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A dynamic model describing heterotrophic culture of chorella and its stability analysis
1. | Department of Applied Mathematics, School of Mathematics and Physics, University of Science and Technology Beijing, Beijing 100083, China |
2. | Department of Biological Science and Technology, School of Chemical and Biological Engineering, University of Science and Technology Beijing, Beijing 100083, China |
3. | Graduate School of Science and Technology, Faculty of Engineering, Shizuoka University, Hamamatsu 432-8561 |
References:
[1] |
E. Beretta and Y. Takeuchi, Qualitative properties of chemostat equations with time delays, Diff. Equ. Dyn. Sys., 2 (1994), 19-40; 263-288. |
[2] |
G. J. Butler and G. S. K. Wolkowicz, A mathematical model of the chemostat with a general class of functions describing nutrient uptake, SIAM J. Appl. Math., 45 (1985), 138-151.
doi: 10.1137/0145006. |
[3] |
F. Chen and Y. Jiang, "Microalgal Biotechnology," Chinese Light Industry Press, Beijing, 1999. |
[4] |
L. Chen, "Nonlinear Biological Dynamical Systems," Science Press, Beijing, 1993. |
[5] |
A. Cunningham and P. Maas, Time lag and nutrient storage effects in the transient growth response of Chlamydomonas reinhardii in nitrogen-limited batch and continuous culture, J. Gen. Microbiol., 104 (1978), 227-231. |
[6] |
A. Cunningham and R. M. Nisbet, Time lag and co-operativity in the transient growth dynamics of microalgae, J. Theor. Biol., 84 (1980), 189-203.
doi: 10.1016/S0022-5193(80)80003-8. |
[7] |
S. F. Ellermeyer, S. S. Pilyugin and Ray Redheffer, Persistence criteria for a chemostat with variable nutrient input, J. Diff. Eq., 171 (2001), 132-147. |
[8] |
H. Endo, H. Hosoya and T. Koibuchi, Growth yields of Chlorella regularis in dark-heterotrophic continuous cultures using acetate, J. Ferment. Technol., 55 (1977), 369-379. |
[9] |
J. K. Hale, "Ordinary Differential Equations," Second edition, Robert E. Krieger Publishing Company, Inc., Huntington, New York, 1980. |
[10] |
S. Han, Z. Zhang and H. Liu, Effects of Chlorella growth factor on physiological function, Chinese J. Biochem. Pharmaceutics, 25 (2004), 5-7. |
[11] |
S. R. Hansen and S. P. Hubbell, Single-nutrient microbial competition: Qualitative agreement between experimental and theoretically forecast outcomes, Science, 207 (1980), 1491-1493.
doi: 10.1126/science.6767274. |
[12] |
F. Khacik, Process for isolation, purification, and recrystallization of lutein from saponified marigold oleoresin and uses thereof: US patent, 5382714, 1995-01-17. |
[13] |
J. T. Landrum and R. A. Bone, Lutein, zeaxanthin, and the macular pigment, Arch. Biochem. Biophys., 385 (2001), 28-40.
doi: 10.1006/abbi.2000.2171. |
[14] |
J. A. Leon and D. B. Tumpson, Competition between two species for two complementary or substitutable resources, J. Theor. Biol., 50 (1975), 185-201.
doi: 10.1016/0022-5193(75)90032-6. |
[15] |
L. W. Levy, Trans-xanthophyll ester concentrates of enhanced purity and method of making same: US patent, 6191293, 2001-02-20. |
[16] |
B. Li, G. S. K. Wolkowicz and Y. Kuang, Global asymptotic behavior of a Chemostat model with two perfectly complementary resources and distributed delay, SIAM J. Appl. Math., 60 (2000), 2058-2086.
doi: 10.1137/S0036139999359756. |
[17] |
B. Li and H. L. Smith, Global dynamics of microbial competition for two resources with internal storage, J. Math. Biol., 55 (2007), 481-515.
doi: 10.1007/s00285-007-0092-8. |
[18] |
S. Liu, H. Meng, S. Liang, J. Yin and P. Mai, High-density heterotrophic culture of Chlorella Vulgaris in bioreactor, J. South China Univ. Tech., 28 (2000), 81-86. |
[19] |
D. L. Madhavi and D. I. Kagan, Process for the isolation of mixed carotenoids from plants: US patent, 6380442, 2002-04-30. |
[20] |
A. Narang and S. S. Pilyugin, Towards an integrated physiological theory of microbial growth: From subcellular variables to population dynamics, Math. Biosci. Eng., 2 (2005), 169-206. |
[21] |
J. C. Ogbonna, H. Masui and H. Tanaka, Sequential heterotrophic / autotrophic cultivation - An efficient method of producing Chlorella biomass for health food and animal feed, J. Appl. Phycol., 9 (1997), 359-366.
