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Competition for a single resource and coexistence of several species in the chemostat
1. | Université de Tunis El Manar, ENIT, LAMSIN, BP 37, Le Belvédère, 1002 Tunis, Tunisia, Tunisia |
2. | IRSTEA, UMR Itap, 361 rue Jean-François Breton, 34196 Montpellier, France, and Université de Haute Alsace, LMIA, 4 rue des frères Lumière, 68093 Mulhouse, France |
References:
[1] |
R. Arditi and L. R. Ginzburg, Coupling in predator-prey dynamics: Ratio-dependence, J. Theor. Biol., 139 (1989), 311-326.
doi: 10.1016/S0022-5193(89)80211-5. |
[2] |
C. K. Essajee and R. D. Tanner, The effect of extracellular variables on the stability of the continuous baker's yeast-ethanol fermentation process, Process Biochem., 14 (1979), 16-25. |
[3] |
R. Fekih-Salem, Modèles mathématiques pour la compétition et la coexistence des espèces microbiennes dans un chémostat, Ph.D thesis, University of Montpellier 2 and University of Tunis el Manar, 2013. https://tel.archives-ouvertes.fr/tel-01018600. |
[4] |
R. Fekih-Salem, J. Harmand, C. Lobry, A. Rapaport and T. Sari, Extensions of the chemostat model with flocculation, J. Math. Anal. Appl., 397 (2013), 292-306.
doi: 10.1016/j.jmaa.2012.07.055. |
[5] |
R. Fekih-Salem, T. Sari and N. Abdellatif, Sur un modèle de compétition et de coexistence dans le chémostat, ARIMA J., 14 (2011), 15-30. |
[6] |
B. Haegeman, C. Lobry and J. Harmand, Modeling bacteria flocculation as density-dependent growth, AIChE J., 53 (2007), 535-539.
doi: 10.1002/aic.11077. |
[7] |
B. Haegeman and A. Rapaport, How flocculation can explain coexistence in the chemostat, J. Biol. Dyn., 2 (2008), 1-13.
doi: 10.1080/17513750801942537. |
[8] |
J. K. Hale and A. S. Somolinas, Competition for fluctuating nutrient, J. Math. Biol., 18 (1983), 255-280.
doi: 10.1007/BF00276091. |
[9] |
J. Harmand and J. J. Godon, Density-dependent kinetics models for a simple description of complex phenomena in macroscopic mass-balance modeling of bioreactors, Ecol. Modell., 200 (2007), 393-402.
doi: 10.1016/j.ecolmodel.2006.08.012. |
[10] |
S. B. Hsu, A competition model for a seasonally fluctuating nutrient, J. Math. Biol., 9 (1980), 115-132.
doi: 10.1007/BF00275917. |
[11] |
P. De Leenheer, D. Angeli and E. D. Sontag, Crowding effects promote coexistence in the chemostat, J. Math. Anal. Appl., 319 (2006), 48-60.
doi: 10.1016/j.jmaa.2006.02.036. |
[12] |
C. Lobry and J. Harmand, A new hypothesis to explain the coexistence of $n$ species in the presence of a single resource, C. R. Biol., 329 (2006), 40-46.
doi: 10.1016/j.crvi.2005.10.004. |
[13] |
C. Lobry and F. Mazenc, Effect on persistence of intra-specific competition in competition models, Electron. J. Diff. Eqns., 125 (2007), 1-10. |
[14] |
C. Lobry, F. Mazenc and A. Rapaport, Persistence in ecological models of competition for a single resource, C. R. Acad. Sci. Paris, Ser. I, 340 (2005), 199-204.
doi: 10.1016/j.crma.2004.12.021. |
[15] |
C. Lobry, A. Rapaport and F. Mazenc, Sur un modèle densité-dépendant de compétition pour une ressource, C. R. Biol., 329 (2006), 63-70.
