本文主要研究了几类受不同种噪声干扰的随机恒化器模型的动力学行为, 全文共分为七章, 主要内容如下:
第一章: 绪论, 主要介绍本文的研究背景, 发展现状以及本文的主要工作, 同时还介绍文中所用到的预备知识.
第二章: 主要考虑了一类随机时滞恒化器模型的平稳分布的存在性问题. 通过构造合适的Lyapunov泛函, 利用随机Lyapunov分析方法, 研究了随机时滞恒化器模型的平稳分布的存在性以及解的遍历性, 这有助于我们更好地理解随机时滞生物模型的动力学行为和概率特征.
第三章: 主要研究了一类带有分布时滞和随机扰动的恒化器模型的解的随机性质. 利用线性链技术, 我们将带有弱核的随机时滞恒化器模型转化为包含三个方程的等价系统, 这个等价系统并不包含时滞. 首先, 我们证明了该模型对任何初值都有唯一的全局正解. 进一步, 我们证明了该系统的解是随机最终有界的. 然后我们建立了系统的解以指数速率趋于边界平衡点的充分条件. 同时, 我们还通过构造一些合适的随机Lyapunov函数, 得到了该系统解的遍历性的充分条件. 最后, 我们给出了一些数值模拟来验证理论结果, 并给出了相应的结论和分析.
第四章: 主要考虑了一类具有周期营养输入的随机恒化器模型的周期行为. 对于随机非自治周期恒化器系统而言, 我们首先证明了系统对任意初值都有全局唯一正解. 然后, 我们建立了非平凡正周期解存在的充分条件. 此外, 我们还分析了微生物指数灭绝的条件, 并证明了随机恒化器模型存在唯一的边界周期解, 并且该边界周期解是全局吸引的. 在本文的最后, 我们也给出了一些数值模拟来验证我们的主要结论.
第五章: 我们讨论了一类带有均值回归Ornstein-Uhlenbeck过程和Monod-Haldane型功能性反应函数的随机恒化器模型. 我们首先证明了系统对任意初值都有全局唯一正解. 然后, 我们建立了微生物指数灭绝和均值持久的充分条件. 最后, 我们也给出了数值模拟来验证我们的主要结论. 我们的结果表明, 均值回归过程是一种将环境噪声引入微生物连续培养模型的有效而合理的方法, 并且我们还发现了回归速度和噪声强度对微生物的灭绝和持久有非常重要的影响.
第六章: 我们考虑了一类带有乘性噪声的随机恒化器模型的Forward吸引子. 基于Ornstein-Uhlenbeck过程和随机动力系统理论, 通过适当的变量替换,我们将带有乘性噪声的随机恒化器模型转化为一个由Ornstein-Uhlenbeck过程驱动的恒化器模型, 这个系统将不再含有白噪声. 首先, 我们证明了随机恒化器系统对任意正初值都存在全局唯一正解, 我们还陈述了一些关于紧的Forward吸收集和Forward吸引集存在性的一些结果, 它们的内部结构将为我们提供一些关于微生物长时间行为的有用信息. 最后, 我们对恒化器模型中两种随机性的建模方法进行了比较和分析, 并给出了一些数值模拟来支持我们的理论结果.
第七章: 我们研究了由彩色噪声驱动的随机恒化器模型的Pullback吸引子. 基于Ornstein-Uhlenbeck过程和随机动力系统理论, 我们考虑了一个由彩色噪声驱动的随机恒化器模型. 首先, 我们证明了随机恒化器系统全局正解的存在唯一性, 并给出了随机Pullback吸引子存在性的一些结论. 最后给出了一些总结.

In this thesis, we mainly study the dynamical behavior of some stochastic chemostat models disturbed by different noises. This thesis is divided into seven chapters, and the main contents are as follows:
Chapter 1: We mainly introduce the research background, development status and main work of this paper, and also introduce the preliminaries used in this thesis.
Chapter 2: We mainly consider the existence of stationary distribution for a stochastic delay chemostat model. By constructing suitable Lyapunov functional and using the stochastic Lyapunov analysis method, we investigate the existence of stationary distribution and the ergodicity of a stochastic delayed chemostat model, which can help us better understand the dynamical behavior and statistical characteristics of stochastic delayed biological model.
Chapter 3: Stochastic properties of solution for a chemostat model with a distributed delay and random disturbance are studied. By the linear chain technique, we transform the stochastic chemostat model with weak kernel into an equivalent degenerate system which contains three equations. Firstly, we state that this model has a unique global positive solution for any initial value. Furthermore, we prove that the solution of the system is stochastic ultimately bounded. Then sufficient conditions for solution of the system tends to the boundary equilibrium point at exponential rate are established. Moreover, we also obtain some sufficient conditions for ergodicity of solution of this system by constructing some suitable stochastic Lyapunov functions. Finally, we provide some numerical examples to illustrate theoretical results, and some conclusions and analysis are given.
Chapter 4: We study the stochastic periodic behavior of a chemostat model with periodic nutrient input and random  perturbation. We first prove the existence of global unique positive solution with any initial value for stochastic non-autonomous periodic chemostat system. After that, the sufficient conditions are established for the existence of nontrivial positive $T-$periodic solution. Moreover, we also analyze the conditions for extinction exponentially of microorganism, and we find that there exists a unique boundary periodic solution for stochastic chemostat model, which is globally attractive. At the same time, in the end of this chapter, we also give some numerical simulations to illustrate our main conclusions.
Chapter 5: We mainly construct and analyze a stochastic chemostat model with mean-reverting Ornstein-Uhlenbeck process and Monod-Haldane response function. We first study the existence of global unique positive solution with any initial value for stochastic chemostat system. After that, the sufficient conditions are established for extinction exponentially and persistence in the mean of microorganism. Finally, we also give numerical simulations to illustrate our main conclusions. Our results show that the mean-reverting process is an effective and reasonable method to introduce environmental noise into the continuous culture model of microorganism, and we also find that the reversion speed and volatility intensity have an important influence on the extinction and persistence of microorganism.
Chapter 6: We consider a stochastic chemostat model with multiplicative noise. By appropriate variable substitution, we get a random chemostat system driven by Ornstein-Uhlenbeck process, which will no longer contain white noise. First, we prove the existence and uniqueness of the global positive solution for any positive initial value for random chemostat system, and we also state some results regarding the existence of a compact forward absorbing set as well as a forward attracting one, its internal structure will provide us some useful information about the long-time behavior of microorganism in random chemostat model. Finally, we make some comparison and analysis between both ways of modeling randomness and stochasticity in the chemostat model and show some numerical simulations to support our theoretical results.
Chapter 7: We study the pullback attractor of the stochastic chemostat model driven by color noise. Based on the theory of the Ornstein-Uhlenbeck process and random dynamical system, in this chapter, a random chemostat model driven by colored noise is considered. First, we prove the existence and uniqueness of the global positive solution for random chemostat system, and we also state some conclusions about the existence of random pullback attractor. Finally, some conclusions are given.