O. O. Aalen and H. K. Gjessing, Understanding the shape of the hazard rate: a process point of view (with comments and a rejoinder by the authors), Stat. Sci, vol.16, issue.1, pp.1-22, 2001.

J. F. Andrews, A mathematical model for the continuous culture of microorganisms utilizing inhibitory substrates, Biotechnology and Bioengineering, vol.9, issue.6, pp.707-723, 1968.
DOI : 10.1002/bit.260100602

J. F. Andrews, A dynamic model of the anaerobic digestion process, J. Sanit. Eng. Div. Proc. Am. Soc. Civil Eng. SA, vol.1, pp.95-116, 1969.

J. F. Andrews, Kinetic models of biological waste treatment processes, Biotechnol . Bioeng. Symp, vol.2, pp.5-33, 1971.

R. Arditi and L. R. Ginzburg, How Species Interact ? Altering the Standard View on Trophic Ecology, 2012.

R. Arditi and H. Saïah, Empirical evidence of the role of heterogeneity in ratiodependent consumption, Ecology and Erratum in Ecology, vol.73, issue.74, pp.1544-1551, 1992.

R. Arditi, Y. Tyutyunov, A. Morgulis, V. Govorukhin, and I. Senina, Directed Movement of Predators and the Emergence of Density-Dependence in Predator???Prey Models, Theoretical Population Biology, vol.59, issue.3, pp.207-221, 2001.
DOI : 10.1006/tpbi.2001.1513

A. Asselah, P. A. Ferrari, and P. Groisman, Quasistationary Distributions and Fleming-Viot Processes in Finite Spaces, Journal of Applied Probability, vol.11, issue.02, pp.322-332, 2011.
DOI : 10.1016/j.spa.2003.10.010

URL : https://hal.archives-ouvertes.fr/hal-00692986

M. Böl, A. E. Ehret, A. B. Albero, J. Hellriegel, and R. Krull, Recent advances in mechanical characterisation of biofilm and their significance for material modelling, Critical Reviews in Biotechnology, vol.44, issue.2, pp.145-171, 2013.
DOI : 10.1080/08927019309386244

G. Bastin and D. Dochain, On-line Estimation and Adaptive Control of Bioreactors, Process Measurement and Control, 1990.

M. Benaïm and B. Cloez, A stochastic approximation approach to quasi-stationary distributions on finite spaces, Electronic Communications in Probability, vol.20, issue.0, 2014.
DOI : 10.1214/ECP.v20-3956

M. Benaï-m and C. Lobry, Lotka volterra in fluctuating environment or " how good can be bad, 2014.

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, vol.22, issue.6, pp.1008-1019, 2012.
DOI : 10.1016/j.jprocont.2012.04.012

URL : https://hal.archives-ouvertes.fr/hal-00777051

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, vol.29, issue.7, pp.424-438, 2001.
DOI : 10.1002/bit.10036

J. Blanchet, P. Glynn, and S. Zheng, Theoretical analysis of a stochastic approximation approach for computing quasi-stationary distributions, 2014.

A. Burchard, Substrate degradation by a mutualistic association of two species in the Chemostat, Journal of Mathematical Biology, vol.41, issue.5, pp.465-489, 1994.
DOI : 10.1007/BF00160169

K. Burdzy, R. Ho?yst, and P. March, A Fleming???Viot Particle Representation??of the Dirichlet Laplacian, Communications in Mathematical Physics, vol.214, issue.3, pp.679-703, 2000.
DOI : 10.1007/s002200000294

G. Butler and G. Wolkowicz, A Mathematical Model of the Chemostat with a General Class of Functions Describing Nutrient Uptake, SIAM Journal on Applied Mathematics, vol.45, issue.1, pp.137-151, 1985.
DOI : 10.1137/0145006

G. Butler, S. Hsu, and P. Waltman, A Mathematical Model of the Chemostat with Periodic Washout Rate, SIAM Journal on Applied Mathematics, vol.45, issue.3, pp.435-449, 1985.
DOI : 10.1137/0145025

