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April  2020, 19(4): 2333-2346. doi: 10.3934/cpaa.2020101

## Analysis of an anaerobic digestion model in landfill with mortality term

 1 Interdisciplinary Laboratory for Natural Resources and Environment, Ibn Tofaïl University, Kénitra, Morocco 2 IRIMAS, University of Haute-Alsace, Mulhouse, France, University of Strasbourg, France 3 LBE, University of Montpellier, INRA, Narbonne, France 4 MISTEA, University of Montpellier, INRA, Montpellier SupAgro, Montpellier, France 5 University of Haute-Alsace, IRIMAS UR 7499, F-68100 Mulhouse, France, University of Strasbourg, France

*Corresponding author

Dedicated to Professor Tomás Caraballo on the occasion of his 60-th birthday

Received  August 2019 Revised  October 2019 Published  January 2020

We study a mathematical model of anaerobic digestion with biomass recirculation, dedicated to landfill problems, and analyze its asymptotic behavior. We show that the global attractor is composed of an infinity of non-hyperbolic equilibria. For non-monotonic growth functions, this set is non connected, which impacts the performances of the bioprocess.

Citation: 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
##### References:
 [1] J. Andrews, A mathematical model for the continuous culture of microorganisms utilizing inhibitory substrates, Biotechnology and Bioengineering, 10 (1968), 707–723. [2] J. Arzate, M. Kirstein, F. Ertem, E. Kielhorn, H. Malule, P. Neubauer, M. Cruz-Bournazou and S. Junne, Anaerobic digestion model (AM2) for the description of biogas processes at dynamic feedstock loading rates, Chemie Ingenieur Technik, 89 (2017), 686–695. [3] I. Barbalat, Systèmes d'équations différentielles d'oscillations non linéaires, Rev. Math. Pures Appl., 4 (1959), 267–270. [4] G. Bastin and D. Dochain, On-line Estimation and Adaptive Control of Bioreactors, Dynamics of Microbial Competition, Elsevier Science Publishers, New-York, 1991. [5] B. Benyahia, T. Sari, B. Cherki and J. Harmand, Bifurcation and stability analysis of a two step model for monitoring anaerobic digestion processes, Journal of Process Control, 22 (2012), 1008–1019. [6] O. Bernard, Z. Hadj-Sadok, D. Dochain, A. Genovesi and J. P. Steyer, Dynamical model development and parameter identification for an anaerobic wastewater treatment process, Biotechnology and Bioengineering, 75 (2001), 424–438. [7] J. Carr, Applications of Centre Manifold Theory, Springer-Verlag, 1981. [8] D. Chenu, Modélisation des transferts réactifs de masse et de chaleur dans les installations de stockage de déchets ménagers: application aux installations de type bioréacteur, PhD thesis, Institut National