一类微生物絮凝模型平衡点的稳定性与Hopf分支
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摘要
考虑了一类具有营养循环的微生物絮凝常微分方程动力学模型.该类动力学模型平衡点可呈现前向与后向分支.利用常微分方程相关理论,研究了其边界平衡点的全局稳定性,以及正平衡点的局部稳定性析与Hopf分支存在性.同时,计算机数值模拟结果显示了与理论结果的一致性,并进一步揭示了该类动力系统可能呈现的复杂的振荡性质.
In this paper, we consider a class of ordinary differential equation dynamic model describing flocculation of microorganism and nutrient cycling. The equilibria of the dynamic model can behave forward and backward branches. Using the theory of ordinary differential equations, global stability of the boundary equilibrium, local stability of the positive equilibrium and the existence of Hopf bifurcation are analyzed.At the same time, the numerical simulation results are given.
In this paper, we consider a class of ordinary differential equation dynamic model describing flocculation of microorganism and nutrient cycling. The equilibria of the dynamic model can behave forward and backward branches. Using the theory of ordinary differential equations, global stability of the boundary equilibrium, local stability of the positive equilibrium and the existence of Hopf bifurcation are analyzed.At the same time, the numerical simulation results are given.
引文
[1]Smith H L, Walt man P. The Theory of the Chemostat:Dynamics of Microbial Competition.Cambridge:Cambridge University Press, 1995.
[2]陈兰荪,陈键.非线性生物动力系统.北京:科学出版社,1993.(Chen L S, Chen J. Nonlinear Biologic Dynamic Systems. Beijing:Science Press, 1933.)
[3]阮世贵.恒化器模型的动力学.华中师范大学学报,1997, 31(4):377-397.(Ruan S G. The dynamics of chemostat models. Journal of Central China Normal University:1997, 31(4):377-397.)
[4]曹炳明. CS-1型絮凝剂的制备及其在污水处理方面的应用.工业水处理,1987, 6(7):27-29.(Cao B M. Preparation of CS-1 flocculant and its application in sewage treatment. Industrial Water Treatment, 1987, 6(7):27-29.)
[5]李永明,于水利,唐玉霖.壳聚糖絮凝剂在水处理中的应用研究进展.水处理技术,2011, 37(9):11-18.(Li Y M, Yu S L, Tang Y L. Research and application of chitosan and its derivatives in water treatment. Technology of Water Treatment, 2011, 37(9):11-18.)
[6]於臻绯.天然高分子絮凝剂——壳聚糖在水处理中的应用.上海建设科技,2012,(5):25-27.(Yu Z F. Natural polymer flocculant—Chitosan and its application in water treatment. Shanghai Construction Science and Technology, 2012,(5):25-27.)
[7]刘秉涛,娄渊知,徐菲.聚合氯化铝/壳聚糖复合絮凝剂在活性污泥中的调理作用.环境化学,2007, 26(1):42-45.(Liu B T, Lou Y Z, Xu F. Effect of polymeric aluminum chloride/chitosan composite flocculanton activated sludge. Environmental Chemistry, 2007, 26(1):42-45.)
[8]王莉,冯贵颖,田鹏.新型复合絮凝剂在水处理中的应用.西北农林科技大学学报.2005, 33(3):149-152.(Wang L, Feng G Y, Tian P. Application of composite flocculant in water treatment. Journal of Northwest Science and Technology University of Agriculture and Forestry,2005, 33(3):149-152.)
[9]Yang R, Li H J, Huang M, et al. A review on chitosan—Based flocculants and their applications in water treatment. Water Research, 2016, 95(5):59-89.
[10]Choi H J. Effect of Mg-sericite flocculant for treatment of brewery wastewater. Applied Clay Science, 2015, 115(11):145-149.
[11]Lee C S, John R, Chong M F. A review on application of flocculants in wastewater treatment.Process Safety and Environmental Protection, 2014, 92(6):489-508.