doi: 10.1023/A:1007981930676. |
[22] |
J. C. Ogbonna, S. Tomiyama and H. Tanaka, Heterotrophic cultivation of Euglena gracilis Z for efficient production of $\alpha$-tocopherol, J. Appl. Phycol., 10 (1998), 67-74.
doi: 10.1023/A:1008011201437. |
[23] |
T. Philip, Purification of lutein-fatty acid esters from plant materials: US patent, 4048203, 1977-09-13. |
[24] |
S. S. Pilyugin and P. Waltman, Multiple limit cycles in the chemostat with variable yield, Math. Biosci., 182 (2003), 151-166.
doi: 10.1016/S0025-5564(02)00214-6. |
[25] |
K. Sasaki, K. Watanabe, T. Tanaka, Y. Hotta and S. Nagai, 5-aminolevulinic acid production by Chlorella sp. during heterotrophic cultivation in the dark, World J. Microbiol. Biotech., 11 (1995), 361-362.
doi: 10.1007/BF00367123. |
[26] |
X. Shi, H. Liu, X. Zhang and F. Chen, Production of biomass and lutein by Chlorella protothecoides at various glucose concentrations in heterotrophic cultures, Process Biochem., 34 (1999), 341-347.
doi: 10.1016/S0032-9592(98)00101-0. |
[27] |
X. Shi, X. Zhang and F. Chen, Heterotrophic production of biomass and lutein Chlorella protothecoides on various nitrogen sources, Enzyme Microb. Technol., 27 (2000), 312-318.
doi: 10.1016/S0141-0229(00)00208-8. |
[28] |
H. L. Smith and P. Waltman, "The Theory of the Chemostat. Dynamics of Microbial Competition," Cambridge Studies in Mathematical Biology, 13, Cambridge University Press, Cambridge, 1995.
doi: 10.1017/CBO9780511530043. |
[29] |
H. L. Smith and P. Waltman, Competition for a single limiting resource in continuous culture: The variable-yield model, SIAM J. Appl. Math., 54 (1994), 1113-1131.
doi: 10.1137/S0036139993245344. |
[30] |
L. V. Thinh and D. J. Griffiths, Amino-acid composition of autotrophic and heterotrophic cultures of emerson strain of Chlorella, Plant Cell Physiol., 17 (1976), 193-196. |
[31] |
S. Wang, H. Yan, B. Zhang, L. Lv and H. Lin, Effects of various nitrogen sources and phytohormones on growth and content of lutin in Chlorella sp. USTB01, Sci. Tech. Review, 23 (2005), 37-40. |
[32] |
H. Xia, G. S. K. Wolkowicz and L. Wang, Transient oscillation induced by delayed growth response in the chemostat, J. Math. Biol., 50 (2005), 489-530.
doi: 10.1007/s00285-004-0311-5. |
[33] |
K. Yamaguchi, Recent advances in microalgal bioscience in Japan, with special reference to utilization of biomass and metabolites: A review, J. Appl. Phycol., 8 (1996), 487-502.
doi: 10.1007/BF02186327. |
[34] |
H. Yan, C. Ye and C. Yin, Kinetics of phthalate esters biodegradation by Chlorella pyrenoidosa, Environ. Toxicol. Chem., 14 (1995), 931-938. |
[35] |
H. Yan and G. Pan, Toxicity and bioaccumulation of copper in three green microalgal species, Chemosphere, 49 (2002),471-476.
doi: 10.1016/S0045-6535(02)00285-0. |
[36] |
H. Yan, J. Zhou, H. He, Y. Wei and J. Sun, Isolation and heterotrophic culture of Chlorella sp., J. Univ. Sci. Tech. Beijing, 27 (2005), 408-412. |
[37] |
H. Yan, B. Zhang, S. Wang, Y. Li, S. Liu and S. Yang, Advances in the heterotrophic culture of Chlorella sp., Modern Chem. Indust., 27 (2007), 18-21. |
[38] |
H. Zhang, S. Sun, K. Mai and Y. Liang, Advances in the studies on heterotrophic culture of microalgae, Trans. Oceanology Limnology, (2000), 51-59. |
[39] |
L. Zhang, R. Yang and H. Xiao, The heterotrophic culture of Chlorella and the optimization of growth condition, Guihaia, 24 (2001), 353-357. |
[40] |
H. Zhou, W. Lin and T. Chen, The heterotrophy and applications of Chlorella, Amino Acids Biotic Resources, 27 (2005), 69-73. |
show all references
References:
[1] |
E. Beretta and Y. Takeuchi, Qualitative properties of chemostat equations with time delays, Diff. Equ. Dyn. Sys., 2 (1994), 19-40; 263-288. |
[2] |
G. J. Butler and G. S. K. Wolkowicz, A mathematical model of the chemostat with a general class of functions describing nutrient uptake, SIAM J. Appl. Math., 45 (1985), 138-151.
