S. , J. Liu, and X. Liu, Convergence rate of synchronization of systems with additive noise, Discrete Contin, Dyn. Syst. Ser. B, vol.22, pp.227-245, 2017.

L. Arnold, Random Dynamical Systems, 1998.

J. Barlow, W. Schaffner, F. De-noyelles, B. Peterson, and J. Peterson, Continuous flow nutrient bioassays with natural phytoplankton populations, Bioassay Techniques and Environmental Chemistry, 1973.

F. Campillo and M. , Stochastic modeling of the chemostat, Ecological Modelling, vol.222, issue.15, pp.2676-2689, 2011.
DOI : 10.1016/j.ecolmodel.2011.04.027

URL : https://hal.archives-ouvertes.fr/hal-00641231

F. Campillo and M. , Joannides and I. Larramendy-Valverde, Approximation of the fokker?planck equation of the stochastic chemostat, Mathematics and Computers in Simulation, pp.99-136, 2014.

F. Campillo and M. , Analysis and Approximation of a Stochastic Growth Model with Extinction, Methodology and Computing in Applied Probability, vol.11, issue.1, pp.499-515, 2016.
DOI : 10.1215/kjm/1250523691

URL : https://hal.archives-ouvertes.fr/hal-01817824

T. Caraballo, M. J. Garrido-atienza, and J. López-de-la-cruz, Some Aspects Concerning the Dynamics of Stochastic Chemostats, pp.227-246, 2016.

T. Caraballo, M. J. Garrido-atienza, and J. López-de-la-cruz, Dynamics of some stochastic chemostat models with multiplicative noise, Communications on Pure and Applied Analysis, vol.16, issue.5, pp.16-1893, 2017.
DOI : 10.3934/cpaa.2017092

T. Caraballo, M. J. Garrido-atienza, and J. , López-de-la-Cruz and A. Rapaport, Corrigendum to " some aspects concerning the dynamics of stochastic chemostats, 2017.

T. Caraballo, M. J. Garrido-atienza, B. Schmalfuss, and J. Valero, Asymptotic behaviour of a stochastic semilinear dissipative functional equation without uniqueness of solutions, Discrete and Continuous Dynamical Systems -Series, pp.14-439, 2010.

T. Caraballo and X. Han, Applied Nonautonomous and Random Dynamical Systems, 2016.
DOI : 10.1007/978-3-319-49247-6

URL : https://idus.us.es/xmlui/bitstream/11441/58773/1/Applied%20nonautonomous%20and%20random%20dynamical%20systems.pdf

T. Caraballo, P. E. Kloeden, and B. Schmalfuss, Exponentially Stable Stationary Solutions for Stochastic Evolution Equations and Their Perturbation, Applied Mathematics and Optimization, vol.50, issue.3, pp.183-207, 2004.
DOI : 10.1007/s00245-004-0802-1

URL : https://idus.us.es/xmlui/bitstream/11441/23668/1/file_1.pdf

T. Caraballo and K. Lu, Attractors for stochastic lattice dynamical systems with a multiplicative noise, Frontiers of Mathematics in China, vol.96, issue.1, pp.317-335, 2008.
DOI : 10.1080/17442509608834083

URL : https://idus.us.es/xmlui/bitstream/11441/23654/1/file_1.pdf

T. Caraballo, G. Lukaszewicz, and J. , Pullback attractors for asymptotically compact non-autonomous dynamical systems, Nonlinear Analysis: Theory, Methods & Applications, vol.64, pp.484-498, 2006.
DOI : 10.1016/j.na.2005.03.111

URL : https://idus.us.es/xmlui/bitstream/11441/23702/1/file_1.pdf

I. F. Creed, D. M. Mcknight, B. A. Pellerin, M. B. Green, B. A. Bergamaschi et al., The river as a chemostat: fresh perspectives on dissolved organic matter flowing down the river continuum, Canadian Journal of Fisheries and Aquatic Sciences, vol.8, issue.8, pp.72-1272, 2015.
DOI : 10.1023/A:1024924216148

G. D. 'ans, P. Kokotovic, and D. Gottlieb, A nonlinear regulator problem for a model of biological waste treatment, IEEE Transactions on Automatic Control, vol.16, pp.341-347, 1971.

