American Institute of Mathematical Sciences

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

Competition for a single resource and coexistence of several species in the chemostat

Nahla Abdellatif 1, , Radhouane Fekih-Salem 1, and  Tewfik Sari 2,

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.
Keywords: coexistence, intra-specific, competition, Chemostat, operating diagram..
Mathematics Subject Classification: 92D25, 34D20, 34D4.
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. doi: 10.1016/S0022-5193(89)80211-5. Google Scholar

[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. Google Scholar

[3]

R. Fekih-Salem, Modèles mathématiques pour la compétition et la coexistence des espèces microbiennes dans un chémostat,, Ph.D thesis, (2013). Google Scholar

[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. doi: 10.1016/j.jmaa.2012.07.055. Google Scholar

[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. Google Scholar

[6]

B. Haegeman, C. Lobry and J. Harmand, Modeling bacteria flocculation as density-dependent growth,, AIChE J., 53 (2007), 535. doi: 10.1002/aic.11077. Google Scholar

[7]

B. Haegeman and A. Rapaport, How flocculation can explain coexistence in the chemostat,, J. Biol. Dyn., 2 (2008), 1. doi: 10.1080/17513750801942537. Google Scholar

[8]

J. K. Hale and A. S. Somolinas, Competition for fluctuating nutrient,, J. Math. Biol., 18 (1983), 255. doi: 10.1007/BF00276091. Google Scholar

[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. doi: 10.1016/j.ecolmodel.2006.08.012. Google Scholar

[10]

S. B. Hsu, A competition model for a seasonally fluctuating nutrient,, J. Math. Biol., 9 (1980), 115. doi: 10.1007/BF00275917. Google Scholar

[11]

P. De Leenheer, D. Angeli and E. D. Sontag, Crowding effects promote coexistence in the chemostat,, J. Math. Anal. Appl., 319 (2006), 48. doi: 10.1016/j.jmaa.2006.02.036. Google Scholar

[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. doi: 10.1016/j.crvi.2005.10.004. Google Scholar

[13]

C. Lobry and F. Mazenc, Effect on persistence of intra-specific competition in competition models,, Electron. J. Diff. Eqns., 125 (2007), 1. Google Scholar

[14]

C. Lobry, F. Mazenc and A. Rapaport, Persistence in ecological models of competition for a single resource,, C. R. Acad. Sci. Paris, 340 (2005), 199. doi: 10.1016/j.crma.2004.12.021. Google Scholar

[15]

C. Lobry, A. Rapaport and F. Mazenc, Sur un modèle densité-dépendant de compétition pour une ressource,, C. R. Biol., 329 (2006), 63. doi: 10.1016/j.crvi.2005.11.004. Google Scholar

[16]

A. Rapaport and J. Harmand, Biological control of the chemostat with nonmonotonic response and different removal rates,, Math. Biosci. Eng., 5 (2008), 539. doi: 10.3934/mbe.2008.5.539. Google Scholar

[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. doi: 10.1016/j.crvi.2007.10.004. Google Scholar

[18]

T. Sari, A Lyapunov function for the chemostat with variable yields,, C. R. Acad. Sci. Paris Ser. I, 348 (2010), 747. doi: 10.1016/j.crma.2010.06.008. Google Scholar

[19]

T. Sari, Competitive exclusion for chemostat equations with variable yields,, Acta Appl. Math., 123 (2013), 201. doi: 10.1007/s10440-012-9761-8. Google Scholar

[20]

T. Sari and F. Mazenc, Global dynamics of the chemostat with different removal rates and variable yields,, Math. Biosci. Eng., 8 (2011), 827. doi: 10.3934/mbe.2011.8.827. Google Scholar

[21]

H. L. Smith and P. Waltman, The Theory of the Chemostat: Dynamics of Microbial Competition,, Cambridge University Press, (1995). doi: 10.1017/CBO9780511530043. Google Scholar

[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. doi: 10.1002/aic.690250515. Google Scholar

[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. doi: 10.1137/0152012. Google Scholar

[24]

G. S. K. Wolkowicz and Z. Lu, Direct interference on competition in a chemostat,, J. Biomath, 13 (1998), 282. Google Scholar

[25]

G. S. K. Wolkowicz and X. Q. Zhao, $N$-species competition in a periodic chemostat,, Differential Integral Equations, 11 (1998), 465. Google Scholar

show all references

References:
[1]

R. Arditi and L. R. Ginzburg, Coupling in predator-prey dynamics: Ratio-dependence,, J. Theor. Biol., 139 (1989), 311. doi: 10.1016/S0022-5193(89)80211-5. Google Scholar

[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. Google Scholar

[3]

R. Fekih-Salem, Modèles mathématiques pour la compétition et la coexistence des espèces microbiennes dans un chémostat,, Ph.D thesis, (2013). Google Scholar

