# American Institute of Mathematical Sciences

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

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

Received  November 2010 Revised  August 2011 Published  March 2012

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.
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.

show all references

##### 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.
 [1] Arturo Hidalgo, Lourdes Tello. On a global climate model with non-monotone multivalued coalbedo. Discrete and Continuous Dynamical Systems - S, 2022  doi: 10.3934/dcdss.2022093 [2] Zhao-Xing Yang, Guo-Bao Zhang, Ge Tian, Zhaosheng Feng. Stability of non-monotone non-critical traveling waves in discrete reaction-diffusion equations with time delay. Discrete and Continuous Dynamical Systems - S, 2017, 10 (3) : 581-603. doi: 10.3934/dcdss.2017029 [3] Abraham Solar. Stability of non-monotone and backward waves for delay non-local reaction-diffusion equations. Discrete and Continuous Dynamical Systems, 2019, 39 (10) : 5799-5823. doi: 10.3934/dcds.2019255 [4] Rui Huang, Ming Mei, Kaijun Zhang, Qifeng Zhang. Asymptotic stability of non-monotone traveling waves for time-delayed nonlocal dispersion equations. Discrete and Continuous Dynamical Systems, 2016, 36 (3) : 1331-1353. doi: 10.3934/dcds.2016.36.1331 [5] S. Ouchtout, Z. Mghazli, J. Harmand, A. Rapaport, Z. Belhachmi. Analysis of an anaerobic digestion model in landfill with mortality term. Communications on Pure and Applied Analysis, 2020, 19 (4) : 2333-2346. doi: 10.3934/cpaa.2020101 [6] Sergiu Aizicovici, Simeon Reich. Anti-periodic solutions to a class of non-monotone evolution equations. Discrete and Continuous Dynamical Systems, 1999, 5 (1) : 35-42. doi: 10.3934/dcds.1999.5.35 [7] Alfonso Castro, Benjamin Preskill. Existence of solutions for a semilinear wave equation with non-monotone nonlinearity. Discrete and Continuous Dynamical Systems, 2010, 28 (2) : 649-658. doi: 10.3934/dcds.2010.28.649 [8] Jun Chen, Wenyu Sun, Zhenghao Yang. A non-monotone retrospective trust-region method for unconstrained optimization. Journal of Industrial and Management Optimization, 2013, 9 (4) : 919-944. doi: 10.3934/jimo.2013.9.919 [9] José Caicedo, Alfonso Castro, Arturo Sanjuán. Bifurcation at infinity for a semilinear wave equation with non-monotone nonlinearity. Discrete and Continuous Dynamical Systems, 2017, 37 (4) : 1857-1865. doi: 10.3934/dcds.2017078 [10] Miled El Hajji, Frédéric Mazenc, Jérôme Harmand. A mathematical study of a syntrophic relationship of a model of anaerobic digestion process. Mathematical Biosciences & Engineering, 2010, 7 (3) : 641-656. doi: 10.3934/mbe.2010.7.641 [11] Yuxiang Zhang, Shiwang Ma. Invasion dynamics of a diffusive pioneer-climax model: Monotone and non-monotone cases. Discrete and Continuous Dynamical Systems - B, 2021, 26 (9) : 4767-4788. doi: 10.3934/dcdsb.2020312 [12] Pablo Amster, Manuel Zamora. Periodic solutions for indefinite singular equations with singularities in the spatial variable and non-monotone nonlinearity. Discrete and Continuous Dynamical Systems, 2018, 38 (10) : 4819-4835. doi: 10.3934/dcds.2018211 [13] Anatoli F. Ivanov, Bernhard Lani-Wayda. Periodic solutions for three-dimensional non-monotone cyclic systems with time delays. Discrete and Continuous Dynamical Systems, 2004, 11 (2&3) : 667-692. doi: 10.3934/dcds.2004.11.667 [14] José Caicedo, Alfonso Castro, Rodrigo Duque, Arturo Sanjuán. Existence of $L^p$-solutions for a semilinear wave equation with non-monotone nonlinearity. Discrete and Continuous Dynamical Systems - S, 2014, 7 (6) : 1193-1202. doi: 10.3934/dcdss.2014.7.1193 [15] Yu Liu, Ting Zhang. On weak (measure-valued)-strong uniqueness for compressible MHD system with non-monotone pressure law. Discrete and Continuous Dynamical Systems - B, 2022  doi: 10.3934/dcdsb.2021307 [16] Xilu Wang, Xiaoliang Cheng. Continuous dependence and optimal control of a dynamic elastic-viscoplastic contact problem with non-monotone boundary conditions. Evolution Equations and Control Theory, 2022  doi: 10.3934/eect.2021064 [17] Frederic Mazenc, Gonzalo Robledo, Michael Malisoff. Stability and robustness analysis for a multispecies chemostat model with delays in the growth rates and uncertainties. Discrete and Continuous Dynamical Systems - B, 2018, 23 (4) : 1851-1872. doi: 10.3934/dcdsb.2018098 [18] Zhiming Guo, Zhi-Chun Yang, Xingfu Zou. Existence and uniqueness of positive solution to a non-local differential equation with homogeneous Dirichlet boundary condition---A non-monotone case. Communications on Pure and Applied Analysis, 2012, 11 (5) : 1825-1838. doi: 10.3934/cpaa.2012.11.1825 [19] Zhiqi Lu. Global stability for a chemostat-type model with delayed nutrient recycling. Discrete and Continuous Dynamical Systems - B, 2004, 4 (3) : 663-670. doi: 10.3934/dcdsb.2004.4.663 [20] Ahmad Al-Salman, Ziyad AlSharawi, Sadok Kallel. Extension, embedding and global stability in two dimensional monotone maps. Discrete and Continuous Dynamical Systems - B, 2020, 25 (11) : 4257-4276. doi: 10.3934/dcdsb.2020096

2018 Impact Factor: 1.313