統(tǒng)平衡點的穩(wěn)定性分析,之后在常微分方程系統(tǒng)中引入時滯,以時滯為參數(shù),得到了邊界平衡點全局漸近穩(wěn)定和不穩(wěn)定的充分條件,進一步,在一定條件下,系統(tǒng)在正平衡點處出現(xiàn)穩(wěn)定性開關(guān)現(xiàn)象,可能出現(xiàn)周期解,隨著時滯的增加周期解繼續(xù)存在,證實了全局Hopft分支的存在性.同時也發(fā)現(xiàn)隨著毒素釋放率的增加,正平衡點不穩(wěn)定的區(qū)間在縮小,說明毒素有助于系統(tǒng)的穩(wěn)定.最后,在時滯系統(tǒng)的基礎(chǔ)上引入擴散,考察擴散和時滯的共同影響,得到擴散不能改變平衡點的穩(wěn)定性,即圖靈不穩(wěn)定性不會發(fā)生.分別考察了大的擴散和小的擴散對Hopf分支的影響,在一定條件下能夠產(chǎn)生空間非齊次周期解,進而給出了算法來決定分支周期解的性質(zhì).最后,研究了兩類系統(tǒng)的混沌控制策略.首先研究了具有兩個時滯的浮游生態(tài)系統(tǒng),以兩個時滯為參數(shù)得到系統(tǒng)可能出現(xiàn)穩(wěn)定型開關(guān)現(xiàn)象.當(dāng)參數(shù)變化時,發(fā)現(xiàn)混沌現(xiàn)象可能發(fā)生,進一步,通過數(shù)值模擬發(fā)現(xiàn)時滯和毒素釋放率的增加可以使得系統(tǒng)的混沌現(xiàn)象消失.浮游動物的最大轉(zhuǎn)化率的增加會使得系統(tǒng)從穩(wěn)定狀態(tài)經(jīng)過倍周期分叉,最終導(dǎo)致混沌.其次,研究了一類具有混沌的浮游生態(tài)系統(tǒng)的時滯反饋控制,以時滯為參數(shù),得到了控制系統(tǒng)發(fā)生Hopf分支的條件,即系統(tǒng)在一定條件下,當(dāng)時滯取某些參數(shù)時可以將系統(tǒng)不穩(wěn)定的周期解變成穩(wěn)定周期解或穩(wěn)定平衡點,說明了控制的有效性.
文內(nèi)圖片：
圖片說明：圖３＊２：巧（Ｔ）曲線圖（如圖（ａ）），，邋ｒ邋＝邋０．１邋＜邋ｒ0時系統(tǒng)波圖（如圖Ｗ）．邋Ｔ邋＝邋２．５邋＞戶時逡逑
[Abstract]:As a cross-discipline, biomathematics has developed rapidly in recent years. Biodynamics is a branch of biological mathematics, and the mathematical model plays a very important role in describing the behavior of biological dynamics. The time-delay biodynamic system is a field with rich practical background and wide application. The stability and branch problems of the time-delay power system play a key role in the development of the practical application field, in which the stability shows the structural balance, and the stability of the system is studied in the infinite dimensional space. In particular, the study of global stability is more comprehensive and in-depth show the dynamic nature of the system. The so-called branch refers to the phenomenon that some characteristics of the system change when the parameter changes and passes some critical values. The Hopf bifurcation is a common and important branch, and it mainly studies the phenomenon that the stability of the equilibrium point changes when the parameters change, so as to generate a small-amplitude periodic solution near the equilibrium point. In this paper, we mainly apply the Lyapunov stability theory, the LaSalle invariance principle, the topological degree theory, the central manifold theorem, the standard method and the global branch theorem. The persistence of the local and global Hopf branches and systems is studied. The specific content is as follows: First, we discuss a kind of SIRS model, choose time-delay as the parameter, get the global asymptotic stability of the disease-free equilibrium point, the local asymptotic stability of the local disease equilibrium point and the existence of the Hopf branch, and the persistence of the system. After that, a class of non-self-governing SIR model is studied, and the sufficient conditions for global existence, uniqueness and global stability of the system's positive periodic solution are obtained by using the coincidence degree theory. Taking into account the influence of the latent period of the disease, a kind of SEIRS model is also studied to obtain the global asymptotic stability of the disease-free equilibrium point, the local asymptotic stability of the local disease equilibrium point and the existence of the global Hopf branch, and the sufficient condition of the system is also obtained. Secondly, the complex dynamics of a kind of floating ecosystem were studied. At first, the stability analysis of the equilibrium point of the system of ordinary differential equation is given, and the time-delay is introduced in the system of ordinary differential equation, and the sufficient conditions for global asymptotic stability and instability of the boundary equilibrium point are obtained. The stability switching phenomenon occurs at the positive equilibrium point, and the periodic solution may occur, and the existence of the global Hopft branch is confirmed with the increase of the time delay. It is also found that with the increase of the release rate of the toxin, the interval between the positive equilibrium point and the positive equilibrium point is reduced, indicating that the toxin can contribute to the stability of the system. In the end, diffusion is introduced on the basis of time-delay system, the common influence of diffusion and time-delay is investigated, and the stability of the equilibrium point can not be changed by diffusion, that is, the Turing instability does not occur. The influence of the large diffusion and the small diffusion on the Hopf bifurcation is investigated, and the non-homogeneous periodic solution of the space can be generated under certain conditions, and then the algorithm is given to determine the nature of the solution of the branch period. At last, the hybrid control strategy of two kinds of systems is studied. In this paper, a floating ecosystem with two time-delay is first studied, and a stable switching phenomenon may occur in the system with two time-delay parameters. When the parameters change, it is found that the mixing phenomenon may occur, and further, it is found that the increase of time-delay and the release rate of the toxin can cause the mixing phenomenon of the system to disappear. The increase in the maximum conversion of zooplankton will cause the system to diverge from a steady state through a period of time, resulting in a mixing. Secondly, the time-delay feedback control of a kind of floating ecosystem with hybrid system is studied, with the time-delay as the parameter, the condition of the Hopf bifurcation of the control system is obtained, that is, the system under certain conditions, The unstable periodic solution of the system can be transformed into a stable periodic solution or a stable equilibrium point when certain parameters are delayed at that time, and the effectiveness of the control is explained.
【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】：2016
【分類號】：O175

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