G. Barles and B. Perthame, Concentrations and constrained Hamilton-Jacobi equations arising in adaptive dynamics, Contemp. Math, vol.439, pp.57-68, 2007.
DOI : 10.1090/conm/439/08463

F. Campillo, M. Joannides, and I. Larramendy, 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

N. Champagnat, R. Ferrière, and S. Méléard, From Individual Stochastic Processes to Macroscopic Models in Adaptive Evolution, Stochastic Models, vol.17, issue.sup1, pp.2-44, 2008.
DOI : 10.1080/15326340802437710
URL : https://hal.archives-ouvertes.fr/inria-00345680

N. Champagnat, A microscopic interpretation for adaptive dynamics trait substitution sequence models. Stochastic Process, Appl, vol.116, issue.8, pp.1127-1160, 2006.
URL : https://hal.archives-ouvertes.fr/hal-00015130

N. Champagnat and P. Jabin, The evolutionary limit for models of populations interacting competitively via several resources, Journal of Differential Equations, vol.251, issue.1, pp.179-195, 2011.
DOI : 10.1016/j.jde.2011.03.007
URL : https://hal.archives-ouvertes.fr/inria-00488979

N. Champagnat, P. Jabin, and G. Raoul, Convergence to equilibrium in competitive Lotka???Volterra and chemostat systems, Comptes Rendus Mathematique, vol.348, issue.23-24, pp.23-24, 2010.
DOI : 10.1016/j.crma.2010.11.001
URL : https://hal.archives-ouvertes.fr/inria-00495991

N. Champagnat and S. Méléard, Polymorphic evolution sequence and evolutionary branching, Probability Theory and Related Fields 151, pp.45-94, 2011.

P. Collet, S. Martinez, and S. Méléard, Stochastic models for a chemostat and long time behavior, Adv. Appl. Probab, 2012.
URL : https://hal.archives-ouvertes.fr/hal-00708680

K. Crump and W. 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

A. Dembo and O. Zeitouni, Large Deviations Techniques and Applications, 1993.

U. Dieckmann and R. Law, The dynamical theory of coevolution: a derivation from stochastic ecological processes, Journal of Mathematical Biology, vol.39, issue.5-6, pp.579-612, 1996.
DOI : 10.1007/BF02409751

O. Diekmann, P. Jabin, S. Mischler, and B. Perthame, The dynamics of adaptation: An illuminating example and a Hamilton???Jacobi approach, Theoretical Population Biology, vol.67, issue.4, pp.257-271, 2005.
DOI : 10.1016/j.tpb.2004.12.003

S. N. Ethier and T. G. Kurtz, Markov Processes, characterization and convergence, 1986.

N. Fournier and S. Méléard, A microscopic probabilistic description of a locally regulated population and macroscopic approximations, The Annals of Applied Probability, vol.14, issue.4, pp.1880-1919, 2004.
DOI : 10.1214/105051604000000882

M. I. Freidlin and A. D. Wentzel, Random Perturbations of Dynamical Systems, 1984.

K. Graham and S. Méléard, An upper bound of large deviations for a generalized starshaped loss network. Markov Processes Relat, pp.199-223, 1997.

J. A. Metz, R. M. Nisbet, and S. A. Geritz, How should we define ???fitness??? for general ecological scenarios?, Trends in Ecology & Evolution, vol.7, issue.6, pp.198-202, 1992.
DOI : 10.1016/0169-5347(92)90073-K

J. A. Metz, S. A. Geritz, G. Meszéna, F. A. Jacobs, and J. S. Van-heerwaarden, Adaptive Dynamics, a geometrical study of the consequences of nearly faithful reproduction. Stochastic and Spatial Structures of Dynamical Systems, S.J. van Strien, S.M. Verduyn Lunel, pp.183-231, 1996.

S. Mirrahimi, B. Perthame, and J. Y. Wakano, Evolution of species trait through resource competition, Journal of Mathematical Biology, vol.13, issue.2, 2012.
DOI : 10.1007/s00285-011-0447-z
URL : https://hal.archives-ouvertes.fr/hal-00566888

S. Mirrahimi, B. Perthame, and J. Y. Wakano, Direct competition results from strong competition for limited resource, Journal of Mathematical Biology, vol.49, issue.2
DOI : 10.1007/s00285-013-0659-5
URL : https://hal.archives-ouvertes.fr/hal-00663832

J. Monod, La technique de culture continue, Ann. Inst. Pasteur, vol.79, pp.390-410, 1950.

A. Novick and L. Szilard, Experiments with the Chemostat on Spontaneous Mutations of Bacteria, Proceedings of the National Academy of Sciences, vol.36, issue.12, pp.708-719, 1950.
DOI : 10.1073/pnas.36.12.708

A. Novick and L. Szilard, Description of the Chemostat, Science, vol.112, issue.2920, pp.715-721, 1950.
DOI : 10.1126/science.112.2920.715

H. Smith and . Waltman, The Theory of the Chemostat, 1995.
DOI : 10.1017/CBO9780511530043

M. L. Zeeman, Hopf bifurcations in competitive three-dimensional Lotka???Volterra systems, Dynamics and Stability of Systems, vol.11, issue.3, pp.189-217, 1993.
DOI : 10.1007/BFb0087009