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Effects of spatial structure and diffusion on the performances of the chemostat
1.  UMR INRA/SupAgro 'MISTEA' and EPI INRA/INRIA 'MODEMIC', 2, pl. Viala 34060, Montpellier, France 
2.  UMR Analyse des Systèmes et Biométrie, INRA, EPI INRA/INRIA 'MODEMIC', 2 pl. Viala 34060 Montpellier 
3.  UMR INRA/SupAgro/CIRAD/IRD 'Eco&Sols', 2, pl. Viala 34060, Montpellier, France 
References:
[1] 
C. de Gooijer, W. Bakker, H. Beeftink and J. Tramper, Bioreactors in series: An overview of design procedures and practical applications, Enzyme and Microbial Technology, 18 (1996), 202219. 
[2] 
C. de Gooijer, H. Beeftink and J. Tramper, Optimal design of a series of continuous stirred tank reactors containing immobilised growing cells, Biotechnology Letters, 18 (1996), 397402. 
[3] 
P. Doran, Design of mixing systems for plant cell suspensions in stirred reactors, Biotechnology Progress, 15 (1999), 319335. 
[4] 
A. Dramé, J. Harmand, A. Rapaport and C. Lobry, Multiple steady state profiles in interconnected biological systems, Mathematical and Computer Modelling of Dynamical Systems, 12 (2006), 379393. 
[5] 
A. Dramé, C. Lobry, J. Harmand, A. Rapaport and F. Mazenc, Multiple stable equilibrium profiles in tubular bioreactors, Mathematical and Computer Modelling, 48 (2008), 18401853. 
[6] 
S. Foger, "Elements of Chemical Reaction Engineering," 4^{th} edition, Prentice Hall, New York, 2008. 
[7] 
A. Grobicki and D. Stuckey, Hydrodynamic characteristics of the anaerobic baffled reactor, Water Research, 26 (1992), 371378. 
[8] 
L. Grady, G. Daigger and H. Lim, "Biological Wastewater Treatment,'' 3^{nd} edition, Environmental Science and Pollution Control Series, Marcel Dekker, New York, 1999. 
[9] 
D. Gravel, F. Guichard, M. Loreau and N. Mouquet, Source and sink dynamics in metaecosystems, Ecology, 91 (2010), 21722184. 
[10] 
I. Hanski, "Metapopulation Ecology,'' Oxford University Press, 1999. 
[11] 
J. Harmand, A. Rapaport and A. Trofino, Optimal design of two interconnected bioreactorssome new results, American Institute of Chemical Engineering Journal, 49 (1999), 14331450. 
[12] 
J. Harmand, A. Rapaport and A. Dramé, Optimal design of two interconnected enzymatic reactors, Journal of Process Control, 14 (2004), 785794. 
[13] 
J. Harmand and D. Dochain, Towards a unified approach for the design of interconnected catalytic and autocatalytic reactors, Computers and Chemical Engineering, 30 (2005), 7082. 
[14] 
G. Hill and C. Robinson, Minimum tank volumes for CFST bioreactors in series, The Canadian Journal of Chemical Engineering, 67 (1989), 818824. 
[15] 
W. Hu, K. Wlashchin, M. Betenbaugh, F. Wurm, G. Seth and W. Zhou, "Cellular Bioprocess Technology, Fundamentals and Frontier,'' Lectures Notes, University of Minesota, 2007. 
[16] 
O. Levenspiel, "Chemical Reaction Engineering,'' 3^{nd} edition, Wiley, New York, 1999. 
[17] 
R. Lovitt and J. Wimpenny, The gradostat: A bidirectional compound chemostat and its applications in microbial research, Journal of General Microbiology, 127 (1981), 261268. 
[18] 
K. Luyben and J. Tramper, Optimal design for continuously stirred tank reactors in series using MichaelisMenten kinetics, Biotechnology and Bioengineering, 24 (1982), 12171220. 
[19] 
R. MacArthur and E. Wilson, "The Theory of Island Biogeography,'' Princeton University Press, 1967. 
[20] 
K. Mischaikow, H. Smith and H. Thieme, Asymptotically autonomous semiflows: Chain recurrence and Lyapunov functions, Transactions of the American Mathematical Society, 347 (1995), 16691685. doi: 10.2307/2154964. 
[21] 
J. Monod, La technique de la culture continue: Théorie et applications, Annales de l'Institut Pasteur, 79 (1950), 390410. 
[22] 
S. Nakaoka and Y. Takeuchi, Competition in chemostattype equations with two habitats, Mathematical Bioscience, 201 (2006), 157171. 
[23] 
M. Nelson and H. Sidhu, Evaluating the performance of a cascade of two bioreactors, Chemical Engineering Science, 61 (2006), 31593166. 