doi: 10.1016/j.crvi.2005.11.004. |
[16] |
A. Rapaport and J. Harmand, Biological control of the chemostat with nonmonotonic response and different removal rates, Math. Biosci. Eng., 5 (2008), 539-547.
doi: 10.3934/mbe.2008.5.539. |
[17] |
S. Ruan, A. Ardito, P. Ricciardi and D. L. DeAngelis, Coexistence in competition models with density-dependent mortality, C. R. Biol., 330 (2007), 845-854.
doi: 10.1016/j.crvi.2007.10.004. |
[18] |
T. Sari, A Lyapunov function for the chemostat with variable yields, C. R. Acad. Sci. Paris Ser. I, 348 (2010), 747-751.
doi: 10.1016/j.crma.2010.06.008. |
[19] |
T. Sari, Competitive exclusion for chemostat equations with variable yields, Acta Appl. Math., 123 (2013), 201-219.
doi: 10.1007/s10440-012-9761-8. |
[20] |
T. Sari and F. Mazenc, Global dynamics of the chemostat with different removal rates and variable yields, Math. Biosci. Eng., 8 (2011), 827-840.
doi: 10.3934/mbe.2011.8.827. |
[21] |
H. L. Smith and P. Waltman, The Theory of the Chemostat: Dynamics of Microbial Competition, Cambridge University Press, 1995.
doi: 10.1017/CBO9780511530043. |
[22] |
G. Stephanopoulos, A. G. Frederickson and R. Aris, The growth of competing microbial populations in a CSTR with periodically varying inputs, AIChE J., 25 (1979), 863-872.
doi: 10.1002/aic.690250515. |
[23] |
G. S. K. Wolkowicz and Z. Lu, Global dynamics of a mathematical model of competition in the chemostat: general response functions and differential death rates, SIAM J. Appl. Math., 52 (1992), 222-233.
doi: 10.1137/0152012. |
[24] |
G. S. K. Wolkowicz and Z. Lu, Direct interference on competition in a chemostat, J. Biomath, 13 (1998), 282-291. |
[25] |
G. S. K. Wolkowicz and X. Q. Zhao, $N$-species competition in a periodic chemostat, Differential Integral Equations, 11 (1998), 465-491. |
show all references
References:
[1] |
R. Arditi and L. R. Ginzburg, Coupling in predator-prey dynamics: Ratio-dependence, J. Theor. Biol., 139 (1989), 311-326.
doi: 10.1016/S0022-5193(89)80211-5. |
[2] |
C. K. Essajee and R. D. Tanner, The effect of extracellular variables on the stability of the continuous baker's yeast-ethanol fermentation process, Process Biochem., 14 (1979), 16-25. |
[3] |
R. Fekih-Salem, Modèles mathématiques pour la compétition et la coexistence des espèces microbiennes dans un chémostat, Ph.D thesis, University of Montpellier 2 and University of Tunis el Manar, 2013. https://tel.archives-ouvertes.fr/tel-01018600. |
[4] |
R. Fekih-Salem, J. Harmand, C. Lobry, A. Rapaport and T. Sari, Extensions of the chemostat model with flocculation, J. Math. Anal. Appl., 397 (2013), 292-306.
doi: 10.1016/j.jmaa.2012.07.055. |
[5] |
R. Fekih-Salem, T. Sari and N. Abdellatif, Sur un modèle de compétition et de coexistence dans le chémostat, ARIMA J., 14 (2011), 15-30. |
[6] |
B. Haegeman, C. Lobry and J. Harmand, Modeling bacteria flocculation as density-dependent growth, AIChE J., 53 (2007), 535-539.
doi: 10.1002/aic.11077. |
[7] |
B. Haegeman and A. Rapaport, How flocculation can explain coexistence in the chemostat, J. Biol. Dyn., 2 (2008), 1-13.
doi: 10.1080/17513750801942537. |
[8] |
J. K. Hale and A. S. Somolinas, Competition for fluctuating nutrient, J. Math. Biol., 18 (1983), 255-280.
doi: 10.1007/BF00276091. |
[9] |
J. Harmand and J. J. Godon, Density-dependent kinetics models for a simple description of complex phenomena in macroscopic mass-balance modeling of bioreactors, Ecol. Modell., 200 (2007), 393-402.
