The mathematical analysis of a syntrophic relationship between two microbial species in a chemostat

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2012, 9(3): 627-645. doi: 10.3934/mbe.2012.9.627

The mathematical analysis of a syntrophic relationship between two microbial species in a chemostat

Tewfik Sari ^1, , Miled El Hajji ^2, and Jérôme Harmand ^3,

1.	Irstea, UMR ITAP, 361 rue Jean-François Breton 34196 Montpellier, & Modemic (Inra/Inria), UMR Mistea, 2 place Viala, 34060 Montpellier, France
2.	ISSATSO (Université de Sousse) Cité Taffala, 4003 Sousse, & LAMSIN-ENIT, Université Tunis El-manar BP 37, 1002 Tunis, Tunisia
3.	INRA UR0050, Laboratoire de Biotechnologie de l’Environnement, Avenue des Étangs, 11100 Narbonne, and Modemic (Inra/Inria), UMR Mistea, 2 place Viala, 34060 Montpellier, France

Received November 2011 Revised April 2012 Published July 2012

Abstract
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A mathematical model involving a syntrophic relationship between two populations of bacteria in a continuous culture is proposed. A detailed qualitative analysis is carried out as well as the analysis of the local and global stability of the equilibria. We demonstrate, under general assumptions of monotonicity which are relevant from an applied point of view, the asymptotic stability of the positive equilibrium point which corresponds to the coexistence of the two bacteria. A syntrophic relationship in the anaerobic digestion process is proposed as a real candidate for this model.

Keywords: global asymptotic stability, syntrophic relationship, coexistence, Chemostat, anaerobic digestion..

Mathematics Subject Classification: Primary: 92A15, 92A17; Secondary: 34C15, 34C3.

Citation: Tewfik Sari, Miled El Hajji, Jérôme Harmand. The mathematical analysis of a syntrophic relationship between two microbial species in a chemostat. Mathematical Biosciences & Engineering, 2012, 9 (3) : 627-645. doi: 10.3934/mbe.2012.9.627

References:

[1]	Y. Aota and H. Nakajima, Mutualistic relationships between phytoplankton and bacteria caused by carbon excretion from phytoplankton, Ecological Research, 16 (2001), 289-299. doi: 10.1046/j.1440-1703.2001.00396.x.
[2]	M. M. Ballyk and G. S. K. Wolkowicz, Classical and resource-based competition: A unifying graphical approach, J. Math. Biol., 62 (2011), 81-109. doi: 10.1007/s00285-010-0328-x.
[3]	G. Bratbak and T. F. Thingstad, Phytoplankton-bacteria interactions: An apparent paradox? Analysis of a model system with both competition and commensalism, Ecological Research, 25 (1985), 23-30.
[4]	M. P. Bryant, E. A. Wolin, M. J. Wolin and R. S. Wolfe, Methanobacillus omelianskii, a symbiotic association of two species of bacteria, Arch. Microbiol., 59 (1967), 20-31.
[5]	J. Chase and M. Leibold, "Ecological Niches - Linking Classical and Contemporary Approaches," The University of Chicago Press, 2003.
[6]	M. El Hajji, J. Harmand, H. Chaker and C. Lobry, Association between competition and obligate mutualism in a chemostat, J. Biol. Dynamics, 3 (2009), 635-647.
[7]	M. El Hajji, F. Mazenc and J. Harmand, A mathematical study of a syntrophic relationship of a model of anaerobic digestion process, Math. Biosci. Eng., 7 (2010), 641-656.
[8]	M. El Hajji and A. Rapaport, Practical coexistence of two species in the chemostat-A slow-fast characterization, Math. Biosci., 218 (2009), 33-39.
[9]	M. El Hajji, T. Sari and J. Harmand, Analyse d'un relation syntrophique: Cas d'un chemostat, in "Proceedings of the 5^th Conference on Trends in Applied Mathematics in Tunisia, Algeria, Morocco, Sousse" (eds. M. Hassine and M. Moakher), 23-26 Avril 2011, Centre de Publication Universitaire, Tunisia, (2011), 451-456.
[10]	H. I. Freedman, R. Kumar, A. K. Easton and M. Singh, Mathematical models of predator mutualists, Canadian Appl. Math. Quart., 9 (2001), 99-111.
[11]	C. Katsuyama, S. Nakaoka, Y. Takeuchi, K. Tago, M. Hayatsu and K. Kato, A mathematical model of syntrophic cocultures in the chemostat, J. Theor. Biol., 256 (2009), 644-654.
[12]	R. Kreikenbohm and E. Bohl, A mathematical model of syntrophic cocultures in the chemostat, FEMS Microbiol. Ecol., 38 (1986), 131-140.
[13]	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.
[14]	H. R. Thieme, Convergence results and a Poincaré-Bendixson trichotomy for asymptotically autonomous differential equations, J. Math. Biol., 30 (1992), 755-763.
[15]	The IWA Task Group on Mathematical Modelling for Design and Operation of Biological Wastewater Treatment, Activated sludge models ASM1, ASM2, ASM2d and ASM3, Scientific and Technical Report No. 9, (2000), IWA publishing, 130 pp.

