本文選題：食物鏈隨機模型　切入點：基因調(diào)控網(wǎng)絡隨機模型　出處：《寧夏大學》2017年博士論文　論文類型：學位論文

【摘要】：本文主要對兩類隨機生物模型的動力學行為進行研究.考慮白噪聲、Markov切換、擴散等因素對系統(tǒng)漸近行為的影響,通過構(gòu)造Lyapunov-Krasvoskii函數(shù),運用Ito微積分理論,隨機過程理論及線性矩陣不等式(LMI)技術(shù)等研究這兩類隨機生物模型的漸近行為,包括:解的長時間行為特征,平穩(wěn)分布與耗散性控制等,并利用數(shù)值模擬顯示隨機生物系統(tǒng)的復雜動力學行為.全文由七章內(nèi)容組成,主要內(nèi)容如下:第一章簡要介紹三種群食物鏈模型和基因調(diào)控網(wǎng)絡模型的研究背景,重點回顧這兩類生物數(shù)學模型的發(fā)展歷史與現(xiàn)狀.最后給出本文主要工作和結(jié)構(gòu)安排.第二章簡單介紹Brown運動,Markov過程和耗散性的基礎知識,尤其介紹本文用到的一些引理.第三章研究三種群食物鏈隨機模型解的長期行為特征.通過Markov半群理論,證明模型解的分布密度依L1收斂到一個不變密度或者弱收斂到一個奇異測度.并結(jié)合數(shù)值模擬分析解的漸近行為.第四章研究在Markov狀態(tài)切換下三種群食物鏈隨機模型的漸近行為及平穩(wěn)分布.首先證明全局正解的存在性,獲得依時間平均持久和依概率滅絕的充分條件.在相對弱的白噪聲下,證明解存在唯一的平穩(wěn)分布及其遍歷性.最后通過數(shù)值模擬驗證理論結(jié)果的正確性.第五章研究在隱式Markov狀態(tài)切換下三種群食物鏈隨機模型的耗散控制問題.對于隱式Markov鏈,本章利用Wonham濾波對不可觀察隨機過程進行有效估計,在三種群食物鏈隨機模型中引入H∞控制和無源化控制,分別通過無源化控制和H∞控制對三種群食物鏈隨機模型的穩(wěn)定性進行分析.最后給出數(shù)值模擬.第六章研究在分數(shù)階Brown運動驅(qū)動下受時變時滯和反應擴散等因素影響的基因調(diào)控網(wǎng)絡模型的耗散性控制問題.通過構(gòu)造合適的Lyapunov-Krasovskii函數(shù),利用線性矩陣不等式技術(shù),得到滿足全局耗散性和γ耗散性的充分條件.最后給出兩個數(shù)值算例進行模擬.論文最后對研究結(jié)果進行總結(jié),并提出進一步的研究計劃。
[Abstract]:In this paper, the dynamic behavior of two kinds of stochastic biological models is studied. Considering the influence of white noise switching and diffusion on the asymptotic behavior of the system, the Lyapunov-Krasvoskii function is constructed and the Ito calculus theory is used. Stochastic process theory and linear matrix inequality (LMI) technique are used to study the asymptotic behavior of these two kinds of stochastic biological models, including: long term behavior of solution, stationary distribution and dissipative control, etc. Numerical simulation is used to show the complex dynamic behavior of stochastic biological systems. This paper is composed of seven chapters. The main contents are as follows: chapter 1 briefly introduces the research background of three species food chain model and gene regulation network model. The development history and present situation of these two kinds of mathematical models are reviewed. At last, the main work and structure of this paper are given. In chapter 2, the basic knowledge of Brown motion and dissipation is briefly introduced. Some Lemma used in this paper are introduced in particular. In Chapter 3, the long-term behavior characteristics of the solution of a three-species food chain stochastic model are studied. By means of Markov semigroup theory, It is proved that the distribution density of the solution of the model converges to an invariant density at L1 or weakly converges to a singular measure. The asymptotic behavior of the solution is analyzed by numerical simulation. Chapter 4th studies the stochastic behavior of the food chain of three species under Markov state switching. The asymptotic behavior and stationary distribution of the model. First, the existence of the global positive solution is proved. Sufficient conditions for time-averaged persistence and probabilistic extinction are obtained. It is proved that the solution has a unique stationary distribution and ergodicity. Finally, the correctness of the theoretical results is verified by numerical simulation. Chapter 5th deals with the dissipative control problem of three species food chain stochastic model under implicit Markov state switching. In this chapter, Wonham filter is used to estimate the unobservable stochastic process, and H 鈭，

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