[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. doi: 10.1016/j.jmaa.2012.07.055. Google Scholar

[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. Google Scholar

[6]

B. Haegeman, C. Lobry and J. Harmand, Modeling bacteria flocculation as density-dependent growth,, AIChE J., 53 (2007), 535. doi: 10.1002/aic.11077. Google Scholar

[7]

B. Haegeman and A. Rapaport, How flocculation can explain coexistence in the chemostat,, J. Biol. Dyn., 2 (2008), 1. doi: 10.1080/17513750801942537. Google Scholar

[8]

J. K. Hale and A. S. Somolinas, Competition for fluctuating nutrient,, J. Math. Biol., 18 (1983), 255. doi: 10.1007/BF00276091. Google Scholar

[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. doi: 10.1016/j.ecolmodel.2006.08.012. Google Scholar

[10]

S. B. Hsu, A competition model for a seasonally fluctuating nutrient,, J. Math. Biol., 9 (1980), 115. doi: 10.1007/BF00275917. Google Scholar

[11]

P. De Leenheer, D. Angeli and E. D. Sontag, Crowding effects promote coexistence in the chemostat,, J. Math. Anal. Appl., 319 (2006), 48. doi: 10.1016/j.jmaa.2006.02.036. Google Scholar

[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. doi: 10.1016/j.crvi.2005.10.004. Google Scholar

[13]

C. Lobry and F. Mazenc, Effect on persistence of intra-specific competition in competition models,, Electron. J. Diff. Eqns., 125 (2007), 1. Google Scholar

[14]

C. Lobry, F. Mazenc and A. Rapaport, Persistence in ecological models of competition for a single resource,, C. R. Acad. Sci. Paris, 340 (2005), 199. doi: 10.1016/j.crma.2004.12.021. Google Scholar

[15]

C. Lobry, A. Rapaport and F. Mazenc, Sur un modèle densité-dépendant de compétition pour une ressource,, C. R. Biol., 329 (2006), 63. doi: 10.1016/j.crvi.2005.11.004. Google Scholar

[16]

A. Rapaport and J. Harmand, Biological control of the chemostat with nonmonotonic response and different removal rates,, Math. Biosci. Eng., 5 (2008), 539. doi: 10.3934/mbe.2008.5.539. Google Scholar

[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. doi: 10.1016/j.crvi.2007.10.004. Google Scholar

[18]

T. Sari, A Lyapunov function for the chemostat with variable yields,, C. R. Acad. Sci. Paris Ser. I, 348 (2010), 747. doi: 10.1016/j.crma.2010.06.008. Google Scholar

[19]

T. Sari, Competitive exclusion for chemostat equations with variable yields,, Acta Appl. Math., 123 (2013), 201. doi: 10.1007/s10440-012-9761-8. Google Scholar

[20]

T. Sari and F. Mazenc, Global dynamics of the chemostat with different removal rates and variable yields,, Math. Biosci. Eng., 8 (2011), 827. doi: 10.3934/mbe.2011.8.827. Google Scholar

[21]

H. L. Smith and P. Waltman, The Theory of the Chemostat: Dynamics of Microbial Competition,, Cambridge University Press, (1995). doi: 10.1017/CBO9780511530043. Google Scholar

[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. doi: 10.1002/aic.690250515. Google Scholar

[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. doi: 10.1137/0152012. Google Scholar

[24]

G. S. K. Wolkowicz and Z. Lu, Direct interference on competition in a chemostat,, J. Biomath, 13 (1998), 282. Google Scholar

[25]

G. S. K. Wolkowicz and X. Q. Zhao, $N$-species competition in a periodic chemostat,, Differential Integral Equations, 11 (1998), 465. Google Scholar

[1]

Bachir Bar, Tewfik Sari. The operating diagram for a model of competition in a chemostat with an external lethal inhibitor. Discrete & Continuous Dynamical Systems - B, 2017, 22 (11) : 0-0. doi: 10.3934/dcdsb.2019203
[2]

Sampurna Sengupta, Pritha Das, Debasis Mukherjee. Stochastic non-autonomous Holling type-Ⅲ prey-predator model with predator's intra-specific competition. Discrete & Continuous Dynamical Systems - B, 2018, 23 (8) : 3275-3296. doi: 10.3934/dcdsb.2018244
[3]

Willard S. Keeran, Patrick D. Leenheer, Sergei S. Pilyugin. Feedback-mediated coexistence and oscillations in the chemostat. Discrete & Continuous Dynamical Systems - B, 2008, 9 (2) : 321-351. doi: 10.3934/dcdsb.2008.9.321
[4]