Polytechnique de Toulouse, France, 2007. [9] I. Didi, H. Dib and B. Cherki, A Luenberger-type observer for the AM2 model, Journal of Process Control, 32 (2015), 117-126. [10] D. Dochain, Automatic Control of Bioprocesses Control systems, John Wiley and Sons, 2010. [11] G. Dollé, O. Duran, N. Feyeux, E. Frénod, M. Giacomini and C. Prud'Homme, Mathematical modeling and numerical simulation of a bioreactor landfill using Feel++, ESAIM: Proceedings and Surveys, 55 (2016), 83–110. [12] R. Fekih-Salem, J. Harmand, C. Lobry, A. Rapaport and T. Sari, Extensions of the chemostat model with flocculation, Journal of Mathematical Analysis and Applications, 397 (2013), 292-306.  doi: 10.1016/j.jmaa.2012.07.055. [13] J. Harmand, C. Lobry, A. Rapaport and T. Sari, The Chemostat: Mathematical Theory of Microorganisms Cultures, ISTE Wiley, 2017. [14] S. Hassam, E. Ficara, A. Leva and J. Harmand, A generic and systematic procedure to derive a simplified model from the anaerobic digestion model, No. 1 (ADM1), Biochemical Engineering Journal, 99 (2015), 193–203. [15] M. Hmissi, J. Harmand, V. Alcaraz-Gonzalez and H. Shayeb, Evaluation of alkalinity spatial distribution in an up-flow fixed bed anaerobic digester, Water Science and Technology, 77 (2018), 948–959. [16] S. J. Jang and J. Baglama, Nutrient-plankton models with nutrient recycling, Computers and Mathematics with Applications, 49 (2005), 375–387. doi: 10.1016/j.camwa.2004.03.013. [17] M. Loreau, Material cycling and the stability of ecosystems, The American Naturalist, 143 (1994), 508–513. [18] J. Monod, La technique de la culture continue: Théorie et applications, Ann. Inst. Pasteur, Lille, 79 (1950), 390–410. [19] L. Perko, Differential Equations and Dynamical Systems, Springer, 3rd ed., 2011. doi: 10.1007/978-1-4684-0392-3. [20] A. Rapaport, T. Bayen, M. Sebbah, A. Donoso-Bravo and A. Torrico, Dynamical modelling and optimal control of landfills, Mathematical Models and Methods in Applied Sciences, 26 (2016), 901–929. doi: 10.1142/S0218202516500214. [21] A. Rapaport, T. Nidelet, S. El Aida and J. Harmand, About biomass overyielding of mixed cultures in batch processes, Prepint hal, (2019). [22] A. Rapaport, T. Nidelet and J. Harmand, About biomass overyielding of mixed cultures in batch processes, in, 8th IFAC Conference on Foundations of Systems Biology in Engineering (FOSBE), Valencia, Spain, 15-18 Oct., (2019). [23] M. Rouez, Dégradation anaérobie de déchets solides: Caractérisation, facteurs d'influence et modélisations, PhD thesis, Institut National des Sciences Appliquées, Lyon, France, 2008. [24] W. Walter, Ordinary Differential Equations, Springer, 1998. doi: 10.1007/978-1-4612-0601-9.