F. Campillo and C. Fritsch, Weak convergence of a mass-structured individualbased model, Appl. Math. Optim, vol.37, pp.245-259, 2014.
URL : https://hal.archives-ouvertes.fr/hal-01090727

F. Campillo and C. Lobry, Effect of population size in a predator???prey model, Ecological Modelling, vol.246, pp.1-10, 2012.
DOI : 10.1016/j.ecolmodel.2012.07.015

URL : https://hal.archives-ouvertes.fr/hal-00723793

F. Campillo, M. Joannides, and I. Larramendy, Stochastic Models of the Chemostat, 2010.
URL : https://hal.archives-ouvertes.fr/inria-00537886

T. Caraballo, X. Han, P. Kloeden, and A. Rapaport, Dynamics of non-autonomous chemostat models, Continuous and Distributed Systems II Studies in Systems Decision and Control, pp.103-120, 2015.

J. R. Carey, P. Liedo, D. Orozco, and J. W. Vaupel, Slowing of mortality rates at older ages in large medfly cohorts, Science, vol.258, issue.5081, pp.457-461, 1992.
DOI : 10.1126/science.1411540

N. Champagnat, P. Jabin, and S. Méléard, Adaptation in a stochastic multi-resources chemostat model, Journal de Math??matiques Pures et Appliqu??es, vol.101, issue.6, pp.755-788, 2014.
DOI : 10.1016/j.matpur.2013.10.003

URL : https://hal.archives-ouvertes.fr/hal-00784166

J. Chazottes, P. Collet, and S. Méléard, Sharp asymptotics for the quasistationary distribution of birth-and-death processes, 2014.

B. Cloez and M. Thai, Quantitative results for the Fleming???Viot particle system and quasi-stationary distributions in discrete space, Stochastic Processes and their Applications, vol.126, issue.3, 2013.
DOI : 10.1016/j.spa.2015.09.016

D. E. Contois, Kinetics of Bacterial Growth: Relationship between Population Density and Specific Growth Rate of Continuous Cultures, Journal of General Microbiology, vol.21, issue.1, pp.40-50, 1959.
DOI : 10.1099/00221287-21-1-40

K. S. Crump and W. C. O-'young, Some stochastic features of bacterial constant growth apparatus, Bulletin of Mathematical Biology, vol.14, issue.1, pp.53-66, 1979.
DOI : 10.1007/BF02547924

D. 'acunto, B. Frunzo, L. Klapper, I. Mattei, and M. R. , Modeling multispecies biofilms including new bacterial species invasion, Math. Biosci, vol.259, pp.20-26, 2015.

G. T. Daigger and C. P. Grady, A model for the bio-oxidation process based on product formation concepts, Water Research, vol.11, issue.12, pp.1049-10570043, 1977.
DOI : 10.1016/0043-1354(77)90005-7

D. L. Deangelis and S. Yurek, Equation-free modeling unravels the behavior of complex ecological systems: Fig. 1., Proceedings of the National Academy of Sciences, vol.112, issue.13, pp.3856-3857, 2015.
DOI : 10.1073/pnas.1503154112

D. Quemener, E. Bouchez, and T. , A thermodynamic theory of microbial growth, The ISME Journal, vol.683, issue.8, pp.1747-1751, 2014.
DOI : 10.1038/ismej.2014.7

URL : https://hal.archives-ouvertes.fr/hal-00825781

P. L. Dold, G. A. Ekama, and G. V. Marais, A GENERAL MODEL FOR THE ACTIVATED SLUDGE PROCESS, Prog. Water Technol, vol.12, pp.47-77, 1980.
DOI : 10.1016/B978-1-4832-8438-5.50010-8

E. Hajji, M. Harmand, J. Chaker, H. Lobry, and C. , Association between competition and obligate mutualism in a chemostat, Journal of Biological Dynamics, vol.13, issue.6, pp.635-647, 2009.
DOI : 10.1128/MMBR.68.4.745-770.2004

URL : https://hal.archives-ouvertes.fr/hal-00858541

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, vol.397, issue.1, pp.292-306, 2013.
DOI : 10.1016/j.jmaa.2012.07.055

URL : https://hal.archives-ouvertes.fr/hal-00604633

R. Fekih-salem, A. Rapaport, and T. Sari, Emergence of coexistence and limit cycles in the chemostat model with flocculation for a general class of functional responses (pre-print), 2015.