[12]马放,段姝悦,孔祥震,等.微生物絮凝剂的研究现状及其发展趋势.中国给水排水,2012, 28(2):14-17.(Ma F, Duan S Y, Kong X Z, et al. Research status and development trend of studies on microbial flocculants. China Water and Wastewater, 2012, 28(2):14-17.)
[13]邓述波,余刚,蒋展鹏,等.微生物絮凝剂MBFA9的絮凝机理研究.水处理技术,2001,27(1):22-25.(Deng S B, Yu G, Jiang Z P, et al. Study on flocculation mechanism of microbial flocculant MBFA9. Technology of Water Treatment, 2001, 27(1):22-25.)
[14]张云波,宗明月,刘其友,等.一种微生物絮凝剂在炼油废水中的应用.广西大学学报,2010, 35(3):473-477.(Zhang Y B, Zong M Y, Liu Q Y, et al. The application of a strain of microbial flocculant in the oil refinery wastewater. Journal of Guangxi University, 2010, 35(3):473-477.)
[15]杨圣云,林波,李昌花,等.微生物絮凝剂HXCS-1处理乳化液废水.江西科学,2006, 24(2):147-149.(Yang S Y, Lin B, Li C H, et al. Treating emulsive wastewater with microbial-flocculant HXCS-1.Jiangxi Science, 2006, 24(2):147-149.)
[16]王猛,施宪法,柴晓利.微生物絮凝剂的应用与研究.化工环保,2001,21(6):328-332.(Wang M, Shi X F, Chai X L. Study on bioflocculant and its application. Environmental Protection of Chemical Industry, 2001, 21(6):328-332.)
[17]张志强,林波,夏四清,等.高活性微生物絮凝剂处理靛蓝印染废水的动力学研究.工业水处理,2007, 27(4):44-47.(Zhang Z Q, Lin B,Xia S Q,et al. Kinetic study on the treatment of indigotin printing and dyeing wastewater with the highly activated microbial flocculant. Industrial Water Treatment,2007, 27(4):44-47.)
[18]邰晓东,马万彪,郭松柏,等.微生物絮凝的时滞动力学模型与理论分析.数学的实践与认识,2015, 28(7):198-209.(Tai X D, Ma W B, Guo S B, et al. A class of dynamic delayed model describing flocculation of microorganism and its theoretical analysis. Mathematics in Practice and Theory, 2015, 28(7):198-209.)
[19]付桂芳,马万彪.由微分方程所描述的微生物连续培养动力系统(Ⅰ).微生物学通报,2004, 31(5):136-139.(Fu G F, Ma W B. Chemostat dynamics models described by differential equations(I). Microbiology China, 2004, 31(5):136-139.)
[20]付桂芳,马万彪.由微分方程所描述的微生物连续培养动力系统(Ⅱ).徽生物学通报,2004, 31(6):128-131.(Fu G F, Ma W B. Chemostat dynamics models described by differential equations(Ⅱ). Microbiology China, 2004, 31(6):128-131.)
[21]陈兰荪,孟新柱,焦建军.生物动力学.北京:科学出版社,2009.(Chen L S, Meng X Z, Jiao J J. Biodynamics. Beijing:Science Press, 2009.)
[22]Powell T, Richerson P J. Temporal variation, spatial heterogeneity, and competition for resources in plankton systems:A theoretical model. The American Naturalist, 1985, 125(3):431-464.
[23]Nisbet R M, Gurney W S C. Model of material cycling in a closed ecosystem. Nature, 1976,264(5587):633-635.
[24]Beretta E, Bischi G I, Solimano F. Stability in chemostat equations with delayed nutrient recycling. Journal of Mathematical Biology, 1990, 28(1):99-111.
[25]Bischi G I. Effects of time lags on transient characteristics of a nutrient cycling model. Mathematical Biosciences, 1992, 109(2):151-175.
[26]Freedman H I, Xu Y T. Models of competition in the chemostat with instantaneous and delayed nutrient recycling. Journal of Mathematical Biology, 1993, 31(5):513-527.
[27]He X Z, Ruan S G. Global stability in chemostat-type plankton models with delayed nutrient recycling. Journal of Mathematical Biology, 1998, 37(3):253-271.