doi: 10.1137/0145006. |
[3] |
F. Chen and Y. Jiang, "Microalgal Biotechnology," Chinese Light Industry Press, Beijing, 1999. |
[4] |
L. Chen, "Nonlinear Biological Dynamical Systems," Science Press, Beijing, 1993. |
[5] |
A. Cunningham and P. Maas, Time lag and nutrient storage effects in the transient growth response of Chlamydomonas reinhardii in nitrogen-limited batch and continuous culture, J. Gen. Microbiol., 104 (1978), 227-231. |
[6] |
A. Cunningham and R. M. Nisbet, Time lag and co-operativity in the transient growth dynamics of microalgae, J. Theor. Biol., 84 (1980), 189-203.
doi: 10.1016/S0022-5193(80)80003-8. |
[7] |
S. F. Ellermeyer, S. S. Pilyugin and Ray Redheffer, Persistence criteria for a chemostat with variable nutrient input, J. Diff. Eq., 171 (2001), 132-147. |
[8] |
H. Endo, H. Hosoya and T. Koibuchi, Growth yields of Chlorella regularis in dark-heterotrophic continuous cultures using acetate, J. Ferment. Technol., 55 (1977), 369-379. |
[9] |
J. K. Hale, "Ordinary Differential Equations," Second edition, Robert E. Krieger Publishing Company, Inc., Huntington, New York, 1980. |
[10] |
S. Han, Z. Zhang and H. Liu, Effects of Chlorella growth factor on physiological function, Chinese J. Biochem. Pharmaceutics, 25 (2004), 5-7. |
[11] |
S. R. Hansen and S. P. Hubbell, Single-nutrient microbial competition: Qualitative agreement between experimental and theoretically forecast outcomes, Science, 207 (1980), 1491-1493.
doi: 10.1126/science.6767274. |
[12] |
F. Khacik, Process for isolation, purification, and recrystallization of lutein from saponified marigold oleoresin and uses thereof: US patent, 5382714, 1995-01-17. |
[13] |
J. T. Landrum and R. A. Bone, Lutein, zeaxanthin, and the macular pigment, Arch. Biochem. Biophys., 385 (2001), 28-40.
doi: 10.1006/abbi.2000.2171. |
[14] |
J. A. Leon and D. B. Tumpson, Competition between two species for two complementary or substitutable resources, J. Theor. Biol., 50 (1975), 185-201.
doi: 10.1016/0022-5193(75)90032-6. |
[15] |
L. W. Levy, Trans-xanthophyll ester concentrates of enhanced purity and method of making same: US patent, 6191293, 2001-02-20. |
[16] |
B. Li, G. S. K. Wolkowicz and Y. Kuang, Global asymptotic behavior of a Chemostat model with two perfectly complementary resources and distributed delay, SIAM J. Appl. Math., 60 (2000), 2058-2086.
doi: 10.1137/S0036139999359756. |
[17] |
B. Li and H. L. Smith, Global dynamics of microbial competition for two resources with internal storage, J. Math. Biol., 55 (2007), 481-515.
doi: 10.1007/s00285-007-0092-8. |
[18] |
S. Liu, H. Meng, S. Liang, J. Yin and P. Mai, High-density heterotrophic culture of Chlorella Vulgaris in bioreactor, J. South China Univ. Tech., 28 (2000), 81-86. |
[19] |
D. L. Madhavi and D. I. Kagan, Process for the isolation of mixed carotenoids from plants: US patent, 6380442, 2002-04-30. |
[20] |
A. Narang and S. S. Pilyugin, Towards an integrated physiological theory of microbial growth: From subcellular variables to population dynamics, Math. Biosci. Eng., 2 (2005), 169-206. |
[21] |
J. C. Ogbonna, H. Masui and H. Tanaka, Sequential heterotrophic / autotrophic cultivation - An efficient method of producing Chlorella biomass for health food and animal feed, J. Appl. Phycol., 9 (1997), 359-366.