R. Freter, H. Brickner, J. Fekete, M. Vickerman, and K. Carey, Survival and implantation of Escherichia coli in the intestinal tract, Infect. Immun, vol.39, issue.2, pp.686-703, 1983.

C. Fritsch, J. Harmand, and F. Campillo, A modeling approach of the chemostat, Ecological Modelling, vol.299, pp.1-13, 2015.
DOI : 10.1016/j.ecolmodel.2014.11.021

URL : https://hal.archives-ouvertes.fr/hal-01090651

K. Gernaey, B. Petersen, G. Parmentier, H. Bogaert, J. P. Ottoy et al., Application of dynamic models (ASM1) and simulation to minimize renovation costs of a municipal activated sludge wastewater treatment plant, Proc. of the 1st World Congress of the International Water Association. AGHTM, 2000.

G. Jr, C. P. Harlow, L. J. Riesing, and R. R. , Effects of growth rate and influent substrate concentration on effluent quality from chemostats containing bacteria in pure and mixed culture, Biotechnol. Bioeng, vol.14, pp.391-410, 1972.

S. P. Graef and J. F. Andrews, Stability and control of anaerobic digestion, J. Water Pollut. Control Fed, vol.46, pp.667-682, 1974.

P. Groisman and M. Jonckheere, Simulation of quasi-stationary distributions on countable spaces, Markov Process. Relat. Fields, vol.19, issue.3, pp.521-542, 2013.

B. Haegeman and A. Rapaport, How flocculation can explain coexistence in the chemostat, Journal of Biological Dynamics, vol.7, issue.1, pp.1-13, 2008.
DOI : 10.1016/S0043-1354(98)00392-3

URL : https://hal.archives-ouvertes.fr/hal-00857826

I. Haidar, A. Rapaport, and F. Gérard, Effects of spatial structure and diffusion on the performances of the chemostat, Mathematical Biosciences and Engineering, vol.8, issue.4, pp.953-971, 2011.
DOI : 10.3934/mbe.2011.8.953

URL : https://hal.archives-ouvertes.fr/hal-01001373

J. Harmand and J. Godon, Density-dependent kinetics models for a simple description of complex phenomena in macroscopic mass-balance modeling of bioreactors, Ecological Modelling, vol.200, issue.3-4, pp.3-4, 2007.
DOI : 10.1016/j.ecolmodel.2006.08.012

J. J. Heijnen and J. P. Vandijken, In search of a thermodynamic description of biomass yields for the chemotrophic growth of microorganisms, Biotechnology and Bioengineering, vol.41, issue.8, pp.833-858, 1992.
DOI : 10.1002/bit.260390806

M. Henze, C. P. Grady, W. Gujer, G. V. Marais, and T. Matsuo, Activated Sludge Model No, 1987.

M. Henze, W. Gujer, T. Mino, T. Matsuo, M. C. Wentzel et al., Activated Sludge Model No.2d, ASM2d, Water Science and Technology, vol.39, issue.1, pp.165-182, 1999.
DOI : 10.1016/S0273-1223(98)00829-4

S. Hsu, S. Hubbell, and P. Waltman, A Mathematical Theory for Single-Nutrient Competition in Continuous Cultures of Micro-Organisms, SIAM Journal on Applied Mathematics, vol.32, issue.2, pp.366-383, 1977.
DOI : 10.1137/0132030

Z. R. Hu, S. Sötemann, R. Moodley, M. C. Wentzel, and G. A. Ekama, Experimental investigation of the external nitrification biological nutrient removal activated sludge (ENBNRAS) system, Biotechnology and Bioengineering, vol.25, issue.6, pp.260-273, 2003.
DOI : 10.1002/bit.10664

