微生物发酵中非线性时滞系统的分支及S系统辨识
|
摘要
1,3-丙二醇是一种重要的化工原料,近年来,微生物发酵法生产1,3-丙二醇受到国内外学者的广泛关注.本文以生物化工领域中的一个实际课题—甘油微生物歧化方法生产1：3-丙二醇的间歇和连续发酵—为背景,研究了发酵过程中两类非线性时滞动力系统的动力行为,以及两类非线性动力系统—S系统的性质和参数辨识问题.首先根据发酵过程的振荡现象与生物意义,分别引入了离散时滞和连续时滞,建立了非线性时滞动力系统,运用Hopf分支理论讨论了系统的振荡行为.其次根据发酵过程的特性和动态行为，分别建立符合各自特性的非线性动力系统—S系统,论述了S系统的主要性质并对其中的参数进行了辨识.本课题受到国家自然科学基金、“973计划”及“863计划”项目的资助.该项研究,一方面可以丰富非线性时滞动力系统、S系统理论与算法的研究,另一方面可以为1,3-丙二醇的大规模产业化生产提供参考,因此具有一定的理论意义与应用价值.本论文研究的内容与取得的主要成果可概括如下：
1.已有的工作主要研究单底物单产物的三维理想模型,而在微生物连续发酵实验中产物不止一个.本文考虑产物乙酸和乙醇对微生物生长的抑制作用,以及微生物对底物的消耗和分泌产物的能力主要由前一时刻的底物和产物浓度决定,因此在细胞比生长速率中引入离散时滞,提出了一个五维离散时滞动力系统.以时滞为参数,讨论了时滞对系统的正平衡点渐近稳定性的影响及Hopf分支的存在性.运用泛函微分方程的Hopf分支理论,在稀释速率恒定的条件下,数值模拟了分支值随进料浓度的变化曲线.运用规范形理论和中心流形定理,给出了判断Hopf分支方向和分支周期解稳定性的算法,并数值模拟了分支周期解与相图.
2.考虑产物乙酸和乙醇对微生物生长的抑制作用,和细胞透过细胞膜摄取底物和分泌产物的传递过程并不是实时的,因此将细胞比生长速率看成过去一段时间比生长速率的加权,通过引入强核连续时滞,提出了一个五维连续时滞动力系统.以平均时滞的倒数为参数,讨论了平均时滞对系统正平衡点的局部稳定性的影响.运用Hopf分支的代数判据,给出了存在Hopf分支的操作参数区域.运用Hopf分支定理,给出了判断Hopf分支方向和分支周期解稳定性的计算方法,并数值模拟了分支周期解及相图.最后讨论了模型的过渡行为.
3.根据间歇发酵过程的特性及动态行为,首次提出了一种特殊的S系统,证明了系统解的存在惟一性以及解对参数的连续依赖性.此外,为了确定系统中的参数值,以实验观测数据和计算值的归一化最小二乘误差为性能指标,以上述间歇发酵S系统为约束条件,建立了参数辨识模型,证明了辨识问题解的存在性,并构造求解参数辨识问题的优化算法.数值结果表明实验观测值和计算值之间的平均相对误差由已有文献的26.13%下降到13.95%,降低了12.18个百分点,说明S系统能更好地描述间歇发酵过程.另外,建立了连续发酵过程的S系统及其参数辨识模型.经数值计算,收到良好效果.最后建立了以稀释速率D和进料浓度CSO为控制变量,以终端时刻1,3-丙二醇的最大体积产率为性能指标,以描述连续发酵过程的S系统为约束的终端稳态优化模型,并求得实际问题的稳态最优解.优化结果显示终端时刻产物1,3-丙二醇的最大体积产率由114.3 mmolL-1h-1提高到124.911 mmolL-1h-1,该结论可为将来1,3-丙二醇的大规模生产实践提供参考.
1,3-propanediol (1,3-PD) is an important chemical raw material. In recent years, the production of 1,3-PD by microbial fermentation has been widely investigated. This dissertation investigates the dynamical behavior of nonlinear delay differential systems, properties and parameter identification problem of S systems with the bio-dissimilation of glycerol to 1,3-PD by Klebsiella pneumoniae in the background. First, on the base of the oscillations in the process of the anaerobic continuous fermentation and biological significance, we introduce discrete time delay and distributed time delay into the specific cellular growth rate, respectively, and build two kinds of five-dimensional delay differential systems. The oscillatory dynamic behavior are discussed using Hopf bifurcation theory. Second, two different kinds of S systems are presented to describe the batch and the con-tinuous fermentation. Properties of S systems and their parameter identification problem are discussed as well. The research is supported by national natural science foundation, " 973 program" and "863 program". In addition, the research not only can enrich the theory and the application of nonlinear delay dynamical system and S system, but also can provide reference for the commercial production of 1,3-PD. Hence, this research is very interesting in both theory and practice. The main contributions are summarized as follows:
1. There are many researches based on three-dimensional dynamic model ground on the system of single substrate and single product. However, some other products also take effect on the fermentation process. Taking acetate and ethanol inhibition into account, and the process of substrate taking up and products secreting across the cell membrane, we introduce discrete time delay into the specific cellular growth rate, and present a five-dimensional discrete delay differential system to describe os-cillatory behavior in microbial continuous culture. Moreover, taking time delay as parameter, we discuss the effect of time delay on the stability of the cquilibrium(s) and the existence of Hopf bifurcation. For a given dilution rate, we simulate the changing regularity of bifurcation value varied with substrate concentration in feed medium, using Hopf bifurcation theory and numerical method of functional differen-tial equation. The algorithm for determining the direction of Hopf bifurcation and the stability of periodic solutions is derived, using the theory of normal form and center manifold. The pictures of periodic solutions and phase planes with specific parameters are performed to illustrate the analytical results found.