[28]Ruan S G, He X Z. Global stability in chemostat-type competition models with nutrient recycling.SIAM Journal on Applied Mathematics, 1998, 58(1):170-192.
[29]Ruan S G. Persistence and coexistence in zooplankton-phytoplankton-nutrient models with instantaneous nutrient recycling. Journal of Mathematical Biology, 1993, 31(3):633-654.
[30]Ruan S G. A three-trophic-level model of plankton dynamics with nutrient recycling. Canadian Applied Mathematics Quarterly, 1993, 4(1):529-553.
[31]Ruan S G. Oscillation in plankton models with nutrient recycling. Journal of Theoretical Biology,2001, 208(1):15-26.
[32]修志龙,曾安平.微生物细胞连续培养过程中振荡和混沌行为的研究进展.生物工程进展,1999, 19(6):58-63.(Xiu Z L, Zeng A P. A study progress in oscillatory and chaotic behavior in microbial continuous cultures. Progress in Biotechnology, 1999, 19(6):58-63.)
[33]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.(Ma Y F,Sun L H,Xiu Z L. Theoretical analysis of the oscillatory behavior in process of microbial continuous culture. Journal of Engineering Mathematics, 2003, 20(1):1-6.)
[34]马水峰,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(A), 2003,18(1):1-7.(Ma Y F, Sun L H, Xiu Z L. Effect of distributed delay on dynamic behavior in microbial continuous culture. Applied Mathematics A Journal of Chinese Universities(A), 2003, 18(1):1-7.)
[35]秦元勋,王联,王慕秋.运动稳定性理论及其应用.北京:科学出版社,1981.(Qin Y X, Wang L, Wang M Q. Theory of Motion Stability and Its Applications. Beijing:Science Press, 1981.)
[36]尤秉礼.常微分方程补充教程.北京:人民教育出版社,1981.(You B L. A Supplementary Course for Ordinary Differential Equations. Beijing:People's Education Press, 1981.)
[37]廖晓昕.稳定性的理论方法和应用.武汉:华中科技大学出版社,1999.(Liao X X. Theory, Methods and Applications of Stability. Wuhan:Huazhong University of Science and Technology Press, 1999.)
[38]张锦炎,冯贝叶.常徽分方程几何理论与分支问题.北京:北京大学出版社,2000.(Zhang J Y, Feng B Y. Geometry Theory and Bifurcations of Ordinary Differential Equations.Beijing:Peking University Press, 2000.)
[39]陈关荣,吕金虎.Lorenz系统族的动力学分析、控制与同步.北京:科学出版社,2003.(Chen G R, Lii J H. Dynamic Analysis, Control and Synchronization of Lorenz System Family.Beijing:Science Press, 2003.)
[2]陈兰荪,陈键.非线性生物动力系统.北京:科学出版社,1993.(Chen L S, Chen J. Nonlinear Biologic Dynamic Systems. Beijing:Science Press, 1933.)
[3]阮世贵.恒化器模型的动力学.华中师范大学学报,1997, 31(4):377-397.(Ruan S G. The dynamics of chemostat models. Journal of Central China Normal University:1997, 31(4):377-397.)
[4]曹炳明. CS-1型絮凝剂的制备及其在污水处理方面的应用.工业水处理,1987, 6(7):27-29.(Cao B M. Preparation of CS-1 flocculant and its application in sewage treatment. Industrial Water Treatment, 1987, 6(7):27-29.)
[5]李永明,于水利,唐玉霖.壳聚糖絮凝剂在水处理中的应用研究进展.水处理技术,2011, 37(9):11-18.(Li Y M, Yu S L, Tang Y L. Research and application of chitosan and its derivatives in water treatment. Technology of Water Treatment, 2011, 37(9):11-18.)
[6]於臻绯.天然高分子絮凝剂——壳聚糖在水处理中的应用.上海建设科技,2012,(5):25-27.(Yu Z F. Natural polymer flocculant—Chitosan and its application in water treatment. Shanghai Construction Science and Technology, 2012,(5):25-27.)
