Analysis of a model for the effects of an external toxin on anaerobic digestion

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2012, 9(2): 445-459. doi: 10.3934/mbe.2012.9.445

Analysis of a model for the effects of an external toxin on anaerobic digestion

Marion Weedermann ^1,

1.	Dominican University, 7900 W Division St, River Forest, IL 60305, United States

Received November 2010 Revised August 2011 Published March 2012

Abstract
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Anaerobic digestion has been modeled as a two-stage process using coupled chemostat models with non-monotone growth functions, [9]. This study incorporates the effects of an external toxin. After reducing the model to a 3-dimensional system, global stability of boundary and interior equilibria is proved using differential inequalities and comparisons to the corresponding toxin-free model. Conditions are given under which the behavior of the toxin-free model is preserved. Introduction of the toxin results in additional patterns such as bistabilities of coexistence steady states or of a periodic orbit and an interior steady state.

Keywords: bistability., non-monotone growth, chemostat, Anaerobic digestion, global stability.

Mathematics Subject Classification: Primary: 34D23, 92D25; Secondary: 34D2.

Citation: Marion Weedermann. Analysis of a model for the effects of an external toxin on anaerobic digestion. Mathematical Biosciences & Engineering, 2012, 9 (2) : 445-459. doi: 10.3934/mbe.2012.9.445

References:

[1]	, "Agricultural Biogas Casebook," Great Lakes Regional Biomass Energy Program. Available from: http://www.cglg.org/biomass/pub/AgriculturalBiogasCasebook.pdf.
[2]	D. J. Batstone, J. Keller, I. Angelidaki, S. Kalyhuzhnyi, S. G. Pavlosthathis, A. Rozzi, W. Sanders, H. Siegrist and V. Vavilin, (IWA Task Group on Modeling Anaerobic Digestion Processes), "Anaerobic Digestion Model No.1 (ADM1)," IWA Publishing, London, UK, 2002.
[3]	T. Barkay and I. Wagner-Döbler, Microbial transformations of mercury: Potentials, challenges, and achievements in controlling mercury toxicity in the environment, Adv. Appl. Microbiol., 57 (2005), 1-40. doi: 10.1016/S0065-2164(05)57001-1.
[4]	J. P. Braselton and P. Waltman, A competition model with dynamically allocated inhibitor production, Math. Biosci., 173 (2001), 55-84. doi: 10.1016/S0025-5564(01)00078-5.
[5]	J. D. Bryers, Structured modeling of anaerobic digestion of biomass particulates, Biotechn. Bioeng., 27 (1984), 638-649. doi: 10.1002/bit.260270514.
[6]	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.
[7]	M. Gerardi, "The Microbiology of Anaerobic Digesters,'' Wiley Interscience, Hoboken, 2003. doi: 10.1002/0471468967.
[8]	M. El Hajji, F. Mazenc and J. Harmand, A mathematical study of syntrophic relationship of a model of anaerobic digestion process, Math. Biosci. Eng., 7 (2010), 641-656. doi: 10.3934/mbe.2010.7.641.
[9]	J. Hess and O. Bernard, Design and study of a risk management criterion for an unstable wastewater treatment process, J. of Process Control, 18 (2008), 71-79. doi: 10.1016/j.jprocont.2007.05.005.
[10]	S.-B. Hsu and P. Waltman, A survey of mathematical models of competition with an inhibitor, Math. Biosci., 187 (2004), 53-91. doi: 10.1016/j.mbs.2003.07.004.
[11]	S. Jeyaseelan, A simple mathematical model for anaerobic digestion process, Wat. Sci. Tech., 35 (1997), 185-191. doi: 10.1016/S0273-1223(97)00166-2.
[12]	B. Li, Global asymptotic behaviour of the chemostat: General response function and differential removal rates, SIAM J. Appl. Math., 59 (1999), 411-422. doi: 10.1137/S003613999631100X.
[13]	G. Lyberatos and I. V. Skiadas, Modelling of anaerobic digestion-a review, Global Nest: The Int. J., 1 (1999), 63-76.
[14]	L. Markus, Asymptotically autonomous differential systems, in "Contributions to the Theory of Nonlinear Oscillations,'' Princeton University Press, 3, Princeton, NJ, (1953), 17-29.
[15]	R. S. Oremland, C. W. Culbertson and M. Winfrey, Methylmercurcy decomposition in sediments and bacterial cultures: Involvement of methanogens and sulfate reducers in oxidative demethylation, Appl. Env. Microbiology, 57 (1991), 130-137.
[16]	K. Pak and R. Bartha, Products of mercury demethylation by sulfidogens and methanogens, Bull. Environ. Contam. Toxicol., 61 (1998), 690-694. doi: 10.1007/s001289900816.
[17]	G. E. Powell, Stable coexistence of syntrophic associations in continuous culture, J. Chem. Tech. Biotechnol. B, 35 (1985), 46-50. doi: 10.1002/jctb.280350109.
[18]	H. Smith and P. Waltman, "The Theory of the Chemostat: Dynamics of Microbial Competition,'' Cambridge Studies in Mathematical Biology, 13, Cambridge University Press, Cambridge, 1995.
[19]	H. Thieme, Convergence results and a Poincaré-Bendixson trichotomy for asymptotically autonomous differential equations, J. of Math. Biology, 30 (1992), 755-763.
[20]	G. S. K. Wolkowicz and Z. Lu, Global Dynamics of a mathematical model of competition in the chemostat: General response functions and differential death rates, SIAM J. Appl. Math., 52 (1992), 222-233. doi: 10.1137/0152012.
[21]	L. Zhu, X. Huang and H. Su, Bifurcation for a functional yield chemostat when one competitor produces a toxin, J. Math. Anal. Appl., 329 (2007), 891-903. doi: 10.1016/j.jmaa.2006.06.062.

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References:

[1]	, "Agricultural Biogas Casebook," Great Lakes Regional Biomass Energy Program. Available from: http://www.cglg.org/biomass/pub/AgriculturalBiogasCasebook.pdf.
[2]	D. J. Batstone, J. Keller, I. Angelidaki, S. Kalyhuzhnyi, S. G. Pavlosthathis, A. Rozzi, W. Sanders, H. Siegrist and V. Vavilin, (IWA Task Group on Modeling Anaerobic Digestion Processes), "Anaerobic Digestion Model No.1 (ADM1)," IWA Publishing, London, UK, 2002.
[3]	T. Barkay and I. Wagner-Döbler, Microbial transformations of mercury: Potentials, challenges, and achievements in controlling mercury toxicity in the environment, Adv. Appl. Microbiol., 57 (2005), 1-40. doi: 10.1016/S0065-2164(05)57001-1.
[4]	J. P. Braselton and P. Waltman, A competition model with dynamically allocated inhibitor production, Math. Biosci., 173 (2001), 55-84. doi: 10.1016/S0025-5564(01)00078-5.
[5]	J. D. Bryers, Structured modeling of anaerobic digestion of biomass particulates, Biotechn. Bioeng., 27 (1984), 638-649. doi: 10.1002/bit.260270514.
[6]	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.
[7]	M. Gerardi, "The Microbiology of Anaerobic Digesters,'' Wiley Interscience, Hoboken, 2003. doi: 10.1002/0471468967.
[8]	M. El Hajji, F. Mazenc and J. Harmand, A mathematical study of syntrophic relationship of a model of anaerobic digestion process, Math. Biosci. Eng., 7 (2010), 641-656. doi: 10.3934/mbe.2010.7.641.
[9]	J. Hess and O. Bernard, Design and study of a risk management criterion for an unstable wastewater treatment process, J. of Process Control, 18 (2008), 71-79. doi: 10.1016/j.jprocont.2007.05.005.
[10]	S.-B. Hsu and P. Waltman, A survey of mathematical models of competition with an inhibitor, Math. Biosci., 187 (2004), 53-91. doi: 10.1016/j.mbs.2003.07.004.
[11]	S. Jeyaseelan, A simple mathematical model for anaerobic digestion process, Wat. Sci. Tech., 35 (1997), 185-191. doi: 10.1016/S0273-1223(97)00166-2.
[12]	B. Li, Global asymptotic behaviour of the chemostat: General response function and differential removal rates, SIAM J. Appl. Math., 59 (1999), 411-422. doi: 10.1137/S003613999631100X.
[13]	G. Lyberatos and I. V. Skiadas, Modelling of anaerobic digestion-a review, Global Nest: The Int. J., 1 (1999), 63-76.
[14]	L. Markus, Asymptotically autonomous differential systems, in "Contributions to the Theory of Nonlinear Oscillations,'' Princeton University Press, 3, Princeton, NJ, (1953), 17-29.
[15]	R. S. Oremland, C. W. Culbertson and M. Winfrey, Methylmercurcy decomposition in sediments and bacterial cultures: Involvement of methanogens and sulfate reducers in oxidative demethylation, Appl. Env. Microbiology, 57 (1991), 130-137.
[16]	K. Pak and R. Bartha, Products of mercury demethylation by sulfidogens and methanogens, Bull. Environ. Contam. Toxicol., 61 (1998), 690-694. doi: 10.1007/s001289900816.
[17]	G. E. Powell, Stable coexistence of syntrophic associations in continuous culture, J. Chem. Tech. Biotechnol. B, 35 (1985), 46-50. doi: 10.1002/jctb.280350109.
[18]	H. Smith and P. Waltman, "The Theory of the Chemostat: Dynamics of Microbial Competition,'' Cambridge Studies in Mathematical Biology, 13, Cambridge University Press, Cambridge, 1995.
[19]	H. Thieme, Convergence results and a Poincaré-Bendixson trichotomy for asymptotically autonomous differential equations, J. of Math. Biology, 30 (1992), 755-763.
[20]	G. S. K. Wolkowicz and Z. Lu, Global Dynamics of a mathematical model of competition in the chemostat: General response functions and differential death rates, SIAM J. Appl. Math., 52 (1992), 222-233. doi: 10.1137/0152012.
[21]	L. Zhu, X. Huang and H. Su, Bifurcation for a functional yield chemostat when one competitor produces a toxin, J. Math. Anal. Appl., 329 (2007), 891-903. doi: 10.1016/j.jmaa.2006.06.062.

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