2. We introduce the strong nuclear continuous time delay into the specific cellular growth rate, and present a five-dimensional distributed delay differential system. Taking the inverse of the average delay as parameter, we discuss the effect of time delay on the stability of the equilibrium (s). The operating parameter region is given using the algebra criteria of Hopf bifurcation. The algorithm for determining the direction of Hopf bifurcation and the stability of periodic solutions is derived, using the theory of Hopf bifurcation, and the pictures of periodic solutions and phase planes with specific parameters are performed. Finally we discuss the transition behavior qualitatively.
3. According to the characteristic and dynamic behavior of microbial growth in batch culture, S system is presented to describe the batch fermentation. The existence and uniqueness of solution to system together with the dependence of solution to parameters are discussed. Moreover, in order to identify values of parameters of S system such that the model can simulate the fermentation as exactly as possible, we develop a parameter identification model taking the normalized least-square error between the experimental data and calculated value as the performance index, and the above S system as the constraint. The existence of optimal solution to the parameter identification problem is proved, and an optimization algorithm to solve this parameter identification problem is constructed. Numerical result shows that the average relative error between calculated value and experimental data is only 13.95%, but that of the other system is 26.13%, which demonstrates that S system is better in describing batch fermentation. In addition, we present S system and its parameter identification model in continuous fermentation. Finally, we present the terminal steady-state optimization model in which control variables are the dilution rate D and the substrate concentration Cso, the constraint condition is S system, the performance index is the maximum volume yields of 1,3-PD at terminal moment. Optimization result shows that the maximum volume yields of 1,3-PD at terminal moment increased from 114.3 mmolL-1h-1 to 124.911 mmolL-1h-1. The result provides reference for the commercial production of 1,3-PD.
1.已有的工作主要研究单底物单产物的三维理想模型,而在微生物连续发酵实验中产物不止一个.本文考虑产物乙酸和乙醇对微生物生长的抑制作用,以及微生物对底物的消耗和分泌产物的能力主要由前一时刻的底物和产物浓度决定,因此在细胞比生长速率中引入离散时滞,提出了一个五维离散时滞动力系统.以时滞为参数,讨论了时滞对系统的正平衡点渐近稳定性的影响及Hopf分支的存在性.运用泛函微分方程的Hopf分支理论,在稀释速率恒定的条件下,数值模拟了分支值随进料浓度的变化曲线.运用规范形理论和中心流形定理,给出了判断Hopf分支方向和分支周期解稳定性的算法,并数值模拟了分支周期解与相图.
2.考虑产物乙酸和乙醇对微生物生长的抑制作用,和细胞透过细胞膜摄取底物和分泌产物的传递过程并不是实时的,因此将细胞比生长速率看成过去一段时间比生长速率的加权,通过引入强核连续时滞,提出了一个五维连续时滞动力系统.以平均时滞的倒数为参数,讨论了平均时滞对系统正平衡点的局部稳定性的影响.运用Hopf分支的代数判据,给出了存在Hopf分支的操作参数区域.运用Hopf分支定理,给出了判断Hopf分支方向和分支周期解稳定性的计算方法,并数值模拟了分支周期解及相图.最后讨论了模型的过渡行为.
3.根据间歇发酵过程的特性及动态行为,首次提出了一种特殊的S系统,证明了系统解的存在惟一性以及解对参数的连续依赖性.此外,为了确定系统中的参数值,以实验观测数据和计算值的归一化最小二乘误差为性能指标,以上述间歇发酵S系统为约束条件,建立了参数辨识模型,证明了辨识问题解的存在性,并构造求解参数辨识问题的优化算法.数值结果表明实验观测值和计算值之间的平均相对误差由已有文献的26.13%下降到13.95%,降低了12.18个百分点,说明S系统能更好地描述间歇发酵过程.另外,建立了连续发酵过程的S系统及其参数辨识模型.经数值计算,收到良好效果.最后建立了以稀释速率D和进料浓度CSO为控制变量,以终端时刻1,3-丙二醇的最大体积产率为性能指标,以描述连续发酵过程的S系统为约束的终端稳态优化模型,并求得实际问题的稳态最优解.优化结果显示终端时刻产物1,3-丙二醇的最大体积产率由114.3 mmolL-1h-1提高到124.911 mmolL-1h-1,该结论可为将来1,3-丙二醇的大规模生产实践提供参考.
1,3-propanediol (1,3-PD) is an important chemical raw material. In recent years, the production of 1,3-PD by microbial fermentation has been widely investigated. This dissertation investigates the dynamical behavior of nonlinear delay differential systems, properties and parameter identification problem of S systems with the bio-dissimilation of glycerol to 1,3-PD by Klebsiella pneumoniae in the background. First, on the base of the oscillations in the process of the anaerobic continuous fermentation and biological significance, we introduce discrete time delay and distributed time delay into the specific cellular growth rate, respectively, and build two kinds of five-dimensional delay differential systems. The oscillatory dynamic behavior are discussed using Hopf bifurcation theory. Second, two different kinds of S systems are presented to describe the batch and the con-tinuous fermentation. Properties of S systems and their parameter identification problem are discussed as well. The research is supported by national natural science foundation, " 973 program" and "863 program". In addition, the research not only can enrich the theory and the application of nonlinear delay dynamical system and S system, but also can provide reference for the commercial production of 1,3-PD. Hence, this research is very interesting in both theory and practice. The main contributions are summarized as follows:
1. There are many researches based on three-dimensional dynamic model ground on the system of single substrate and single product. However, some other products also take effect on the fermentation process. Taking acetate and ethanol inhibition into account, and the process of substrate taking up and products secreting across the cell membrane, we introduce discrete time delay into the specific cellular growth rate, and present a five-dimensional discrete delay differential system to describe os-cillatory behavior in microbial continuous culture. Moreover, taking time delay as parameter, we discuss the effect of time delay on the stability of the cquilibrium(s) and the existence of Hopf bifurcation. For a given dilution rate, we simulate the changing regularity of bifurcation value varied with substrate concentration in feed medium, using Hopf bifurcation theory and numerical method of functional differen-tial equation. The algorithm for determining the direction of Hopf bifurcation and the stability of periodic solutions is derived, using the theory of normal form and center manifold. The pictures of periodic solutions and phase planes with specific parameters are performed to illustrate the analytical results found.
