两类随机生物模型的渐近行为及耗散控制

发布时间：2018-03-12 11:38

  本文选题：食物链随机模型　切入点：基因调控网络随机模型　出处：《宁夏大学》2017年博士论文　论文类型：学位论文

【摘要】：本文主要对两类随机生物模型的动力学行为进行研究.考虑白噪声、Markov切换、扩散等因素对系统渐近行为的影响,通过构造Lyapunov-Krasvoskii函数,运用Ito微积分理论,随机过程理论及线性矩阵不等式(LMI)技术等研究这两类随机生物模型的渐近行为,包括:解的长时间行为特征,平稳分布与耗散性控制等,并利用数值模拟显示随机生物系统的复杂动力学行为.全文由七章内容组成,主要内容如下:第一章简要介绍三种群食物链模型和基因调控网络模型的研究背景,重点回顾这两类生物数学模型的发展历史与现状.最后给出本文主要工作和结构安排.第二章简单介绍Brown运动,Markov过程和耗散性的基础知识,尤其介绍本文用到的一些引理.第三章研究三种群食物链随机模型解的长期行为特征.通过Markov半群理论,证明模型解的分布密度依L1收敛到一个不变密度或者弱收敛到一个奇异测度.并结合数值模拟分析解的渐近行为.第四章研究在Markov状态切换下三种群食物链随机模型的渐近行为及平稳分布.首先证明全局正解的存在性,获得依时间平均持久和依概率灭绝的充分条件.在相对弱的白噪声下,证明解存在唯一的平稳分布及其遍历性.最后通过数值模拟验证理论结果的正确性.第五章研究在隐式Markov状态切换下三种群食物链随机模型的耗散控制问题.对于隐式Markov链,本章利用Wonham滤波对不可观察随机过程进行有效估计,在三种群食物链随机模型中引入H∞控制和无源化控制,分别通过无源化控制和H∞控制对三种群食物链随机模型的稳定性进行分析.最后给出数值模拟.第六章研究在分数阶Brown运动驱动下受时变时滞和反应扩散等因素影响的基因调控网络模型的耗散性控制问题.通过构造合适的Lyapunov-Krasovskii函数,利用线性矩阵不等式技术,得到满足全局耗散性和γ耗散性的充分条件.最后给出两个数值算例进行模拟.论文最后对研究结果进行总结,并提出进一步的研究计划。
[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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