doi: 10.1016/j.ecolmodel.2006.08.012. |
[10] |
S. B. Hsu, A competition model for a seasonally fluctuating nutrient, J. Math. Biol., 9 (1980), 115-132.
doi: 10.1007/BF00275917. |
[11] |
P. De Leenheer, D. Angeli and E. D. Sontag, Crowding effects promote coexistence in the chemostat, J. Math. Anal. Appl., 319 (2006), 48-60.
doi: 10.1016/j.jmaa.2006.02.036. |
[12] |
C. Lobry and J. Harmand, A new hypothesis to explain the coexistence of $n$ species in the presence of a single resource, C. R. Biol., 329 (2006), 40-46.
doi: 10.1016/j.crvi.2005.10.004. |
[13] |
C. Lobry and F. Mazenc, Effect on persistence of intra-specific competition in competition models, Electron. J. Diff. Eqns., 125 (2007), 1-10. |
[14] |
C. Lobry, F. Mazenc and A. Rapaport, Persistence in ecological models of competition for a single resource, C. R. Acad. Sci. Paris, Ser. I, 340 (2005), 199-204.
doi: 10.1016/j.crma.2004.12.021. |
[15] |
C. Lobry, A. Rapaport and F. Mazenc, Sur un modèle densité-dépendant de compétition pour une ressource, C. R. Biol., 329 (2006), 63-70.
doi: 10.1016/j.crvi.2005.11.004. |
[16] |
A. Rapaport and J. Harmand, Biological control of the chemostat with nonmonotonic response and different removal rates, Math. Biosci. Eng., 5 (2008), 539-547.
doi: 10.3934/mbe.2008.5.539. |
[17] |
S. Ruan, A. Ardito, P. Ricciardi and D. L. DeAngelis, Coexistence in competition models with density-dependent mortality, C. R. Biol., 330 (2007), 845-854.
doi: 10.1016/j.crvi.2007.10.004. |
[18] |
T. Sari, A Lyapunov function for the chemostat with variable yields, C. R. Acad. Sci. Paris Ser. I, 348 (2010), 747-751.
doi: 10.1016/j.crma.2010.06.008. |
[19] |
T. Sari, Competitive exclusion for chemostat equations with variable yields, Acta Appl. Math., 123 (2013), 201-219.
doi: 10.1007/s10440-012-9761-8. |
[20] |
T. Sari and F. Mazenc, Global dynamics of the chemostat with different removal rates and variable yields, Math. Biosci. Eng., 8 (2011), 827-840.
doi: 10.3934/mbe.2011.8.827. |
[21] |
H. L. Smith and P. Waltman, The Theory of the Chemostat: Dynamics of Microbial Competition, Cambridge University Press, 1995.
doi: 10.1017/CBO9780511530043. |
[22] |
G. Stephanopoulos, A. G. Frederickson and R. Aris, The growth of competing microbial populations in a CSTR with periodically varying inputs, AIChE J., 25 (1979), 863-872.
doi: 10.1002/aic.690250515. |
[23] |
G. S. K. Wolkowicz and Z. Lu, Global dynamics of a mathematical model of competition in the chemostat: general response functions and differential death rates, SIAM J. Appl. Math., 52 (1992), 222-233.
doi: 10.1137/0152012. |
[24] |
G. S. K. Wolkowicz and Z. Lu, Direct interference on competition in a chemostat, J. Biomath, 13 (1998), 282-291. |
[25] |
G. S. K. Wolkowicz and X. Q. Zhao, $N$-species competition in a periodic chemostat, Differential Integral Equations, 11 (1998), 465-491. |
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