2. We introduce the strong nuclear continuous time delay into the specific cellular growth rate, and present a five-dimensional distributed delay differential system. Taking the inverse of the average delay as parameter, we discuss the effect of time delay on the stability of the equilibrium (s). The operating parameter region is given using the algebra criteria of Hopf bifurcation. The algorithm for determining the direction of Hopf bifurcation and the stability of periodic solutions is derived, using the theory of Hopf bifurcation, and the pictures of periodic solutions and phase planes with specific parameters are performed. Finally we discuss the transition behavior qualitatively.
3. According to the characteristic and dynamic behavior of microbial growth in batch culture, S system is presented to describe the batch fermentation. The existence and uniqueness of solution to system together with the dependence of solution to parameters are discussed. Moreover, in order to identify values of parameters of S system such that the model can simulate the fermentation as exactly as possible, we develop a parameter identification model taking the normalized least-square error between the experimental data and calculated value as the performance index, and the above S system as the constraint. The existence of optimal solution to the parameter identification problem is proved, and an optimization algorithm to solve this parameter identification problem is constructed. Numerical result shows that the average relative error between calculated value and experimental data is only 13.95%, but that of the other system is 26.13%, which demonstrates that S system is better in describing batch fermentation. In addition, we present S system and its parameter identification model in continuous fermentation. Finally, we present the terminal steady-state optimization model in which control variables are the dilution rate D and the substrate concentration Cso, the constraint condition is S system, the performance index is the maximum volume yields of 1,3-PD at terminal moment. Optimization result shows that the maximum volume yields of 1,3-PD at terminal moment increased from 114.3 mmolL-1h-1 to 124.911 mmolL-1h-1. The result provides reference for the commercial production of 1,3-PD.
引文
[1]陈兰荪.数学生态学模型与研究方法.北京：科学出版社,1998.
[2]S.B. Hsu. S.P. Hubbcll, P. Waltman. A mathematical theory of Single-nutrient Competi-tion in Continuous Culture of Microorganisms. SIAM J.Appl.Math.,1977,32(2):366-383.
[3]J.M. Cushing. Integro-diffcrential equationsand delay models in population dynamics. in: Lecture Notes in Biomathematics, Vol.20, Springer, Berlin,1977.
[4]Y. Kuang. Basic properties of mathematical population models. J.Biomath.,2002,17: 129-142.
[5]R.K. Finn, R.E. Wilson. Population dynamics of a continuous propagator for microorgan-isms. J.Agric.food.chem.,1953,2:66-69.
[6]J. Caperson. Time lag in population grwth response of isochrysis galbana to a variable nitrate environment. Ecology,1969,50:188-192.
[7]S. Rusan, G.S.K. Wolkowicz. Bifurcation analysis of a chemostat model with discrete delays. J.Math.Anal.Appl.,1996,204:188-192.
[8]G.S.K. Wolkowicz, H. Xia. Global asymptotic behavior of a chemostat model with discrete delays. SIAM J.Appl.Math.,1997,57:1019-1043.
[9]G.S.K. Wolkowicz, H. Xia, S. Rusan. Competition in the chemostat:adistributed delay model and its global asymptotic behavior. SIAM J.Appl.Math.,1997,57:1281-1310.
[10]H.I. Freedman, Y.T. Xu. Model of competition in the chemostat with in stantaneous and delayed nutrient recycling. J.Math.Biol.,1993,31:513-527.
[11]J.K. Hale. Theory of Runctional Differential Equations. New York:springker-Verlag,1977.
[12]李森林,温立志.泛函微分方程.长沙：湖南科学技术出版社,1987.
[13]秦元勋等.带有时滞的动力系统的运动稳定性(第二版).北京：科学出版社,1989.
[14]Y. Kuang. Delay Differential Equations with Application to Population Dynamics. New York:Academic Press,1993.
[15]H.Y. Hu, Z.H. Wang. Dynamics of Controlled Mechanical Systems with Delayed Feedback. Berlin:Springer-Verlag,2002.
[16]胡海岩,王在华.非线性时滞动力系统的研究进展.力学进展,1999,29(4)：501-512.
[17]G. Stcpan. Retarded Dynamical Systems:Stability and Characteristic Functions. Essex: Longman Scientific and Technical,1989.
[18]叶彦谦著.极限环论.上海：上海科学技术出版社,1984.
[19]张锦炎.常微分方程几何理论与分支问题.北京：北京大学出版社,1981.
[20]李炳熙著.高维动力系统的周期轨道：理论和应用.上海科学技术出版社,1984.
[21]B.D. Hassard, N.D. Kazarinoff, and Y.H. Wan. Theory and Applications of Hopf bifurca-tion. Cambridge:Cambridge University Press, Cambridge,1981.