Jianquan Li, Zuren Feng, Juan Zhang, Jie Lou. A competition model of the chemostat with an external inhibitor. Mathematical Biosciences & Engineering, 2006, 3 (1) : 111-123. doi: 10.3934/mbe.2006.3.111
[5]

Hua Nie, Yuan Lou, Jianhua Wu. Competition between two similar species in the unstirred chemostat. Discrete & Continuous Dynamical Systems - B, 2016, 21 (2) : 621-639. doi: 10.3934/dcdsb.2016.21.621
[6]

Hua Nie, Feng-Bin Wang. Competition for one nutrient with recycling and allelopathy in an unstirred chemostat. Discrete & Continuous Dynamical Systems - B, 2015, 20 (7) : 2129-2155. doi: 10.3934/dcdsb.2015.20.2129
[7]

Hua Nie, Sze-Bi Hsu, Jianhua Wu. Coexistence solutions of a competition model with two species in a water column. Discrete & Continuous Dynamical Systems - B, 2015, 20 (8) : 2691-2714. doi: 10.3934/dcdsb.2015.20.2691
[8]

Georg Hetzer, Tung Nguyen, Wenxian Shen. Coexistence and extinction in the Volterra-Lotka competition model with nonlocal dispersal. Communications on Pure & Applied Analysis, 2012, 11 (5) : 1699-1722. doi: 10.3934/cpaa.2012.11.1699
[9]

Hua Nie, Sze-bi Hsu, Jianhua Wu. A competition model with dynamically allocated toxin production in the unstirred chemostat. Communications on Pure & Applied Analysis, 2017, 16 (4) : 1373-1404. doi: 10.3934/cpaa.2017066
[10]

Charlotte Beauthier, Joseph J. Winkin, Denis Dochain. Input/state invariant LQ-optimal control: Application to competitive coexistence in a chemostat. Evolution Equations & Control Theory, 2015, 4 (2) : 143-158. doi: 10.3934/eect.2015.4.143
[11]

Benlong Xu, Hongyan Jiang. Invasion and coexistence of competition-diffusion-advection system with heterogeneous vs homogeneous resources. Discrete & Continuous Dynamical Systems - B, 2018, 23 (10) : 4255-4266. doi: 10.3934/dcdsb.2018136
[12]

Sze-Bi Hsu, Cheng-Che Li. A discrete-delayed model with plasmid-bearing, plasmid-free competition in a chemostat. Discrete & Continuous Dynamical Systems - B, 2005, 5 (3) : 699-718. doi: 10.3934/dcdsb.2005.5.699
[13]

Satoshi Kosugi, Yoshihisa Morita, Shoji Yotsutani. A complete bifurcation diagram of the Ginzburg-Landau equation with periodic boundary conditions. Communications on Pure & Applied Analysis, 2005, 4 (3) : 665-682. doi: 10.3934/cpaa.2005.4.665
[14]

Lisa C Flatley, Robert S MacKay, Michael Waterson. Optimal strategies for operating energy storage in an arbitrage or smoothing market. Journal of Dynamics & Games, 2016, 3 (4) : 371-398. doi: 10.3934/jdg.2016020
[15]

Eric Ruggieri, Sebastian J. Schreiber. The Dynamics of the Schoener-Polis-Holt model of Intra-Guild Predation. Mathematical Biosciences & Engineering, 2005, 2 (2) : 279-288. doi: 10.3934/mbe.2005.2.279
[16]

Tinggui Chen, Yanhui Jiang. Research on operating mechanism for creative products supply chain based on game theory. Discrete & Continuous Dynamical Systems - S, 2015, 8 (6) : 1103-1112. doi: 10.3934/dcdss.2015.8.1103
[17]

Zhipeng Qiu, Jun Yu, Yun Zou. The asymptotic behavior of a chemostat model. Discrete & Continuous Dynamical Systems - B, 2004, 4 (3) : 721-727. doi: 10.3934/dcdsb.2004.4.721
[18]

Meng Fan, Bingbing Zhang, Michael Yi Li. Mechanisms for stable coexistence in an insect community. Mathematical Biosciences & Engineering, 2010, 7 (3) : 603-622. doi: 10.3934/mbe.2010.7.603
[19]

Frédéric Mazenc, Michael Malisoff, Patrick D. Leenheer. On the stability of periodic solutions in the perturbed chemostat. Mathematical Biosciences & Engineering, 2007, 4 (2) : 319-338. doi: 10.3934/mbe.2007.4.319
[20]

Gonzalo Robledo. Feedback stabilization for a chemostat with delayed output. Mathematical Biosciences & Engineering, 2009, 6 (3) : 629-647. doi: 10.3934/mbe.2009.6.629

2018 Impact Factor: 1.313

Tools

Metrics

  • PDF downloads (9)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences