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The stampacchia maximum principle for stochastic partial differential equations forced by lévy noise
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 |
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.
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. |





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