[22]库比切克,马雷克.分叉理论和耗散结构的计算方法.科学出版社,1990.
[23]S.K. Scott. Chemical Chaos. Clarenden Press, Oxford,1991.
[24]R.Y.K. Yang, J. Su. Bioprocess Engineering,1993,9:97-102.
[25]R.J. Olsen, I.R. Epstein. Bifurcation-Analysis of Chemical-Reaction Mechanisms. Ⅰ: Steady-State Bifurcation Structure. J.Chem.Phys,94,4(1991):3083-3095.
[26]R.J. Olsen, I.R. Epstein. Bifurcation-Analysis of Chemical-Reaction Mechanisms. Ⅱ:Hopf-Bifurcation Analysis. J.Chem.Phys,1993,98(4):2805-2822.
[27]E.Doedel, X. Wang, T. Falrgrieve. AUTO:software for continuation and bifurcation prob-lems in ordinary differential equations. Technical report, California Institute of Technol-ogy,1986.
[28]K. Engelborghs, T. Luzyanina, Q Sammaey. DDE-Biftool v 2.00 user manual:a Matlab package for bifurcation analysis of delay differential equations. Technical Report TW-330. Department of Computer Science. K.U. Leuven, Leuven, Belgium,2001.
[29]B.W. Kooi, M.P Boer, et al. Resistance of a food chain to invasion by a top predator. Math.Bios.,1999,157,217-236.
[30]Y.C. Zhang, M.A. Henson. Bifurcation analysis of continuous biochemical reactor models. Biotechnol.Prog.,2001,17:647-660.
[31]修志龙.1,3-丙二醇的微生物法生产分析.现代化工,1999,19(3)：33-35.
[32]Biebl H, Menzel K, Zeng A P, et al. Microbial production of 1,3-propanediol. Applied Microbiology and Biotechnology,1999,52(3):289-297.
[33]A.P. Zeng, A. Rose, et al. Multiple product inhibition and growth modeling of Clostrid-ium butyricum and Klebsiella pneumoniae in glycerol fermentation. Biotechnology and Bioengeering.1994,44(8):902-911.
[34]A.P. Zeng. A kinetic model for product fermation of microbial and mamalian cells. Biotech-nology and Biocngeering,1995,46(4):314-324.
[35]修志龙.微生物发酵法生产1,3-丙二醇的研究进展.微生物通报,2000：27(4)：300-302.
[36]修志龙,曾安平,安利佳.甘油生物转化为1,3-丙二醇过程的动力学数学模拟和多稳态研究.大连理工大学学报,2000,40(4)：428-433.
[37]A.P. Zeng, W.D. Dcckwer. A kinetic model for substrate and energy consumption of mi-crobia growth under substrate-sufficient conditions. Biotechnology Progress,1995,11(1): 71-79.
[38]高彩霞,冯恩民,王宗涛,修志龙.微生物间歇发酵生产1,3-丙二醇过程辨识与优化.大连理工大学学报,2006,46(5)：771-774.
[39]王宝光：刘铭,杜晨宇等.微生物法生产1,3-丙二醇过程的代谢工程研究进展.过程工程学报,2006,6(1)：144-148.
[40]修志龙：曾安平,安利佳.甘油生物连续歧化过程的过渡行为及其数学模拟.高校化学工程学报,2000,14(1)：53-58.
[41]马永峰.微生物连续培养模型中非线性行为的分析与模拟(博士学位论文).大连：大连理工大学,2004.
[42]C.X. Gao, E.M. Feng. Z.T. Wang, et al. Nonlinear dynamical systems of bio-dissimilation of glycerol to 1,3-propanediol and their optimal controls. Journal of Industrial and Man-agement Optiminization.2005,1(3):377-388.
[43]C.X. Gao, E.M. Feng, Z.L. Xiu. Identification and optimization of the nonlinear impul-sive system in microbial fed-batch fermentation. Dynamics of Continuous Discrete and Impulsive systems-series A-Mathmatical Analysis,2006,13:625-632.
[44]C.X. Gao, K.Z. Li. E.M. Feng. Nonlinear impulsive system of fed-batch culture in fermen-tative production and its properties. Chaos, Solitons & Fractals.2006,28(1):271-277.
[45]C.X. Gao, Y.H. Lang, et al. Nonlinear impulsive system of microbial production in fed-batch culture and its optimal control. Journal of Appllied Mathematics and Computing, 2005,19(1-2):203-214.
[46]李晓红,冯恩民,修志龙.微生物连续培养非线性动力系统的性质及最优性条件.工程数学学报,2006,23(1)：7-12.
[47]李晓红,冯恩民：修志龙.微生物间歇发酵非线性动力系统的性质及最优控制.运筹学学报,2005,9(4)：89-96.
[48]李晓红,冯恩民,修志龙.微生物连续发酵稳定模型的算法与收敛性.清华大学学报(自然科学版).2007,47(2)：1907-1909.
[49]李晓红,冯恩民,修志龙.微生物连续培养过程平衡点的稳定分析.高校应用数学学报B辑.2005,20(4)：377-383.
[50]H.Y. Wang, E.M. Feng, Z.L. Xiu. Optimality condition of the nonlinear impulsive system in fed-batch fermentation. Nonlinear Analysis TMA,2008,68(1):12-23.
[51]G. Wang, E.M. Feng, Z.L. Xiu. Vector measure for explicit nonlinear impulsive system of glycerol bioconversion in fed-batch cultures and its parameter identification. Applied Mathematics and Computation,2007,188(2):1151-1160.
