2016, 13(4): 631-652. doi: 10.3934/mbe.2016012

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

Received  April 2015 Revised  March 2016 Published  May 2016

We study a model of the chemostat with several species in competition for a single resource. We take into account the intra-specific interactions between individuals of the same population of micro-organisms and we assume that the growth rates are increasing and the dilution rates are distinct. Using the concept of steady-state characteristics, we present a geometric characterization of the existence and stability of all equilibria. Moreover, we provide necessary and sufficient conditions on the control parameters of the system to have a positive equilibrium. Using a Lyapunov function, we give a global asymptotic stability result for the competition model of several species. The operating diagram describes the asymptotic behavior of this model with respect to control parameters and illustrates the effect of the intra-specific competition on the coexistence region of the species.
Citation: Nahla Abdellatif, Radhouane Fekih-Salem, Tewfik Sari. Competition for a single resource and coexistence of several species in the chemostat. Mathematical Biosciences & Engineering, 2016, 13 (4) : 631-652. doi: 10.3934/mbe.2016012
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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