[24] 
A. Novick and L. Szilard, Description of the chemostat, Science, 112 (1950), 715716. 
[25] 
A. Rapaport, J. Harmand and F. Mazenc, Coexistence in the design of a series of two chemostats, Nonlinear Analysis, Real World Applications, 9 (2008), 10521067. 
[26] 
E. Roca, C. Ghommidh, J.M. Navarro and J.M. Lema, Hydraulic model of a gaslift bioreactor with flocculating yeast, Bioprocess and Biosystems Engineering, 12 (1995), 269272. 
[27] 
G. Roux, B. Dahhou and I. Queinnec, Adaptive nonlinear control of a continuous stirred tank bioreactor, Journal of Process Control, 4 (1994), 121126. 
[28] 
A. Saddoud, T. Sari, A. Rapaport, R. Lortie, J. Harmand and E. Dubreucq, A mathematical study of an enzymatic hydrolysis of a cellulosic substrate in non homogeneous reactors, Proceedings of the IFAC Computer Applications in Biotechnology Conference (CAB 2010), Leuven, Belgium, July 79, 2010. 
[29] 
A. Scheel and E. Van Vleck, Lattice differential equations embedded into reactiondiffusion systems, Proceedings of the Royal Society Edinburgh Section A, 139 (2009), 193207. 
[30] 
H. Smith and P. Waltman, "The Theory of Chemostat. Dynamics of Microbial Competition,'' Cambridge Studies in Mathematical Biology, 13, Cambridge University Press, Cambridge, 1995. doi: 10.1017/CBO9780511530043. 
[31] 
G. Stephanopoulos and A. Fredrickson, Effect of inhomogeneities on the coexistence of competing microbial populations, Biotechnology and Bioengineering, 21 (1979), 14911498. 
[32] 
R. Schwartz, A. Juo and K. McInnes, Estimating parameters for a dualporosity model to describe nonequilibrium, reactive transport in a finetextured soil, Journal of Hydrology, 229 (2000), 149167. 
[33] 
C. Tsakiroglou and M. Ioannidis, Dualporosity modelling of the pore structure and transport properties of a contaminated soil, European Journal of Soil Science, 59 (2008), 744761. 
[34] 
F. ValdesParada, J. AlvarezRamirez and A. OchoaTapia, An approximate solution for a transient twophase stirred tank bioreactor with nonlinear kinetics, Biotechnology Progress, 21 (2005), 14201428. 
[35] 
K. Van't Riet and J. Tramper, "Basic Bioreactor Design,'' Marcel Dekker, New York, 1991. 
show all references
References:
[1] 
C. de Gooijer, W. Bakker, H. Beeftink and J. Tramper, Bioreactors in series: An overview of design procedures and practical applications, Enzyme and Microbial Technology, 18 (1996), 202219. 
[2] 
C. de Gooijer, H. Beeftink and J. Tramper, Optimal design of a series of continuous stirred tank reactors containing immobilised growing cells, Biotechnology Letters, 18 (1996), 397402. 
[3] 
P. Doran, Design of mixing systems for plant cell suspensions in stirred reactors, Biotechnology Progress, 15 (1999), 319335. 
[4] 
A. Dramé, J. Harmand, A. Rapaport and C. Lobry, Multiple steady state profiles in interconnected biological systems, Mathematical and Computer Modelling of Dynamical Systems, 12 (2006), 379393. 
[5] 
A. Dramé, C. Lobry, J. Harmand, A. Rapaport and F. Mazenc, Multiple stable equilibrium profiles in tubular bioreactors, Mathematical and Computer Modelling, 48 (2008), 18401853. 
[6] 
S. Foger, "Elements of Chemical Reaction Engineering," 4^{th} edition, Prentice Hall, New York, 2008. 
[7] 
A. Grobicki and D. Stuckey, Hydrodynamic characteristics of the anaerobic baffled reactor, Water Research, 26 (1992), 371378. 
[8] 
L. Grady, G. Daigger and H. Lim, "Biological Wastewater Treatment,'' 3^{nd} edition, Environmental Science and Pollution Control Series, Marcel Dekker, New York, 1999. 
[9] 
D. Gravel, F. Guichard, M. Loreau and N. Mouquet, Source and sink dynamics in metaecosystems, Ecology, 91 (2010), 21722184. 
[10] 
I. Hanski, "Metapopulation Ecology,'' Oxford University Press, 1999. 
[11] 
J. Harmand, A. Rapaport and A. Trofino, Optimal design of two interconnected bioreactorssome new results, American Institute of Chemical Engineering Journal, 49 (1999), 14331450. 