[52]G. Wang, E.M. Feng, Z.L. Xiu. Nonlinear hybrid kinetic system of microbial bioconversion in fed-batch culture. Nonlinear Analysis:Hybrid Systems,2008,2(1):65-73.
[53]G. Wang, E.M. Feng, Z.L. Xiu. Vector measure as controls for explicit nonlinear impulsive system of fed-batch culture. Journal of Mathematical Analysis and Applications,2009, 351(1):120-127.
[54]G. Wang, E.M. Feng, Z.L. Xiu. Modelling and parameter identification of microbial bi-conversion in fed-batch cultures. Journal of Process Control,2008,18(5):458-464.
[55]C.Y. Liu, Z.H. Gong, E.M. Feng, H.C. Yin. Optimal switching control for microbial fed-batch culture. Nonlinear Analysis:Hybrid Systems,2008,2(4):1168-1174.
[56]马知恩,周义仓.常微分方程定性与稳定性方法.北京：科学出版社,2001.
[57]匡蛟勋.泛函微分方程的数值处理.北京：科学出版社,1999.
[58]徐英祥.时滞微分系统的若干分歧问题与其数值分析.长春：吉林大学,2005,1-11,38-44.
[59]陆启韶.常微分方程的定性方法和分叉.北京：北京航空航天大学出版社,1989.
[60]M.A. Savageau. Biochemical Systems Analysis:A study of Function and Design in Molec-ular. Biology. Reading, MA:Addison-Wesley,1976.
[61]王树青,元英进.生化过程自动化技术.北京：化学工业出版社,1999.
[62]V. Hatzimanikatis, C.A. Floudas, J.E. Bailey. Analysis and design of metabolic reaction networks via mixed-integer linear optimization. AIChE Journal,1996,42(5):1277-1292.
[63]V. Hatzimanikatis, C.A. Floudas, J.E. Bailey. Optimization of regulatory architectures in metabolic reaction networks. Biotechnology and Bioengineering,1996,52(4):485-500.
[64]J.P. Dean, G.A. Dervakos. Redesigning metabolic networks using mathematical program-ming. Biotechnology and Bioengineering,1998,58(2-3):267-271.
[65]Y.J. Chang, N.V. Sahinidis. Optimization of metabolic pathways under stability consid-erations. Computers and Chemical Engineering,2005,29(3):447-458.
[66]E.O. Voit. Optimization of integrated biochemical systems. Biotechnology and Bioengi-neering,1992,40(5):572-582.
[67]N.V. Torres, E.O. Voit, et al. Optimization of nonlinear biotcchnological processes with linear programming:Application to citric acid production by Aspergillus niger. Biotech-nology and Bioengineering.1996,49(3):247-258.
[68]N.V. Torres, E.O. Voit, C. Gonzalez-Alcon, et al. An indirect optimization method for bio-chemical systems:Description of method and application to the maximization of the rate of ethanol, glycerol and carbohydrate production in Saccharomyces cerevisiae. Biotech-nology and Bioengineering,1997,55(5):758-772.
[69]A.P. Zeng, W.D. Deckwer. A kinetic model for substrate and energy consumption of micro-bial growth under substrate-sufficient conditions. Biotechnol.Prog.,1995,11(1), pp.71-79.
[70]A.P. Zeng, W.D. Deckwer. Mathematical modeling and analysis of glucose and glutamine utilization and regulation in cultures of continuous mammalian cells. Biotechnol. Bioeng. 1995.47(3), pp.334-346.
[71]A.P. Zeng. A kinetic model for product formation of microbial and mammalian cells. Biotechnol. Bioeng.1995,46, pp.314-324.
[72]X. Liao. Theory Method and Application of Stability. Huazhong Univ. of Technology Press, Wuhan,1999.
[73]K. Menzel, A.P. Zeng, H. Biebl, et al. Kenetic, Dynamics, and Pathway Studies of Glycerol Metabolism by Klebsiella pneumoniae in Anaerobic Continuous Culture:I. The Phenom-ena and Characterization of Oscillation and Hysteresis. Biotechnol. Bioeng.1996,52: 549-560.
[74]A.P. Zeng, K. Menzel, W.D. Deckwer. Kenetic, dynamic, and pathway studies of glycerol metabolism by klebsiella pneumoniae in anaerobic continuous culture:Ⅱ. analysis of metabolic rates and pathways under oscillation and steady-state conditions. Biotechnol. Bioeng.1996,52:561-571.
[75]孙丽华,郭庆广,修志龙.一类具有时滞的生化模型的Hopf分支.生物数学学报,2002,17(3)：286-292.
[76]孙丽华,宋炳辉,修志龙.微生物连续培养过程中动态行为研究.大连理工大学学报,2003,43(4)：433-437.
[77]马永峰,孙丽华,修志龙.微生物连续培养过程中振荡的理论分析.工程数学学报,2003,20(1)：1-6.
[78]马永峰,孙丽华,修志龙.连续时滞对微生物连续培养过程中动态行为的影响.高校应用数学学报,2003,18(1)：1-7.
[79]S.G. Ruan, J.J. Wei. On the Zeros of Transcendental Functions with Applications to Sta-bility of Delay Differential Equations with Two Delays. Dynamics of Continuous, Discrete and Impulsive Systems.2003,10:863-874.
[80]K. Menzel, A.P. Zeng, W.D. Deckwer. High concentration and productivity of 1,3-propanediol from continuous fermentation of glycerol by Klebsiella pneumoniae, Enzyme Microb. Technol.1997,20(2), pp.82-86.
[81]高彩霞.非线性脉冲动力系统的最优控制及应用：(博士学位论文).大连：大连理工大学,2005.