R. E. Hungate, Microbial ecology of the rumen, Bacteriol. Rev, vol.24, issue.4, pp.353-364, 1960.

U. Jeppsson, C. Rosen, J. Alex, J. Copp, K. V. Gernaey et al., Towards a benchmark simulation model for plant-wide control strategy performance evaluation of WWTPs, Water Science and Technology, vol.53, issue.1, pp.287-295, 2006.
DOI : 10.2166/wst.2006.031

U. Jeppsson, J. Alex, D. J. Batstone, L. Benedetti, J. Comas et al., ?, Water Science & Technology, vol.68, issue.1, pp.1-15, 2013.
DOI : 10.2166/wst.2013.246

C. Jost, G. Devulder, J. A. Vucetich, R. O. Peterson, and R. Arditi, The wolves of Isle Royale display scale-invariant satiation and ratio-dependent predation on moose, Journal of Animal Ecology, vol.4, issue.5, pp.809-816, 2005.
DOI : 10.1098/rspb.1999.0673

URL : https://hal.archives-ouvertes.fr/hal-00093186

C. Jost, Predator-prey theory: hidden twins in ecology and microbiology, Oikos, vol.90, issue.1, pp.202-208, 2000.
DOI : 10.1034/j.1600-0706.2000.900125.x

P. J. Kim and N. D. Price, Genetic Co-Occurrence Network across Sequenced Microbes, PLoS Computational Biology, vol.10, issue.12, 2011.
DOI : 10.1371/journal.pcbi.1002340.s009

R. Kleerebezem and M. C. Van-loosdrecht, A Generalized Method for Thermodynamic State Analysis of Environmental Systems, Critical Reviews in Environmental Science and Technology, vol.41, issue.1, pp.1-54, 2010.
DOI : 10.1002/bit.10455

R. Kreikenbohm and E. Bohl, A mathematical model of syntrophic cocultures in the chemostat, FEMS Microbiology Letters, vol.38, issue.3, pp.131-140, 1986.
DOI : 10.1111/j.1574-6968.1986.tb01722.x

T. G. Kurtz, Solutions of ordinary differential equations as limits of pure jump markov processes, Journal of Applied Probability, vol.3, issue.01, pp.49-58, 1970.
DOI : 10.1017/S0021900200026929

M. Land, L. Hauser, S. Jun, I. Nookaew, M. R. Leuze et al., Insights from 20??years of bacterial genome sequencing, Functional & Integrative Genomics, vol.68, issue.2, pp.141-161, 2015.
DOI : 10.1007/s10142-015-0433-4

P. Lessard and B. Beck, Dynamic Modeling of Primary Sedimentation, Journal of Environmental Engineering, vol.114, issue.4, pp.753-7690733, 1988.
DOI : 10.1061/(ASCE)0733-9372(1988)114:4(753)

C. Lobry and J. Harmand, A new hypothesis to explain the coexistence of n species in the presence of a single resource, Comptes Rendus Biologies, vol.329, issue.1, pp.40-46, 2006.
DOI : 10.1016/j.crvi.2005.10.004

URL : https://hal.archives-ouvertes.fr/hal-01001131

C. Lobry, F. Mazenc, and A. Rapaport, Persistence in ecological models of competition for a single resource, Comptes Rendus Mathematique, vol.340, issue.3, 2005.
DOI : 10.1016/j.crma.2004.12.021

URL : https://hal.archives-ouvertes.fr/hal-01001120

C. Lobry, A. Rapaport, and T. Sari, Stability loss delay in the chemostat with a slowly varying washout rate, Proc. of the 6th Vienna International Conference on Mathematical Modelling AGESIM/ASIM, pp.1582-1586, 2009.
URL : https://hal.archives-ouvertes.fr/hal-01019296

C. Lobry, La compétition dans le chémostat, In: Des Nombres et des Mondes, pp.119-187, 2013.

A. J. Lotka, Contribution to the Theory of Periodic Reactions, The Journal of Physical Chemistry, vol.14, issue.3, pp.271-274, 1910.
DOI : 10.1021/j150111a004

A. J. Lotka, Natural Selection as a Physical Principle, Proceedings of the National Academy of Sciences, vol.8, issue.6, pp.151-154, 1922.
DOI : 10.1073/pnas.8.6.151