show all references

##### References:
 [1] J. Andrews, A mathematical model for the continuous culture of microorganisms utilizing inhibitory substrates, Biotechnology and Bioengineering, 10 (1968), 707–723. [2] J. Arzate, M. Kirstein, F. Ertem, E. Kielhorn, H. Malule, P. Neubauer, M. Cruz-Bournazou and S. Junne, Anaerobic digestion model (AM2) for the description of biogas processes at dynamic feedstock loading rates, Chemie Ingenieur Technik, 89 (2017), 686–695. [3] I. Barbalat, Systèmes d'équations différentielles d'oscillations non linéaires, Rev. Math. Pures Appl., 4 (1959), 267–270. [4] G. Bastin and D. Dochain, On-line Estimation and Adaptive Control of Bioreactors, Dynamics of Microbial Competition, Elsevier Science Publishers, New-York, 1991. [5] B. Benyahia, T. Sari, B. Cherki and J. Harmand, Bifurcation and stability analysis of a two step model for monitoring anaerobic digestion processes, Journal of Process Control, 22 (2012), 1008–1019. [6] O. Bernard, Z. Hadj-Sadok, D. Dochain, A. Genovesi and J. P. Steyer, Dynamical model development and parameter identification for an anaerobic wastewater treatment process, Biotechnology and Bioengineering, 75 (2001), 424–438. [7] J. Carr, Applications of Centre Manifold Theory, Springer-Verlag, 1981. [8] D. Chenu, Modélisation des transferts réactifs de masse et de chaleur dans les installations de stockage de déchets ménagers: application aux installations de type bioréacteur, PhD thesis, Institut National Polytechnique de Toulouse, France, 2007. [9] I. Didi, H. Dib and B. Cherki, A Luenberger-type observer for the AM2 model, Journal of Process Control, 32 (2015), 117-126. [10] D. Dochain, Automatic Control of Bioprocesses Control systems, John Wiley and Sons, 2010. [11] G. Dollé, O. Duran, N. Feyeux, E. Frénod, M. Giacomini and C. Prud'Homme, Mathematical modeling and numerical simulation of a bioreactor landfill using Feel++, ESAIM: Proceedings and Surveys, 55 (2016), 83–110. [12] R. Fekih-Salem, J. Harmand, C. Lobry, A. Rapaport and T. Sari, Extensions of the chemostat model with flocculation, Journal of Mathematical Analysis and Applications, 397 (2013), 292-306.  doi: 10.1016/j.jmaa.2012.07.055. [13] J. Harmand, C. Lobry, A. Rapaport and T. Sari, The Chemostat: Mathematical Theory of Microorganisms Cultures, ISTE Wiley, 2017. [14] S. Hassam, E. Ficara, A. Leva and J. Harmand, A generic and systematic procedure to derive a simplified model from the anaerobic digestion model, No. 1 (ADM1), Biochemical Engineering Journal, 99 (2015), 193–203. [15] M. Hmissi, J. Harmand, V. Alcaraz-Gonzalez and H. Shayeb, Evaluation of alkalinity spatial distribution in an up-flow fixed bed anaerobic digester, Water Science and Technology, 77 (2018), 948–959. [16] S. J. Jang and J. Baglama, Nutrient-plankton models with nutrient recycling, Computers and Mathematics with Applications, 49 (2005), 375–387. doi: 10.1016/j.camwa.2004.03.013. [17] M. Loreau, Material cycling and the stability of ecosystems, The American Naturalist, 143 (1994), 508–513. [18] J. Monod, La technique de la culture continue: Théorie et applications, Ann. Inst. Pasteur, Lille, 79 (1950), 390–410. [19] L. Perko, Differential Equations and Dynamical Systems, Springer, 3rd ed., 2011. doi: 10.1007/978-1-4684-0392-3. [20] A. Rapaport, T. Bayen, M. Sebbah, A. Donoso-Bravo and A. Torrico, Dynamical modelling and optimal control of landfills, Mathematical Models and Methods in Applied Sciences, 26 (2016), 901–929. doi: 10.1142/S0218202516500214. [21] A. Rapaport, T. Nidelet, S. El Aida and J. Harmand, About biomass overyielding of mixed cultures in batch processes, Prepint hal, (2019). [22] A. Rapaport, T. Nidelet and J. Harmand, About biomass overyielding of mixed cultures in batch processes, in, 8th IFAC Conference on Foundations of Systems Biology in Engineering (FOSBE), Valencia, Spain, 15-18 Oct., (2019). [23] M. Rouez, Dégradation anaérobie de déchets solides: Caractérisation, facteurs d'influence et modélisations, PhD thesis, Institut National des Sciences Appliquées, Lyon, France, 2008. [24] W. Walter, Ordinary Differential Equations, Springer, 1998. doi: 10.1007/978-1-4612-0601-9.
Overall scheme of the anaerobic degradation of organic matter
Graphs of Monod and Haldane functions
Graph of the Haldane function considered in the example
Graph of $S(\cdot)$ for initial conditions $X_0 = 340$ to $X_0 = 360$ with a step of 5
Threshold on $X_0$ as a function of $K_d$ for model parameters indicated in Table 1
Biogas production as a function of $X_0$ for $K_d = 0.02$
$\lambda^-$, $\lambda^+$ (on top) ${\rm Biogas}^-$ and ${\rm Biogas}^+$ (on bottom) as functions of $K_d$
 $K_h$ $Y$ $f_1$ $f_2$ $\alpha$ 0.176 0.05 0.7 0.76 0.9
 $K_h$ $Y$ $f_1$ $f_2$ $\alpha$ 0.176 0.05 0.7 0.76 0.9
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