Stability In Terms Of Two Measures And Global Stability Of A Kind Of Microorganism Fermentation System

1,3-Propanediol is a very important chemical material. With its special symmetrical struc-ture, it is used to synthesize many polymers with good characteristics. Since 1980s, producing 1,3-propanediol by microbial fermentation has attracted a great deal of scholastic attention all of the world. This dissertation is on the background of producing 1,3-propanediol in Klebsiella pneumoniae using glycerol as substrate. In this dissertation, the fed-batch culture and continu-ous fermentation are studied. This dissertation formulates the main properties and stabilities of the fermentation's dynamical systems. For the model of fed-batch culture with impulses at fixed times, the "critical solution" is defined and its stability property is proved. For continuous fer-mentation, the five-dimensional and eight-dimensional hybrid dynamical system are introduced to study the properties and the stabilities in terms of two measures. The existence and bound-edness of the positive solution and the stability property of the eight-dimensional continuous fermentation system perfect the stability of the fermentation dynamics. The dissertation includes five chapters, which is summarized as follows:1. Chapter 1, as the introduction, generalizes the current development and the theoretical sig-nificance of nonlinear dynamical system, present the current concerns and development of stability. And then this chapter summarize briefly the application and the research of the current trend of microbial fermentation to 1,3-propanediol and its corresponding fermentation dynamics.2. Chapter 2, as preliminaries, introduces the basic definition and properties of the related differential system, stability and non-smooth analysis.3. The chapter 3 focuses on the dynamical system of microbial bioconversion in fed-batch culture:the system with impulses at fixed times. The non-existence of the equilibrium is proved and the "critical solution" is defined. The stability of "critical solution" is obtained. 4. Considering that the continuous fermentation is based on the batch fermentation, the chapter 4 focuses on the whole process of the continuous fermentation:batch and contin-uous fermentations. One kind of five-dimensional hybrid system is introduced to describe the extracellular substances. Its stability properties and practical stability in terms of two measures are researched. Lastly, the stability of the five-dimensional hybrid dynamical system of the continuous fermentation is obtained.5. Considering 1,3-propanediol is produced within a cell and the intracellular substances play significant roles in the synthesis of goal product 1,3-propanediol, the chapter 5 con-siders an eight-dimensional system with consideration of active transport of glycerol and passive transport of 1,3-PD based on the discussion in chapter 4. Last, the dissertation discusses the existence and uniqueness of the solution, linear-growth property, the exis-tence and boundedness of the positive solution by introducing upper Hamilton function, the existence of the equilibrium, the local and global stability.