S. Méléard and D. Villemonais, Quasi-stationary distributions and population processes, Probability Surveys, vol.9, issue.0, pp.340-410, 2012.
DOI : 10.1214/11-PS191

P. L. Mccarty, Thermodynamics of biological synthesis and growth, Air Water Pollut, vol.9, issue.10, pp.621-639, 1965.

D. Mollison, Dependence of epidemic and population velocities on basic parameters, Mathematical Biosciences, vol.107, issue.2, pp.255-2870025, 1991.
DOI : 10.1016/0025-5564(91)90009-8

J. Monod, Recherches sur la croissance des cellules bactériennes, 1942.

P. D. Moral and L. Miclo, Branching and interacting particle systems approximations of feynman-kac formulae with applications to non-linear filtering, In: Séminaire de Probabilités, XXXIV. Lecture Notes in Math, vol.22, issue.1, pp.1-145, 2000.
DOI : 10.1007/BFb0006577

H. Moser, Structure and Dynamics of Bacterial Populations Maintained in the Chemostat, Cold Spring Harbor Symposia on Quantitative Biology, vol.22, issue.0, 1958.
DOI : 10.1101/SQB.1957.022.01.015

J. D. Murray, E. A. Stanley, and D. L. Brown, On the Spatial Spread of Rabies among Foxes, Proceedings of the Royal Society B: Biological Sciences, vol.229, issue.1255, pp.111-150, 1255.
DOI : 10.1098/rspb.1986.0078

M. I. Nelson and H. S. Sidhu, Analysis of the activated sludge model (number 1), Applied Mathematics Letters, vol.22, issue.5, pp.629-635, 2009.
DOI : 10.1016/j.aml.2008.05.003

D. R. Noguera and C. Picioreanu, Results from the multi-species Benchmark Problem 3 (BM3) using two-dimensional models, Water Sci. Technol, vol.49, pp.11-12, 2004.

R. Pearl and L. J. Reed, On the Rate of Growth of the Population of the United States since 1790 and Its Mathematical Representation, Proceedings of the National Academy of Sciences, vol.6, issue.6, pp.275-288, 1920.
DOI : 10.1073/pnas.6.6.275

C. Picioreanu, M. C. Van-loosdrecht, and J. J. Heijnen, Mathematical modeling of biofilm structure with a hybrid differential-discrete cellular automaton approach, 1<101::AID-BIT11>3.0.CO;2-M, pp.101-1161097, 1998.
DOI : 10.1002/(SICI)1097-0290(19980405)58:1<101::AID-BIT11>3.0.CO;2-M

S. Pilyugin and P. Waltman, The Simple Chemostat with Wall Growth, SIAM Journal on Applied Mathematics, vol.59, issue.5, pp.1552-1572, 1999.
DOI : 10.1137/S0036139997326181

J. C. Poggiale, J. Michalski, and R. Arditi, Emergence of donor control in patchy predator-prey systems, Bulletin of Mathematical Biology, vol.60, issue.6, pp.1149-1166, 1998.
DOI : 10.1016/S0092-8240(98)90006-4

P. Pollett, Quasi-Stationary Distributions: A Bibliography, 2015.

J. I. Prosser, B. J. Bohannan, T. P. Curtis, R. J. Ellis, M. K. Firestone et al., The role of ecological theory in microbial ecology, Nature Reviews Microbiology, vol.25, issue.5, pp.384-392, 2007.
DOI : 10.1038/nrmicro1643

A. Rapaport, I. Haidar, and J. Harmand, Global dynamics of the buffered chemostat for a general class of growth functions, J. Math. Biol, pp.285-299, 2014.

P. J. Reilly, Stability of commensalistic systems, Biotechnology and Bioengineering, vol.32, issue.10, pp.1373-1392, 1974.
DOI : 10.1002/bit.260161006

B. E. Rittmann and P. L. Mccarty, Environmental Biotechnology: Principles and Applications, 2001.

J. A. Roels, Application of macroscopic principles to microbial metabolism, Biotechnology and Bioengineering, vol.47, issue.12, pp.2457-2514, 1980.
DOI : 10.1002/bit.260221202

T. Sari and J. Harmand, Maintenance does not affect the stability of a twotiered microbial 'food chain', hal-01026149. https, 2014.