F. Flandoli and B. Schmalfuss, Random attractors for the 3D stochastic navier-stokes equation with multiplicative white noise, Stochastics and Stochastic Reports, pp.59-80, 1996.

J. Grasman, M. D. Gee, and O. A. Herwaarden, Breakdown of a Chemostat Exposed to Stochastic Noise, Journal of Engineering Mathematics, vol.35, issue.3-4, pp.53-291, 2005.
DOI : 10.1007/978-3-662-02377-8

T. Caraballo, M. Garrido-atienza, J. , and A. Rapaport,

J. Harmand, C. Lobry, A. Rapaport, and T. Sari, The Chemostat: Mathematical Theory of Micro-organisms Cultures, Chemical Engineering Series, 2017.
DOI : 10.1002/9781119437215

D. J. Higham, An Algorithmic Introduction to Numerical Simulation of Stochastic Differential Equations, SIAM Review, vol.43, issue.3, pp.525-546, 2001.
DOI : 10.1137/S0036144500378302

S. B. 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.32-366, 1977.
DOI : 10.1137/0132030

L. Imhof and S. Walcher, Exclusion and persistence in deterministic and stochastic chemostat models, Journal of Differential Equations, vol.217, issue.1, pp.26-53, 2005.
DOI : 10.1016/j.jde.2005.06.017

H. W. Jannasch, Steady state and the chemostat in ecology1,1, Limnology and Oceanography, vol.19, issue.4, pp.716-720, 1974.
DOI : 10.4319/lo.1974.19.4.0716

J. Kalff and R. Knoechel, Phytoplankton and their Dynamics in Oligotrophic and Eutrophic Lakes, Annual Review of Ecology and Systematics, vol.9, issue.1, pp.475-495, 1978.
DOI : 10.1146/annurev.es.09.110178.002355

J. W. , LaRivì ere, Microbial ecology of liquid waste treatment, Advances in Microbial Ecology, pp.215-259, 1977.

E. Rurangwa and M. C. Verdegem, Microorganisms in recirculating aquaculture systems and their management, Reviews in Aquaculture, vol.25, issue.3, pp.117-130, 2015.
DOI : 10.1016/S0144-8609(01)00071-1

H. L. Smith and P. Waltman, The theory of the chemostat: dynamics of microbial competition, 1995.
DOI : 10.1017/CBO9780511530043

L. Wang and D. Jiang, Periodic solution for the stochastic chemostat with general response function, Physica A: Statistical Mechanics and its Applications, pp.486-378, 2017.

L. Wang, D. Jiang, and D. O-'regan, The periodic solutions of a stochastic chemostat model with periodic washout rate, Communications in Nonlinear Science and Numerical Simulation, vol.37, pp.1-13, 2016.
DOI : 10.1016/j.cnsns.2016.01.002

C. Xu and S. Yuan, An analogue of break-even concentration in a simple stochastic chemostat model, Applied Mathematics Letters, vol.48, pp.62-68, 2015.
DOI : 10.1016/j.aml.2015.03.012

C. Xu, S. Yuan, and T. Zhang, Asymptotic behavior of a chemostat model with stochastic perturbation on the dilution rate, Abstract and Applied Analysis, pp.2013-2014, 2013.

D. Zhao and S. Yuan, Critical result on the break-even concentration in a single-species stochastic chemostat model, Journal of Mathematical Analysis and Applications, vol.434, issue.2, pp.1336-1345, 2016.
DOI : 10.1016/j.jmaa.2015.09.070

D. Zhao and S. Yuan, Break-even concentration and periodic behavior of a stochastic chemostat model with seasonal fluctuation, Communications in Nonlinear Science and Numerical Simulation, vol.46, pp.46-62, 2017.
DOI : 10.1016/j.cnsns.2016.10.014

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