[12] 
J. Harmand, A. Rapaport and A. Dramé, Optimal design of two interconnected enzymatic reactors, Journal of Process Control, 14 (2004), 785794. 
[13] 
J. Harmand and D. Dochain, Towards a unified approach for the design of interconnected catalytic and autocatalytic reactors, Computers and Chemical Engineering, 30 (2005), 7082. 
[14] 
G. Hill and C. Robinson, Minimum tank volumes for CFST bioreactors in series, The Canadian Journal of Chemical Engineering, 67 (1989), 818824. 
[15] 
W. Hu, K. Wlashchin, M. Betenbaugh, F. Wurm, G. Seth and W. Zhou, "Cellular Bioprocess Technology, Fundamentals and Frontier,'' Lectures Notes, University of Minesota, 2007. 
[16] 
O. Levenspiel, "Chemical Reaction Engineering,'' 3^{nd} edition, Wiley, New York, 1999. 
[17] 
R. Lovitt and J. Wimpenny, The gradostat: A bidirectional compound chemostat and its applications in microbial research, Journal of General Microbiology, 127 (1981), 261268. 
[18] 
K. Luyben and J. Tramper, Optimal design for continuously stirred tank reactors in series using MichaelisMenten kinetics, Biotechnology and Bioengineering, 24 (1982), 12171220. 
[19] 
R. MacArthur and E. Wilson, "The Theory of Island Biogeography,'' Princeton University Press, 1967. 
[20] 
K. Mischaikow, H. Smith and H. Thieme, Asymptotically autonomous semiflows: Chain recurrence and Lyapunov functions, Transactions of the American Mathematical Society, 347 (1995), 16691685. doi: 10.2307/2154964. 
[21] 
J. Monod, La technique de la culture continue: Théorie et applications, Annales de l'Institut Pasteur, 79 (1950), 390410. 
[22] 
S. Nakaoka and Y. Takeuchi, Competition in chemostattype equations with two habitats, Mathematical Bioscience, 201 (2006), 157171. 
[23] 
M. Nelson and H. Sidhu, Evaluating the performance of a cascade of two bioreactors, Chemical Engineering Science, 61 (2006), 31593166. 
[24] 
A. Novick and L. Szilard, Description of the chemostat, Science, 112 (1950), 715716. 
[25] 
A. Rapaport, J. Harmand and F. Mazenc, Coexistence in the design of a series of two chemostats, Nonlinear Analysis, Real World Applications, 9 (2008), 10521067. 
[26] 
E. Roca, C. Ghommidh, J.M. Navarro and J.M. Lema, Hydraulic model of a gaslift bioreactor with flocculating yeast, Bioprocess and Biosystems Engineering, 12 (1995), 269272. 
[27] 
G. Roux, B. Dahhou and I. Queinnec, Adaptive nonlinear control of a continuous stirred tank bioreactor, Journal of Process Control, 4 (1994), 121126. 
[28] 
A. Saddoud, T. Sari, A. Rapaport, R. Lortie, J. Harmand and E. Dubreucq, A mathematical study of an enzymatic hydrolysis of a cellulosic substrate in non homogeneous reactors, Proceedings of the IFAC Computer Applications in Biotechnology Conference (CAB 2010), Leuven, Belgium, July 79, 2010. 
[29] 
A. Scheel and E. Van Vleck, Lattice differential equations embedded into reactiondiffusion systems, Proceedings of the Royal Society Edinburgh Section A, 139 (2009), 193207. 
[30] 
H. Smith and P. Waltman, "The Theory of Chemostat. Dynamics of Microbial Competition,'' Cambridge Studies in Mathematical Biology, 13, Cambridge University Press, Cambridge, 1995. doi: 10.1017/CBO9780511530043. 
[31] 
G. Stephanopoulos and A. Fredrickson, Effect of inhomogeneities on the coexistence of competing microbial populations, Biotechnology and Bioengineering, 21 (1979), 14911498. 
[32] 
R. Schwartz, A. Juo and K. McInnes, Estimating parameters for a dualporosity model to describe nonequilibrium, reactive transport in a finetextured soil, Journal of Hydrology, 229 (2000), 149167. 
[33] 
C. Tsakiroglou and M. Ioannidis, Dualporosity modelling of the pore structure and transport properties of a contaminated soil, European Journal of Soil Science, 59 (2008), 744761. 
[34] 
F. ValdesParada, J. AlvarezRamirez and A. OchoaTapia, An approximate solution for a transient twophase stirred tank bioreactor with nonlinear kinetics, Biotechnology Progress, 21 (2005), 14201428. 
[35] 
K. Van't Riet and J. Tramper, "Basic Bioreactor Design,'' Marcel Dekker, New York, 1991. 
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