[82]K. Price, R. Storn. Minimizing the real function of the ICEC'96 contest by differential evolutions. IEEE International Conference on Evolutionary Computation, Nagoya.1996: 842-844.
[83]Leandro dos Santos Coelho, Mariani V C. Improved differential evolution algorithms for handing economic dispatch optimization with generator constraints [J]. Energy Conversion and Management.2007,48:1631-1639.
[84]C.W. Chen, D.Z. Chen, G.Z. Cao. An improved differential evolution algorithm in training and encoding prior knowledge into feedforward networks with application in chemistry [J]. Chemometrics and Intelligent Laboratory Systems.2002,64:27-43.
[85]李颖,徐桂之,饶利芸等.微分进化算法在头部电阻抗成像中的应用[J].中国生物医学工程学报.2005,24(6)：672-675,694.
[86]M.A. Savageau. Biochemical systems analysis Ⅱ:The steady-state solutions for an n-pool system using a power-law apporoximation. Journal of Theoretical Biology.1969,25(3): 370-379.
[87]Z.L. Xiu, A.P. Zeng, W.D. Deckwer. Multiplicity and stability analysis of microorganisms in continuous culture:effects of metabolic overflow and growth inhibition, Biotechnol. Bioeng.1998,57:251-261.
[88]徐恭贤.一类生化过程的优化及控制方法研究：(博士学位论文).大连：大连理工大学,2007.
[2]S.B. Hsu. S.P. Hubbcll, P. Waltman. A mathematical theory of Single-nutrient Competi-tion in Continuous Culture of Microorganisms. SIAM J.Appl.Math.,1977,32(2):366-383.
[3]J.M. Cushing. Integro-diffcrential equationsand delay models in population dynamics. in: Lecture Notes in Biomathematics, Vol.20, Springer, Berlin,1977.
[4]Y. Kuang. Basic properties of mathematical population models. J.Biomath.,2002,17: 129-142.
[5]R.K. Finn, R.E. Wilson. Population dynamics of a continuous propagator for microorgan-isms. J.Agric.food.chem.,1953,2:66-69.
[6]J. Caperson. Time lag in population grwth response of isochrysis galbana to a variable nitrate environment. Ecology,1969,50:188-192.
[7]S. Rusan, G.S.K. Wolkowicz. Bifurcation analysis of a chemostat model with discrete delays. J.Math.Anal.Appl.,1996,204:188-192.
[8]G.S.K. Wolkowicz, H. Xia. Global asymptotic behavior of a chemostat model with discrete delays. SIAM J.Appl.Math.,1997,57:1019-1043.
[9]G.S.K. Wolkowicz, H. Xia, S. Rusan. Competition in the chemostat:adistributed delay model and its global asymptotic behavior. SIAM J.Appl.Math.,1997,57:1281-1310.
[10]H.I. Freedman, Y.T. Xu. Model of competition in the chemostat with in stantaneous and delayed nutrient recycling. J.Math.Biol.,1993,31:513-527.
[11]J.K. Hale. Theory of Runctional Differential Equations. New York:springker-Verlag,1977.
[12]李森林,温立志.泛函微分方程.长沙：湖南科学技术出版社,1987.
[13]秦元勋等.带有时滞的动力系统的运动稳定性(第二版).北京：科学出版社,1989.
[14]Y. Kuang. Delay Differential Equations with Application to Population Dynamics. New York:Academic Press,1993.
[15]H.Y. Hu, Z.H. Wang. Dynamics of Controlled Mechanical Systems with Delayed Feedback. Berlin:Springer-Verlag,2002.
[16]胡海岩,王在华.非线性时滞动力系统的研究进展.力学进展,1999,29(4)：501-512.
[17]G. Stcpan. Retarded Dynamical Systems:Stability and Characteristic Functions. Essex: Longman Scientific and Technical,1989.
[18]叶彦谦著.极限环论.上海：上海科学技术出版社,1984.
[19]张锦炎.常微分方程几何理论与分支问题.北京：北京大学出版社,1981.
[20]李炳熙著.高维动力系统的周期轨道：理论和应用.上海科学技术出版社,1984.
[21]B.D. Hassard, N.D. Kazarinoff, and Y.H. Wan. Theory and Applications of Hopf bifurca-tion. Cambridge:Cambridge University Press, Cambridge,1981.
[22]库比切克,马雷克.分叉理论和耗散结构的计算方法.科学出版社,1990.
[23]S.K. Scott. Chemical Chaos. Clarenden Press, Oxford,1991.
[24]R.Y.K. Yang, J. Su. Bioprocess Engineering,1993,9:97-102.
[25]R.J. Olsen, I.R. Epstein. Bifurcation-Analysis of Chemical-Reaction Mechanisms. Ⅰ: Steady-State Bifurcation Structure. J.Chem.Phys,94,4(1991):3083-3095.
[26]R.J. Olsen, I.R. Epstein. Bifurcation-Analysis of Chemical-Reaction Mechanisms. Ⅱ:Hopf-Bifurcation Analysis. J.Chem.Phys,1993,98(4):2805-2822.
[27]E.Doedel, X. Wang, T. Falrgrieve. AUTO:software for continuation and bifurcation prob-lems in ordinary differential equations. Technical report, California Institute of Technol-ogy,1986.
[28]K. Engelborghs, T. Luzyanina, Q Sammaey. DDE-Biftool v 2.00 user manual:a Matlab package for bifurcation analysis of delay differential equations. Technical Report TW-330. Department of Computer Science. K.U. Leuven, Leuven, Belgium,2001.
[29]B.W. Kooi, M.P Boer, et al. Resistance of a food chain to invasion by a top predator. Math.Bios.,1999,157,217-236.
