Practical coexistence of two species in the chemostat – A slow–fast characterization

Aris, 1977, Dynamics of a chemostat in which two organisms compete for a common substrate, Biotechnol. Bioeng., 19, 1375, 10.1002/bit.260190910 Butler, 1995, A mathematical model of the chemostat with a general class of functions describing nutrient uptake, SIAM J. Appl. Math., 45, 138, 10.1137/0145006 Butler, 1985, A mathematical model of the chemostat with periodic washout rate, SIAM J. Appl. Math., 45, 435, 10.1137/0145025 Cenens, 2000, Modeling the competition between floc-forming and filamentous bacteria in activated sludge waste water treatment systems – II. A prototype mathematical model based on kinetic selection and filamentous backbone theory, Water Res., 34, 2535, 10.1016/S0043-1354(99)00422-4 Duchêne, 1996, Kinetics scheme reduction, attractive invariant manifold and slow/fast dynamical systems, Chem. Eng. Sci., 51, 4661, 10.1016/0009-2509(96)00310-7 Fenichel, 1979, Geometric singular perturbation theory for ordinary differential equations, J. Differ. Equations, 31, 53, 10.1016/0022-0396(79)90152-9 Haegeman, 2008, How flocculation can explain coexistence in the chemostat, J. Biol. Dyn., 2, 1, 10.1080/17513750801942537 Hansen, 1980, Single nutrient microbial competition: agreement between experimental and theoretical forecast outcomes, Science, 207, 1491, 10.1126/science.6767274 Hardin, 1960, The competition exclusion principle, Science, 131, 1292, 10.1126/science.131.3409.1292 Hsu, 1977, A mathematical theory for single-nutrient competition in continuous cultures of micro-organisms, SIAM J. Appl. Math., 32, 366, 10.1137/0132030 C.K.R.T. Jones, in: Geometric Singular Perturbation Theory, Lecture Notes in Mathematics ‘Dynamical Systems: Monteactini Terme’, vol. 1609, 1994, pp. 44. Khalil, 1996 Lobry, 2004, Persistence in ecological models of competition for a single resource, C. R. Acad. Sci. Paris Ser. I, 340, 199, 10.1016/j.crma.2004.12.021 Lobry, 2006, A new hypothesis to explain the coexistence of n species in the presence of a single resource, C. R. Biol., 329, 40, 10.1016/j.crvi.2005.10.004 Lobry, 2006, Sur un modèle densité-dépendant de compétition pour une resource, C. R. Biol., 329, 63, 10.1016/j.crvi.2005.11.004 A. Rapaport, D. Dochain, J. Harmand, Long run coexistence in the chemostat with multiple species, J. Theor. Biol., to appear. Rapaport, 2008, Coexistence in the design of a series of two chemostats, Nonlinear Anal. Real World Appl., 9, 1052, 10.1016/j.nonrwa.2007.02.003 Smith, 1981, Competitive coexistence in an oscillating chemostat, SIAM J. Appl. Math., 40, 498, 10.1137/0140042 Smith, 1995, The theory of the chemostat. Dynamics of microbial competition, vol. 13 Stephanopoulos, 1979, A stochastic analysis of the growth of competing microbial populations in a continuous biochemical reactor, Math. Biosci., 45, 99, 10.1016/0025-5564(79)90098-1 Stephanopoulos, 1979, Effect of inhomogeneities on the coexistence of competing microbial populations, Biotechnol. Bioeng., 21, 1491, 10.1002/bit.260210817 Tikhonov, 1952, Systems of differential equations containing a small parameter multiplying the derivative, Math. Sb., 31, 575