American Institute of Mathematical Sciences

Advanced
  • Previous Article
    Convergence of nonautonomous multivalued problems with large diffusion to ordinary differential inclusions
  • CPAA Home
  • This Issue
  • Next Article
    The stampacchia maximum principle for stochastic partial differential equations forced by lévy noise
April  2020, 19(4): 2333-2346. doi: 10.3934/cpaa.2020101

Analysis of an anaerobic digestion model in landfill with mortality term

S. Ouchtout 1,2, , Z. Mghazli 1, , J. Harmand 3, , A. Rapaport 4,, and  Z. Belhachmi 5,

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.

Keywords: Bioprocesses, biogas production, ordinary differential equations, non-hyperbolic equilibrium.
Mathematics Subject Classification: Primary: 37N25, 93D20; Secondary: 34C11, 34D35.
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.

Figure 1.  Overall scheme of the anaerobic degradation of organic matter
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 2.  Graphs of Monod and Haldane functions
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 3.  Graph of the Haldane function considered in the example
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 4.  Graph of $ S(\cdot) $ for initial conditions $ X_0 = 340 $ to $ X_0 = 360 $ with a step of 5
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 5.  Threshold on $ X_0 $ as a function of $ K_d $ for model parameters indicated in Table 1
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 6.  Biogas production as a function of $ X_0 $ for $ K_d = 0.02 $
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 7.  $ \lambda^- $, $ \lambda^+ $ (on top) $ {\rm Biogas}^- $ and $ {\rm Biogas}^+ $ (on bottom) as functions of $ K_d $
Figure Options

Download full-size image

Download as PowerPoint slide

Table 1. 
$ K_h$ $Y $ $f_1 $ $ f_2$ $ \alpha$
0.176 0.05 0.7 0.76 0.9
Table Options

Download as excel

$ K_h$ $Y $ $f_1 $ $ f_2$ $ \alpha$
0.176 0.05 0.7 0.76 0.9
[1]

Pablo G. Barrientos, Artem Raibekas. Robustly non-hyperbolic transitive symplectic dynamics. Discrete and Continuous Dynamical Systems, 2018, 38 (12) : 5993-6013. doi: 10.3934/dcds.2018259
[2]

Wen Si, Fenfen Wang, Jianguo Si. Almost-periodic perturbations of non-hyperbolic equilibrium points via Pöschel-Rüssmann KAM method. Communications on Pure and Applied Analysis, 2020, 19 (1) : 541-585. doi: 10.3934/cpaa.2020027
[3]

Zhicong Liu. SRB attractors with intermingled basins for non-hyperbolic diffeomorphisms. Discrete and Continuous Dynamical Systems, 2013, 33 (4) : 1545-1562. doi: 10.3934/dcds.2013.33.1545
[4]

Vanderlei Horita, Marcelo Viana. Hausdorff dimension for non-hyperbolic repellers II: DA diffeomorphisms. Discrete and Continuous Dynamical Systems, 2005, 13 (5) : 1125-1152. doi: 10.3934/dcds.2005.13.1125
[5]

Katrin Gelfert. Non-hyperbolic behavior of geodesic flows of rank 1 surfaces. Discrete and Continuous Dynamical Systems, 2019, 39 (1) : 521-551. doi: 10.3934/dcds.2019022
[6]

Anna Cima, Armengol Gasull, Víctor Mañosa. Parrondo's dynamic paradox for the stability of non-hyperbolic fixed points. Discrete and Continuous Dynamical Systems, 2018, 38 (2) : 889-904. doi: 10.3934/dcds.2018038
[7]

Inmaculada Baldomá, Ernest Fontich, Pau Martín. Gevrey estimates for one dimensional parabolic invariant manifolds of non-hyperbolic fixed points. Discrete and Continuous Dynamical Systems, 2017, 37 (8) : 4159-4190. doi: 10.3934/dcds.2017177
[8]

Bernard Dacorogna, Alessandro Ferriero. Regularity and selecting principles for implicit ordinary differential equations. Discrete and Continuous Dynamical Systems - B, 2009, 11 (1) : 87-101. doi: 10.3934/dcdsb.2009.11.87
[9]

Zvi Artstein. Averaging of ordinary differential equations with slowly varying averages. Discrete and Continuous Dynamical Systems - B, 2010, 14 (2) : 353-365. doi: 10.3934/dcdsb.2010.14.353
[10]

Yongluo Cao, Stefano Luzzatto, Isabel Rios. Some non-hyperbolic systems with strictly non-zero Lyapunov exponents for all invariant measures: Horseshoes with internal tangencies. Discrete and Continuous Dynamical Systems, 2006, 15 (1) : 61-71. doi: 10.3934/dcds.2006.15.61
[11]

Serge Nicaise. Stability and asymptotic properties of dissipative evolution equations coupled with ordinary differential equations. Mathematical Control and Related Fields, 2021  doi: 10.3934/mcrf.2021057
[12]

Stefano Maset. Conditioning and relative error propagation in linear autonomous ordinary differential equations. Discrete and Continuous Dynamical Systems - B, 2018, 23 (7) : 2879-2909. doi: 10.3934/dcdsb.2018165
[13]

W. Sarlet, G. E. Prince, M. Crampin. Generalized submersiveness of second-order ordinary differential equations. Journal of Geometric Mechanics, 2009, 1 (2) : 209-221. doi: 10.3934/jgm.2009.1.209
[14]

Aeeman Fatima, F. M. Mahomed, Chaudry Masood Khalique. Conditional symmetries of nonlinear third-order ordinary differential equations. Discrete and Continuous Dynamical Systems - S, 2018, 11 (4) : 655-666. doi: 10.3934/dcdss.2018040
[15]

Ping Lin, Weihan Wang. Optimal control problems for some ordinary differential equations with behavior of blowup or quenching. Mathematical Control and Related Fields, 2018, 8 (3&4) : 809-828. doi: 10.3934/mcrf.2018036
[16]

Jean Mawhin, James R. Ward Jr. Guiding-like functions for periodic or bounded solutions of ordinary differential equations. Discrete and Continuous Dynamical Systems, 2002, 8 (1) : 39-54. doi: 10.3934/dcds.2002.8.39
[17]

Hongwei Lou, Weihan Wang. Optimal blowup/quenching time for controlled autonomous ordinary differential equations. Mathematical Control and Related Fields, 2015, 5 (3) : 517-527. doi: 10.3934/mcrf.2015.5.517
[18]

Alex Bihlo, James Jackaman, Francis Valiquette. On the development of symmetry-preserving finite element schemes for ordinary differential equations. Journal of Computational Dynamics, 2020, 7 (2) : 339-368. doi: 10.3934/jcd.2020014
[19]

Iasson Karafyllis, Lars Grüne. Feedback stabilization methods for the numerical solution of ordinary differential equations. Discrete and Continuous Dynamical Systems - B, 2011, 16 (1) : 283-317. doi: 10.3934/dcdsb.2011.16.283
[20]

Bin Wang, Arieh Iserles. Dirichlet series for dynamical systems of first-order ordinary differential equations. Discrete and Continuous Dynamical Systems - B, 2014, 19 (1) : 281-298. doi: 10.3934/dcdsb.2014.19.281

2021 Impact Factor: 1.273

Tools

Metrics

  • PDF downloads (336)
  • HTML views (100)
  • Cited by (1)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy