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Properties of the chemostat model with aggregated biomass

Published online by Cambridge University Press:  27 March 2018

ALAIN RAPAPORT
ALAIN RAPAPORT*
Affiliation:
MISTEA, University of Montpellier, INRA, Montpellier SupAgro, France email: alain.rapaport@inra.fr
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Abstract

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We revisit the well-known chemostat model, considering that bacteria can be attached together in aggregates or flocs. We distinguish explicitly free and attached compartments in the model and give sufficient conditions for coexistence of these two forms. We then study the case of fast attachment and detachment and show how it is related to density-dependent growth functions. Finally, we give some insights concerning the cases of multi-specific flocs and different removal rates.

Keywords

92B0592D2537N2534A34

Information

Type
Papers
Information
European Journal of Applied Mathematics , Volume 29 , Special Issue 6: Biofilms , December 2018 , pp. 972 - 990
Copyright
Copyright © Cambridge University Press 2018 

References

[1] Ballyk, M., Jones, D. & Smith, H. (2008) The Biofilm Model of Freter: A Review, Structured Population Models in Biology and Epidemiology, Lecture Notes in Mathematics, Magal, P. & Ruan, S. (editors), Springer-Verlag, New-York pp. 265–302.Google Scholar
[2] Ballyk, M. & Smith, H. (1999) A model of microbial growth in a plug flow reactor with wall attachment. Math. Biosci. 158, 95–126.Google Scholar
[3] Berlin, A. & Kislenko, V. (1995) Kinetic models of suspension flocculation by polymers. Colloids Surf. A: Physicochem. Eng. Asp. 104, 67–72.Google Scholar
[4] Contois, D. (1959) Kinetics of bacterial growth: Relationship between population density and specific growth rate of continuous cultures. J. Gen. Microbiol. 21, 40–50.Google Scholar
[5] Costeron, J. (1995) Overview of microbial biofilms. J. Indust. Microbiol. 15, l37–140.Google Scholar
[6] De Leenheer, P., Angeli, D. & Sontag, E. (2006) Crowding effects promote coexistence in the chemostat. J. Math. Anal. Appl. 319 (1), 48–60.Google Scholar
[7] Fekih-Salem, R. (2013) Modéles mathématiques pour la compétition et la coexistence des espéces microbiennes dans un chémostat. PhD thesis, University of Montpellier II and University of Tunis el Manar. https://tel.archives-ouvertes.fr/tel-01018600.Google Scholar
[8] Fekih-Salem, R., Harmand, J., Lobry, C., Rapaport, A. & Sari, T. (2013) Extensions of the chemostat model with flocculation. J. Math. Anal. Appl. 397, 292–306.Google Scholar
[9] Fekih-Salem, R., Rapaport, A. & Sari, T. (2016) Emergence of coexistence and limit cycles in the chemostat model with flocculation for a general class of functional responses. Appl. Math. Model. 40, 7656–7677.Google Scholar
[10] Freter, R., Brickner, H., Fekete, J., Vickerman, M. & Carey, K. (1983) Survival and implantation of escherichia coli in the intestinal tract. Infect. Immun. 39, 686–703.Google Scholar
[11] Haegeman, B., Lobry, C. & Harmand, J. (2007) Modeling bacteria flocculation as density-dependent growth. AIChE J. 53 (2), 535–539.Google Scholar
[12] Haegeman, B. & Rapaport, A. (2008) How flocculation can explain coexistence in the chemostat. J. Biol. Dyn. 2, 1–13.Google Scholar
[13] Harmand, J. & Godon, J. J. (2007) Density-dependent kinetics models for a simple description of complex phenomena in macroscopic mass-balance modeling of bioreactors. Ecol. Modelling 200 (3–4), 393–402.Google Scholar
[14] Harmand, J., Lobry, C., Rapaport, A. & Sari, T. (2017) The Chemostat, Mathematical Theory of the Continuous Culture of Micro-Organisms, Wiley-ISTE, London.Google Scholar
[15] Heffernan, B., Murphy, C. & Casey, E. (2009) Comparison of planktonic and biofilm cultures of Pseudomonas fluorescens DSM 8341 cells grown on fluoroacetate. Appl. Environ. Microbiol. 75, 2899–2907.Google Scholar
[16] IWA Task Group on Biofilm Modeling (2006) Mathematical Modeling of Biofilms. IWA Publishing, London.Google Scholar
[17] Jones, D., Kojouharov, H., Le, D. & Smith, H. (2003) The Freter model: A simple model of biofilm formation. J. Math. Biol. 47, 137–152.Google Scholar
[18] Khalil, H. (1996) Nonlinear Systems, Prentice Hall, Upper Saddle River (NJ).Google Scholar
[19] Lobry, C., Mazenc, F. & Rapaport, A. (2005) Persistence in ecological models of competition for a single resource. C. R. Math. 340 (3), 199–204.Google Scholar
[20] Lobry, C., Sari, T. & Touhami, S. (1995) On Tikhonov's theorem for convergence of solutions of slow and fast systems. Electron. J. Differ. Equ. 19, 1–22.Google Scholar
[21] Mischaikow, M., Smith, H. & Thieme, H. (1995) Asymptotically autonomous semiflows: Chain recurrence and Lyapunov functions. Trans. Am. Math. Soc. 347 (5), 1669–1685.Google Scholar
[22] Pilyugin, S. & Waltman, P. (1999) The simple chemostat with wall growth. SIAM J. Appl. Math. 59, 1552–1572.Google Scholar
[23] Smith, H. & Waltman, P. (1995) The Theory of the Chemostat: Dynamics of Microbial Competition, Vol. 13, Cambridge University Press, New-York.Google Scholar
[24] Stemmons, E. & Smith, H. (2000) Competition in a chemostat with wall attachment. SIAM J. Appl. Math. 61, 567–595.Google Scholar
[25] Tang, B., Sitomer, A. & Jackson, T. (1997) Population dynamics and competition in chemostat models with adaptive nutrient uptake. J. Math. Biol. 35, 453–479.Google Scholar
[26] Thomas, D., Judd, S. & Fawcett, N. (1999) Flocculation modelling: A review. Water Res. 33, 1579–1592.Google Scholar