T. Sari, M. E. Hajji, and J. Harmand, The mathematical analysis of a syntrophic relationship between two microbial species in a chemostat, Mathematical Biosciences and Engineering, vol.9, issue.3, pp.627-645, 2012.
DOI : 10.3934/mbe.2012.9.627

URL : https://hal.archives-ouvertes.fr/hal-00765311

M. Sbarciog, M. Loccufier, and E. Noldus, Determination of appropriate operating strategies for anaerobic digestion systems, Biochemical Engineering Journal, vol.51, issue.3, pp.180-188, 2010.
DOI : 10.1016/j.bej.2010.06.016

G. Stephanopoulos, R. Aris, and A. Fredrickson, A stochastic analysis of the growth of competing microbial populations in a continuous biochemical reactor, Mathematical Biosciences, vol.45, issue.1-2, 1979.
DOI : 10.1016/0025-5564(79)90098-1

I. Takács, G. G. Patry, and D. Nolasco, A dynamic model of the clarification-thickening process, Water Research, vol.25, issue.10, pp.1263-12710043, 1991.
DOI : 10.1016/0043-1354(91)90066-Y

B. Tang, A. Sitomer, and T. Jackson, Population dynamics and competition in chemostat models with adaptive nutrient uptake, Journal of Mathematical Biology, vol.35, issue.4, pp.453-479, 1997.
DOI : 10.1007/s002850050061

D. N. Thomas, S. J. Judd, and N. Fawcett, Flocculation modelling: a review, Water Research, vol.33, issue.7, pp.1579-1592, 1999.
DOI : 10.1016/S0043-1354(98)00392-3

Y. Tyutyunov, L. Titova, and R. Arditi, Predator interference emerging from trophotaxis in predator???prey systems: An individual-based approach, Ecological Complexity, vol.5, issue.1, pp.48-58, 2008.
DOI : 10.1016/j.ecocom.2007.09.001

C. Venter and D. Cohen, The Century of Biology, New Perspectives Quarterly, vol.21, issue.4, pp.73-77, 2004.
DOI : 10.1111/j.1540-5842.2004.00701.x

P. Verhulst, Notice sur la loi que la population poursuit dans son accroissement, Corresp. Math. Phys, vol.10, pp.113-121, 1838.

D. Villemonais, Interacting Particle Systems and Yaglom Limit Approximation of Diffusions with Unbounded Drift, Electronic Journal of Probability, vol.16, issue.0, pp.1663-1692, 2011.
DOI : 10.1214/EJP.v16-925

URL : https://hal.archives-ouvertes.fr/hal-00481580

M. J. Wade, R. W. Pattinson, N. G. Parker, and J. Dolfing, Emergent behaviour in a chlorophenol-mineralising three-tiered microbial ???food web???, Journal of Theoretical Biology, vol.389, pp.171-186, 2016.
DOI : 10.1016/j.jtbi.2015.10.032

M. Weedermann, G. S. Wolkowicz, and J. Sasara, Optimal biogas production in a model for anaerobic digestion, Nonlinear Dynamics, vol.52, issue.3, 2015.
DOI : 10.1007/s11071-015-2051-z

G. S. Wolkowicz, Successful invasion of a food web in a chemostat, Mathematical Biosciences, vol.93, issue.2, pp.249-2680025, 1989.
DOI : 10.1016/0025-5564(89)90025-4

A. Xu, J. Dolfing, T. P. Curtis, G. Montague, and E. Martin, Maintenance affects the stability of a two-tiered microbial ???food chain????, Journal of Theoretical Biology, vol.276, issue.1, pp.35-41, 2011.
DOI : 10.1016/j.jtbi.2011.01.026

URL : https://hal.archives-ouvertes.fr/hal-00682412

H. Ye, Equation-free mechanistic ecosystem forecasting using empirical dynamic modeling, Proceedings of the National Academy of Sciences, vol.112, issue.13, pp.1569-1576, 2015.
DOI : 10.1073/pnas.1417063112