[30]Y.C. Zhang, M.A. Henson. Bifurcation analysis of continuous biochemical reactor models. Biotechnol.Prog.,2001,17:647-660.
[31]修志龙.1,3-丙二醇的微生物法生产分析.现代化工,1999,19(3)：33-35.
[32]Biebl H, Menzel K, Zeng A P, et al. Microbial production of 1,3-propanediol. Applied Microbiology and Biotechnology,1999,52(3):289-297.
[33]A.P. Zeng, A. Rose, et al. Multiple product inhibition and growth modeling of Clostrid-ium butyricum and Klebsiella pneumoniae in glycerol fermentation. Biotechnology and Bioengeering.1994,44(8):902-911.
[34]A.P. Zeng. A kinetic model for product fermation of microbial and mamalian cells. Biotech-nology and Biocngeering,1995,46(4):314-324.
[35]修志龙.微生物发酵法生产1,3-丙二醇的研究进展.微生物通报,2000：27(4)：300-302.
[36]修志龙,曾安平,安利佳.甘油生物转化为1,3-丙二醇过程的动力学数学模拟和多稳态研究.大连理工大学学报,2000,40(4)：428-433.
[37]A.P. Zeng, W.D. Dcckwer. A kinetic model for substrate and energy consumption of mi-crobia growth under substrate-sufficient conditions. Biotechnology Progress,1995,11(1): 71-79.
[38]高彩霞,冯恩民,王宗涛,修志龙.微生物间歇发酵生产1,3-丙二醇过程辨识与优化.大连理工大学学报,2006,46(5)：771-774.
[39]王宝光：刘铭,杜晨宇等.微生物法生产1,3-丙二醇过程的代谢工程研究进展.过程工程学报,2006,6(1)：144-148.
[40]修志龙：曾安平,安利佳.甘油生物连续歧化过程的过渡行为及其数学模拟.高校化学工程学报,2000,14(1)：53-58.
[41]马永峰.微生物连续培养模型中非线性行为的分析与模拟(博士学位论文).大连：大连理工大学,2004.
[42]C.X. Gao, E.M. Feng. Z.T. Wang, et al. Nonlinear dynamical systems of bio-dissimilation of glycerol to 1,3-propanediol and their optimal controls. Journal of Industrial and Man-agement Optiminization.2005,1(3):377-388.
[43]C.X. Gao, E.M. Feng, Z.L. Xiu. Identification and optimization of the nonlinear impul-sive system in microbial fed-batch fermentation. Dynamics of Continuous Discrete and Impulsive systems-series A-Mathmatical Analysis,2006,13:625-632.
[44]C.X. Gao, K.Z. Li. E.M. Feng. Nonlinear impulsive system of fed-batch culture in fermen-tative production and its properties. Chaos, Solitons & Fractals.2006,28(1):271-277.
[45]C.X. Gao, Y.H. Lang, et al. Nonlinear impulsive system of microbial production in fed-batch culture and its optimal control. Journal of Appllied Mathematics and Computing, 2005,19(1-2):203-214.
[46]李晓红,冯恩民,修志龙.微生物连续培养非线性动力系统的性质及最优性条件.工程数学学报,2006,23(1)：7-12.
[47]李晓红,冯恩民：修志龙.微生物间歇发酵非线性动力系统的性质及最优控制.运筹学学报,2005,9(4)：89-96.
[48]李晓红,冯恩民,修志龙.微生物连续发酵稳定模型的算法与收敛性.清华大学学报(自然科学版).2007,47(2)：1907-1909.
[49]李晓红,冯恩民,修志龙.微生物连续培养过程平衡点的稳定分析.高校应用数学学报B辑.2005,20(4)：377-383.
[50]H.Y. Wang, E.M. Feng, Z.L. Xiu. Optimality condition of the nonlinear impulsive system in fed-batch fermentation. Nonlinear Analysis TMA,2008,68(1):12-23.
[51]G. Wang, E.M. Feng, Z.L. Xiu. Vector measure for explicit nonlinear impulsive system of glycerol bioconversion in fed-batch cultures and its parameter identification. Applied Mathematics and Computation,2007,188(2):1151-1160.
[52]G. Wang, E.M. Feng, Z.L. Xiu. Nonlinear hybrid kinetic system of microbial bioconversion in fed-batch culture. Nonlinear Analysis:Hybrid Systems,2008,2(1):65-73.
[53]G. Wang, E.M. Feng, Z.L. Xiu. Vector measure as controls for explicit nonlinear impulsive system of fed-batch culture. Journal of Mathematical Analysis and Applications,2009, 351(1):120-127.
[54]G. Wang, E.M. Feng, Z.L. Xiu. Modelling and parameter identification of microbial bi-conversion in fed-batch cultures. Journal of Process Control,2008,18(5):458-464.
[55]C.Y. Liu, Z.H. Gong, E.M. Feng, H.C. Yin. Optimal switching control for microbial fed-batch culture. Nonlinear Analysis:Hybrid Systems,2008,2(4):1168-1174.
[56]马知恩,周义仓.常微分方程定性与稳定性方法.北京：科学出版社,2001.
[57]匡蛟勋.泛函微分方程的数值处理.北京：科学出版社,1999.
[58]徐英祥.时滞微分系统的若干分歧问题与其数值分析.长春：吉林大学,2005,1-11,38-44.
[59]陆启韶.常微分方程的定性方法和分叉.北京：北京航空航天大学出版社,1989.
[60]M.A. Savageau. Biochemical Systems Analysis:A study of Function and Design in Molec-ular. Biology. Reading, MA:Addison-Wesley,1976.
