# American Institute of Mathematical Sciences

2010, 7(3): 641-656. doi: 10.3934/mbe.2010.7.641

## A mathematical study of a syntrophic relationship of a model of anaerobic digestion process

 1 UMR Analyses des Systèmes et Biométrie, INRA 02 Place, INRA-INRIA MERE research team, Viala, 34060 Montpellier, France, France 2 LBE-INRA, UR050, Avenue des Étangs, 11100 Narbonne & INRA-INRIA MERE research team, UMR Analyses des Systèmes et Biométrie, INRA 02 Place Viala, 34060 Montpellier, France

Received  October 2009 Revised  March 2010 Published  June 2010

A mathematical model involving the syntrophic relationship of two major populations of bacteria (acetogens and methanogens), each responsible for a stage of the methane fermentation process is proposed. A detailed qualitative analysis is carried out. The local and global stability analyses of the equilibria are performed. We demonstrate, under general assumptions of monotonicity, relevant from an applied point of view, the global asymptotic stability of a positive equilibrium point which corresponds to the coexistence of acetogenic and methanogenic bacteria.
Citation: 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
 [1] Tewfik Sari, Miled El Hajji, Jérôme Harmand. The mathematical analysis of a syntrophic relationship between two microbial species in a chemostat. Mathematical Biosciences & Engineering, 2012, 9 (3) : 627-645. doi: 10.3934/mbe.2012.9.627 [2] 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 [3] S. Ouchtout, Z. Mghazli, J. Harmand, A. Rapaport, Z. Belhachmi. Analysis of an anaerobic digestion model in landfill with mortality term. Communications on Pure & Applied Analysis, 2020, 19 (4) : 2333-2346. doi: 10.3934/cpaa.2020101 [4] Wei Feng, Michael T. Cowen, Xin Lu. Coexistence and asymptotic stability in stage-structured predator-prey models. Mathematical Biosciences & Engineering, 2014, 11 (4) : 823-839. doi: 10.3934/mbe.2014.11.823 [5] Juan Pablo Aparicio, Carlos Castillo-Chávez. Mathematical modelling of tuberculosis epidemics. Mathematical Biosciences & Engineering, 2009, 6 (2) : 209-237. doi: 10.3934/mbe.2009.6.209 [6] Telma Silva, Adélia Sequeira, Rafael F. Santos, Jorge Tiago. Existence, uniqueness, stability and asymptotic behavior of solutions for a mathematical model of atherosclerosis. Discrete & Continuous Dynamical Systems - S, 2016, 9 (1) : 343-362. doi: 10.3934/dcdss.2016.9.343 [7] Geoffrey Beck, Sebastien Imperiale, Patrick Joly. Mathematical modelling of multi conductor cables. Discrete & Continuous Dynamical Systems - S, 2015, 8 (3) : 521-546. doi: 10.3934/dcdss.2015.8.521 [8] Nirav Dalal, David Greenhalgh, Xuerong Mao. Mathematical modelling of internal HIV dynamics. Discrete & Continuous Dynamical Systems - B, 2009, 12 (2) : 305-321. doi: 10.3934/dcdsb.2009.12.305 [9] Oliver Penrose, John W. Cahn. On the mathematical modelling of cellular (discontinuous) precipitation. Discrete & Continuous Dynamical Systems - A, 2017, 37 (2) : 963-982. doi: 10.3934/dcds.2017040 [10] Alexander V. Budyansky, Kurt Frischmuth, Vyacheslav G. Tsybulin. Cosymmetry approach and mathematical modeling of species coexistence in a heterogeneous habitat. Discrete & Continuous Dynamical Systems - B, 2019, 24 (2) : 547-561. doi: 10.3934/dcdsb.2018196 [11] Liumei Wu, Baojun Song, Wen Du, Jie Lou. Mathematical modelling and control of echinococcus in Qinghai province, China. Mathematical Biosciences & Engineering, 2013, 10 (2) : 425-444. doi: 10.3934/mbe.2013.10.425 [12] Roderick Melnik, B. Lassen, L. C Lew Yan Voon, M. Willatzen, C. Galeriu. Accounting for nonlinearities in mathematical modelling of quantum dot molecules. Conference Publications, 2005, 2005 (Special) : 642-651. doi: 10.3934/proc.2005.2005.642 [13] Luis L. Bonilla, Vincenzo Capasso, Mariano Alvaro, Manuel Carretero, Filippo Terragni. On the mathematical modelling of tumor-induced angiogenesis. Mathematical Biosciences & Engineering, 2017, 14 (1) : 45-66. doi: 10.3934/mbe.2017004 [14] M.A.J Chaplain, G. Lolas. Mathematical modelling of cancer invasion of tissue: dynamic heterogeneity. Networks & Heterogeneous Media, 2006, 1 (3) : 399-439. doi: 10.3934/nhm.2006.1.399 [15] Aditya S. Khanna, Dobromir T. Dimitrov, Steven M. Goodreau. What can mathematical models tell us about the relationship between circular migrations and HIV transmission dynamics?. Mathematical Biosciences & Engineering, 2014, 11 (5) : 1065-1090. doi: 10.3934/mbe.2014.11.1065 [16] Christos V. Nikolopoulos. Mathematical modelling of a mushy region formation during sulphation of calcium carbonate. Networks & Heterogeneous Media, 2014, 9 (4) : 635-654. doi: 10.3934/nhm.2014.9.635 [17] Alexandre Cornet. Mathematical modelling of cardiac pulse wave reflections due to arterial irregularities. Mathematical Biosciences & Engineering, 2018, 15 (5) : 1055-1076. doi: 10.3934/mbe.2018047 [18] Mostafa Adimy, Laurent Pujo-Menjouet. Asymptotic behavior of a singular transport equation modelling cell division. Discrete & Continuous Dynamical Systems - B, 2003, 3 (3) : 439-456. doi: 10.3934/dcdsb.2003.3.439 [19] Chunqing Wu, Patricia J.Y. Wong. Global asymptotical stability of the coexistence fixed point of a Ricker-type competitive model. Discrete & Continuous Dynamical Systems - B, 2015, 20 (9) : 3255-3266. doi: 10.3934/dcdsb.2015.20.3255 [20] Kousuke Kuto. Stability and Hopf bifurcation of coexistence steady-states to an SKT model in spatially heterogeneous environment. Discrete & Continuous Dynamical Systems - A, 2009, 24 (2) : 489-509. doi: 10.3934/dcds.2009.24.489

2018 Impact Factor: 1.313