[7]刘秉涛,娄渊知,徐菲.聚合氯化铝/壳聚糖复合絮凝剂在活性污泥中的调理作用.环境化学,2007, 26(1):42-45.(Liu B T, Lou Y Z, Xu F. Effect of polymeric aluminum chloride/chitosan composite flocculanton activated sludge. Environmental Chemistry, 2007, 26(1):42-45.)
[8]王莉,冯贵颖,田鹏.新型复合絮凝剂在水处理中的应用.西北农林科技大学学报.2005, 33(3):149-152.(Wang L, Feng G Y, Tian P. Application of composite flocculant in water treatment. Journal of Northwest Science and Technology University of Agriculture and Forestry,2005, 33(3):149-152.)
[9]Yang R, Li H J, Huang M, et al. A review on chitosan—Based flocculants and their applications in water treatment. Water Research, 2016, 95(5):59-89.
[10]Choi H J. Effect of Mg-sericite flocculant for treatment of brewery wastewater. Applied Clay Science, 2015, 115(11):145-149.
[11]Lee C S, John R, Chong M F. A review on application of flocculants in wastewater treatment.Process Safety and Environmental Protection, 2014, 92(6):489-508.
[12]马放,段姝悦,孔祥震,等.微生物絮凝剂的研究现状及其发展趋势.中国给水排水,2012, 28(2):14-17.(Ma F, Duan S Y, Kong X Z, et al. Research status and development trend of studies on microbial flocculants. China Water and Wastewater, 2012, 28(2):14-17.)
[13]邓述波,余刚,蒋展鹏,等.微生物絮凝剂MBFA9的絮凝机理研究.水处理技术,2001,27(1):22-25.(Deng S B, Yu G, Jiang Z P, et al. Study on flocculation mechanism of microbial flocculant MBFA9. Technology of Water Treatment, 2001, 27(1):22-25.)
[14]张云波,宗明月,刘其友,等.一种微生物絮凝剂在炼油废水中的应用.广西大学学报,2010, 35(3):473-477.(Zhang Y B, Zong M Y, Liu Q Y, et al. The application of a strain of microbial flocculant in the oil refinery wastewater. Journal of Guangxi University, 2010, 35(3):473-477.)
[15]杨圣云,林波,李昌花,等.微生物絮凝剂HXCS-1处理乳化液废水.江西科学,2006, 24(2):147-149.(Yang S Y, Lin B, Li C H, et al. Treating emulsive wastewater with microbial-flocculant HXCS-1.Jiangxi Science, 2006, 24(2):147-149.)
[16]王猛,施宪法,柴晓利.微生物絮凝剂的应用与研究.化工环保,2001,21(6):328-332.(Wang M, Shi X F, Chai X L. Study on bioflocculant and its application. Environmental Protection of Chemical Industry, 2001, 21(6):328-332.)
[17]张志强,林波,夏四清,等.高活性微生物絮凝剂处理靛蓝印染废水的动力学研究.工业水处理,2007, 27(4):44-47.(Zhang Z Q, Lin B,Xia S Q,et al. Kinetic study on the treatment of indigotin printing and dyeing wastewater with the highly activated microbial flocculant. Industrial Water Treatment,2007, 27(4):44-47.)
[18]邰晓东,马万彪,郭松柏,等.微生物絮凝的时滞动力学模型与理论分析.数学的实践与认识,2015, 28(7):198-209.(Tai X D, Ma W B, Guo S B, et al. A class of dynamic delayed model describing flocculation of microorganism and its theoretical analysis. Mathematics in Practice and Theory, 2015, 28(7):198-209.)
[19]付桂芳,马万彪.由微分方程所描述的微生物连续培养动力系统(Ⅰ).微生物学通报,2004, 31(5):136-139.(Fu G F, Ma W B. Chemostat dynamics models described by differential equations(I). Microbiology China, 2004, 31(5):136-139.)
[20]付桂芳,马万彪.由微分方程所描述的微生物连续培养动力系统(Ⅱ).徽生物学通报,2004, 31(6):128-131.(Fu G F, Ma W B. Chemostat dynamics models described by differential equations(Ⅱ). Microbiology China, 2004, 31(6):128-131.)
[21]陈兰荪,孟新柱,焦建军.生物动力学.北京:科学出版社,2009.(Chen L S, Meng X Z, Jiao J J. Biodynamics. Beijing:Science Press, 2009.)
[22]Powell T, Richerson P J. Temporal variation, spatial heterogeneity, and competition for resources in plankton systems:A theoretical model. The American Naturalist, 1985, 125(3):431-464.
[23]Nisbet R M, Gurney W S C. Model of material cycling in a closed ecosystem. Nature, 1976,264(5587):633-635.
[24]Beretta E, Bischi G I, Solimano F. Stability in chemostat equations with delayed nutrient recycling. Journal of Mathematical Biology, 1990, 28(1):99-111.
[25]Bischi G I. Effects of time lags on transient characteristics of a nutrient cycling model. Mathematical Biosciences, 1992, 109(2):151-175.
[26]Freedman H I, Xu Y T. Models of competition in the chemostat with instantaneous and delayed nutrient recycling. Journal of Mathematical Biology, 1993, 31(5):513-527.
[27]He X Z, Ruan S G. Global stability in chemostat-type plankton models with delayed nutrient recycling. Journal of Mathematical Biology, 1998, 37(3):253-271.
[28]Ruan S G, He X Z. Global stability in chemostat-type competition models with nutrient recycling.SIAM Journal on Applied Mathematics, 1998, 58(1):170-192.
[29]Ruan S G. Persistence and coexistence in zooplankton-phytoplankton-nutrient models with instantaneous nutrient recycling. Journal of Mathematical Biology, 1993, 31(3):633-654.
[30]Ruan S G. A three-trophic-level model of plankton dynamics with nutrient recycling. Canadian Applied Mathematics Quarterly, 1993, 4(1):529-553.
[31]Ruan S G. Oscillation in plankton models with nutrient recycling. Journal of Theoretical Biology,2001, 208(1):15-26.
[32]修志龙,曾安平.微生物细胞连续培养过程中振荡和混沌行为的研究进展.生物工程进展,1999, 19(6):58-63.(Xiu Z L, Zeng A P. A study progress in oscillatory and chaotic behavior in microbial continuous cultures. Progress in Biotechnology, 1999, 19(6):58-63.)
[33]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.(Ma Y F,Sun L H,Xiu Z L. Theoretical analysis of the oscillatory behavior in process of microbial continuous culture. Journal of Engineering Mathematics, 2003, 20(1):1-6.)
[34]马水峰,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(A), 2003,18(1):1-7.(Ma Y F, Sun L H, Xiu Z L. Effect of distributed delay on dynamic behavior in microbial continuous culture. Applied Mathematics A Journal of Chinese Universities(A), 2003, 18(1):1-7.)
[35]秦元勋,王联,王慕秋.运动稳定性理论及其应用.北京:科学出版社,1981.(Qin Y X, Wang L, Wang M Q. Theory of Motion Stability and Its Applications. Beijing:Science Press, 1981.)
[36]尤秉礼.常微分方程补充教程.北京:人民教育出版社,1981.(You B L. A Supplementary Course for Ordinary Differential Equations. Beijing:People's Education Press, 1981.)
[37]廖晓昕.稳定性的理论方法和应用.武汉:华中科技大学出版社,1999.(Liao X X. Theory, Methods and Applications of Stability. Wuhan:Huazhong University of Science and Technology Press, 1999.)
[38]张锦炎,冯贝叶.常徽分方程几何理论与分支问题.北京:北京大学出版社,2000.(Zhang J Y, Feng B Y. Geometry Theory and Bifurcations of Ordinary Differential Equations.Beijing:Peking University Press, 2000.)
[39]陈关荣,吕金虎.Lorenz系统族的动力学分析、控制与同步.北京:科学出版社,2003.(Chen G R, Lii J H. Dynamic Analysis, Control and Synchronization of Lorenz System Family.Beijing:Science Press, 2003.)