[61]王树青,元英进.生化过程自动化技术.北京：化学工业出版社,1999.
[62]V. Hatzimanikatis, C.A. Floudas, J.E. Bailey. Analysis and design of metabolic reaction networks via mixed-integer linear optimization. AIChE Journal,1996,42(5):1277-1292.
[63]V. Hatzimanikatis, C.A. Floudas, J.E. Bailey. Optimization of regulatory architectures in metabolic reaction networks. Biotechnology and Bioengineering,1996,52(4):485-500.
[64]J.P. Dean, G.A. Dervakos. Redesigning metabolic networks using mathematical program-ming. Biotechnology and Bioengineering,1998,58(2-3):267-271.
[65]Y.J. Chang, N.V. Sahinidis. Optimization of metabolic pathways under stability consid-erations. Computers and Chemical Engineering,2005,29(3):447-458.
[66]E.O. Voit. Optimization of integrated biochemical systems. Biotechnology and Bioengi-neering,1992,40(5):572-582.
[67]N.V. Torres, E.O. Voit, et al. Optimization of nonlinear biotcchnological processes with linear programming:Application to citric acid production by Aspergillus niger. Biotech-nology and Bioengineering.1996,49(3):247-258.
[68]N.V. Torres, E.O. Voit, C. Gonzalez-Alcon, et al. An indirect optimization method for bio-chemical systems:Description of method and application to the maximization of the rate of ethanol, glycerol and carbohydrate production in Saccharomyces cerevisiae. Biotech-nology and Bioengineering,1997,55(5):758-772.
[69]A.P. Zeng, W.D. Deckwer. A kinetic model for substrate and energy consumption of micro-bial growth under substrate-sufficient conditions. Biotechnol.Prog.,1995,11(1), pp.71-79.
[70]A.P. Zeng, W.D. Deckwer. Mathematical modeling and analysis of glucose and glutamine utilization and regulation in cultures of continuous mammalian cells. Biotechnol. Bioeng. 1995.47(3), pp.334-346.
[71]A.P. Zeng. A kinetic model for product formation of microbial and mammalian cells. Biotechnol. Bioeng.1995,46, pp.314-324.
[72]X. Liao. Theory Method and Application of Stability. Huazhong Univ. of Technology Press, Wuhan,1999.
[73]K. Menzel, A.P. Zeng, H. Biebl, et al. Kenetic, Dynamics, and Pathway Studies of Glycerol Metabolism by Klebsiella pneumoniae in Anaerobic Continuous Culture:I. The Phenom-ena and Characterization of Oscillation and Hysteresis. Biotechnol. Bioeng.1996,52: 549-560.
[74]A.P. Zeng, K. Menzel, W.D. Deckwer. Kenetic, dynamic, and pathway studies of glycerol metabolism by klebsiella pneumoniae in anaerobic continuous culture:Ⅱ. analysis of metabolic rates and pathways under oscillation and steady-state conditions. Biotechnol. Bioeng.1996,52:561-571.
[75]孙丽华,郭庆广,修志龙.一类具有时滞的生化模型的Hopf分支.生物数学学报,2002,17(3)：286-292.
[76]孙丽华,宋炳辉,修志龙.微生物连续培养过程中动态行为研究.大连理工大学学报,2003,43(4)：433-437.
[77]马永峰,孙丽华,修志龙.微生物连续培养过程中振荡的理论分析.工程数学学报,2003,20(1)：1-6.
[78]马永峰,孙丽华,修志龙.连续时滞对微生物连续培养过程中动态行为的影响.高校应用数学学报,2003,18(1)：1-7.
[79]S.G. Ruan, J.J. Wei. On the Zeros of Transcendental Functions with Applications to Sta-bility of Delay Differential Equations with Two Delays. Dynamics of Continuous, Discrete and Impulsive Systems.2003,10:863-874.
[80]K. Menzel, A.P. Zeng, W.D. Deckwer. High concentration and productivity of 1,3-propanediol from continuous fermentation of glycerol by Klebsiella pneumoniae, Enzyme Microb. Technol.1997,20(2), pp.82-86.
[81]高彩霞.非线性脉冲动力系统的最优控制及应用：(博士学位论文).大连：大连理工大学,2005.
[82]K. Price, R. Storn. Minimizing the real function of the ICEC'96 contest by differential evolutions. IEEE International Conference on Evolutionary Computation, Nagoya.1996: 842-844.
[83]Leandro dos Santos Coelho, Mariani V C. Improved differential evolution algorithms for handing economic dispatch optimization with generator constraints [J]. Energy Conversion and Management.2007,48:1631-1639.
[84]C.W. Chen, D.Z. Chen, G.Z. Cao. An improved differential evolution algorithm in training and encoding prior knowledge into feedforward networks with application in chemistry [J]. Chemometrics and Intelligent Laboratory Systems.2002,64:27-43.
[85]李颖,徐桂之,饶利芸等.微分进化算法在头部电阻抗成像中的应用[J].中国生物医学工程学报.2005,24(6)：672-675,694.
[86]M.A. Savageau. Biochemical systems analysis Ⅱ:The steady-state solutions for an n-pool system using a power-law apporoximation. Journal of Theoretical Biology.1969,25(3): 370-379.
[87]Z.L. Xiu, A.P. Zeng, W.D. Deckwer. Multiplicity and stability analysis of microorganisms in continuous culture:effects of metabolic overflow and growth inhibition, Biotechnol. Bioeng.1998,57:251-261.
[88]徐恭贤.一类生化过程的优化及控制方法研究：(博士学位论文).大连：大连理工大学,2007.