doi: 10.1023/A:1007981930676. |
[22] |
J. C. Ogbonna, S. Tomiyama and H. Tanaka, Heterotrophic cultivation of Euglena gracilis Z for efficient production of $\alpha$-tocopherol, J. Appl. Phycol., 10 (1998), 67-74.
doi: 10.1023/A:1008011201437. |
[23] |
T. Philip, Purification of lutein-fatty acid esters from plant materials: US patent, 4048203, 1977-09-13. |
[24] |
S. S. Pilyugin and P. Waltman, Multiple limit cycles in the chemostat with variable yield, Math. Biosci., 182 (2003), 151-166.
doi: 10.1016/S0025-5564(02)00214-6. |
[25] |
K. Sasaki, K. Watanabe, T. Tanaka, Y. Hotta and S. Nagai, 5-aminolevulinic acid production by Chlorella sp. during heterotrophic cultivation in the dark, World J. Microbiol. Biotech., 11 (1995), 361-362.
doi: 10.1007/BF00367123. |
[26] |
X. Shi, H. Liu, X. Zhang and F. Chen, Production of biomass and lutein by Chlorella protothecoides at various glucose concentrations in heterotrophic cultures, Process Biochem., 34 (1999), 341-347.
doi: 10.1016/S0032-9592(98)00101-0. |
[27] |
X. Shi, X. Zhang and F. Chen, Heterotrophic production of biomass and lutein Chlorella protothecoides on various nitrogen sources, Enzyme Microb. Technol., 27 (2000), 312-318.
doi: 10.1016/S0141-0229(00)00208-8. |
[28] |
H. L. Smith and P. Waltman, "The Theory of the Chemostat. Dynamics of Microbial Competition," Cambridge Studies in Mathematical Biology, 13, Cambridge University Press, Cambridge, 1995.
doi: 10.1017/CBO9780511530043. |
[29] |
H. L. Smith and P. Waltman, Competition for a single limiting resource in continuous culture: The variable-yield model, SIAM J. Appl. Math., 54 (1994), 1113-1131.
doi: 10.1137/S0036139993245344. |
[30] |
L. V. Thinh and D. J. Griffiths, Amino-acid composition of autotrophic and heterotrophic cultures of emerson strain of Chlorella, Plant Cell Physiol., 17 (1976), 193-196. |
[31] |
S. Wang, H. Yan, B. Zhang, L. Lv and H. Lin, Effects of various nitrogen sources and phytohormones on growth and content of lutin in Chlorella sp. USTB01, Sci. Tech. Review, 23 (2005), 37-40. |
[32] |
H. Xia, G. S. K. Wolkowicz and L. Wang, Transient oscillation induced by delayed growth response in the chemostat, J. Math. Biol., 50 (2005), 489-530.
doi: 10.1007/s00285-004-0311-5. |
[33] |
K. Yamaguchi, Recent advances in microalgal bioscience in Japan, with special reference to utilization of biomass and metabolites: A review, J. Appl. Phycol., 8 (1996), 487-502.
doi: 10.1007/BF02186327. |
[34] |
H. Yan, C. Ye and C. Yin, Kinetics of phthalate esters biodegradation by Chlorella pyrenoidosa, Environ. Toxicol. Chem., 14 (1995), 931-938. |
[35] |
H. Yan and G. Pan, Toxicity and bioaccumulation of copper in three green microalgal species, Chemosphere, 49 (2002),471-476.
doi: 10.1016/S0045-6535(02)00285-0. |
[36] |
H. Yan, J. Zhou, H. He, Y. Wei and J. Sun, Isolation and heterotrophic culture of Chlorella sp., J. Univ. Sci. Tech. Beijing, 27 (2005), 408-412. |
[37] |
H. Yan, B. Zhang, S. Wang, Y. Li, S. Liu and S. Yang, Advances in the heterotrophic culture of Chlorella sp., Modern Chem. Indust., 27 (2007), 18-21. |
[38] |
H. Zhang, S. Sun, K. Mai and Y. Liang, Advances in the studies on heterotrophic culture of microalgae, Trans. Oceanology Limnology, (2000), 51-59. |
[39] |
L. Zhang, R. Yang and H. Xiao, The heterotrophic culture of Chlorella and the optimization of growth condition, Guihaia, 24 (2001), 353-357. |
[40] |
H. Zhou, W. Lin and T. Chen, The heterotrophy and applications of Chlorella, Amino Acids Biotic Resources, 27 (2005), 69-73. |
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