show all references

References:

[1]	Y. Aota and H. Nakajima, Mutualistic relationships between phytoplankton and bacteria caused by carbon excretion from phytoplankton, Ecological Research, 16 (2001), 289-299. doi: 10.1046/j.1440-1703.2001.00396.x.
[2]	M. M. Ballyk and G. S. K. Wolkowicz, Classical and resource-based competition: A unifying graphical approach, J. Math. Biol., 62 (2011), 81-109. doi: 10.1007/s00285-010-0328-x.
[3]	G. Bratbak and T. F. Thingstad, Phytoplankton-bacteria interactions: An apparent paradox? Analysis of a model system with both competition and commensalism, Ecological Research, 25 (1985), 23-30.
[4]	M. P. Bryant, E. A. Wolin, M. J. Wolin and R. S. Wolfe, Methanobacillus omelianskii, a symbiotic association of two species of bacteria, Arch. Microbiol., 59 (1967), 20-31.
[5]	J. Chase and M. Leibold, "Ecological Niches - Linking Classical and Contemporary Approaches," The University of Chicago Press, 2003.
[6]	M. El Hajji, J. Harmand, H. Chaker and C. Lobry, Association between competition and obligate mutualism in a chemostat, J. Biol. Dynamics, 3 (2009), 635-647.
[7]	M. El Hajji, F. Mazenc and J. Harmand, A mathematical study of a syntrophic relationship of a model of anaerobic digestion process, Math. Biosci. Eng., 7 (2010), 641-656.
[8]	M. El Hajji and A. Rapaport, Practical coexistence of two species in the chemostat-A slow-fast characterization, Math. Biosci., 218 (2009), 33-39.
[9]	M. El Hajji, T. Sari and J. Harmand, Analyse d'un relation syntrophique: Cas d'un chemostat, in "Proceedings of the 5^th Conference on Trends in Applied Mathematics in Tunisia, Algeria, Morocco, Sousse" (eds. M. Hassine and M. Moakher), 23-26 Avril 2011, Centre de Publication Universitaire, Tunisia, (2011), 451-456.
[10]	H. I. Freedman, R. Kumar, A. K. Easton and M. Singh, Mathematical models of predator mutualists, Canadian Appl. Math. Quart., 9 (2001), 99-111.
[11]	C. Katsuyama, S. Nakaoka, Y. Takeuchi, K. Tago, M. Hayatsu and K. Kato, A mathematical model of syntrophic cocultures in the chemostat, J. Theor. Biol., 256 (2009), 644-654.
[12]	R. Kreikenbohm and E. Bohl, A mathematical model of syntrophic cocultures in the chemostat, FEMS Microbiol. Ecol., 38 (1986), 131-140.
[13]	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.
[14]	H. R. Thieme, Convergence results and a Poincaré-Bendixson trichotomy for asymptotically autonomous differential equations, J. Math. Biol., 30 (1992), 755-763.
[15]	The IWA Task Group on Mathematical Modelling for Design and Operation of Biological Wastewater Treatment, Activated sludge models ASM1, ASM2, ASM2d and ASM3, Scientific and Technical Report No. 9, (2000), IWA publishing, 130 pp.

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