American Institute of Mathematical Sciences

Advanced
August  2019, 24(8): 3755-3764. doi: 10.3934/dcdsb.2018314

A new proof of the competitive exclusion principle in the chemostat

Alain Rapaport 1,, and  Mario Veruete 2,

1. 

MISTEA, Univ. Montpellier, Inra, Montpellier SupAgro, 2, place Pierre Viala, 34060 Montpellier, France
2. 

IMAG, Univ. Montpellier, CNRS, Place Eugène Bataillon, 34090 Montpellier, France

* Corresponding author

Received  April 2018 Revised  June 2018 Published  October 2018

We give an new proof of the well-known competitive exclusion principle in the chemostat model with $N$ species competing for a single resource, for any set of increasing growth functions. The proof is constructed by induction on the number of the species, after being ordered. It uses elementary analysis and comparisons of solutions of ordinary differential equations.

Keywords: Chemostat model, competitive exclusion, global stability, comparison principle, induction.
Mathematics Subject Classification: 34K20, 92D25.
Citation: Alain Rapaport, Mario Veruete. A new proof of the competitive exclusion principle in the chemostat. Discrete & Continuous Dynamical Systems - B, 2019, 24 (8) : 3755-3764. doi: 10.3934/dcdsb.2018314
References:
[1]

A. Ajbar and K. Alhumaizi, Dynamics of the Chemostat: A Bifurcation Theory Approach, CRC Press, Boca Raton, FL, 2012.  Google Scholar

[2]

R. A. Armstrong and R. McGehee, Competitive exclusion, American Naturalist, 115 (1980), 151-170.  doi: 10.1086/283553.  Google Scholar

[3]

G. J. Butler and G. Wolkowicz, A mathematical model of the chemostat with a general class of functions describing nutrient uptake, SIAM Journal on Applied Mathematics, 45 (1985), 138-151.  doi: 10.1137/0145006.  Google Scholar

[4]

P. de Leenheer, B. Li and H. L. Smith, Competition in the chemostat: Some remarks, Canadian Applied Mathematics Quarterly, 11 (2003), 229-248.   Google Scholar

[5]

G. F. Gause, Experimental studies on the struggle for existence, Journal of Experimental Biology, 9 (1932), 389-402.   Google Scholar

[6]

G. F. Gause, The Struggle for Existence, Dover, 1934. translated from Russian in 1971. Google Scholar

[7]

G. Hardin, The competitive exclusion principle, Science, 131 (1960), 1292-1297.   Google Scholar

[8]

J. Harmand, C. Lobry, A. Rapaport and T. Sari, The Chemostat: Mathematical Theory of Microorganism Cultures, ISTE-Wiley, 2017.  Google Scholar

[9]

S. Hsu, Limiting behavior for competing species, SIAM Journal on Applied Mathematics, 34 (1978), 760-763.  doi: 10.1137/0134064.  Google Scholar

[10]

S. Hsu, S. Hubbell and P. Waltman, A mathematical theory for single-nutrient competition in continuous cultures of microorganisms, SIAM Journal on Applied Mathematics, 32 (1977), 366-383.  doi: 10.1137/0132030.  Google Scholar

[11]

B. Li, Global asymptotic behavior of the chemostat: General response functions and differential removal rates, SIAM Journal on Applied Mathematics, 59 (1998), 411-422.  doi: 10.1137/S003613999631100X.  Google Scholar

[12]

W. Murdoch, C. Briggs and R. Nisbet, Consumer-Resource Dynamics, Princeton University Press, 2003. Google Scholar

[13]

A. Rapaport, D. Dochain and J. Harmand, Long run coexistence in the chemostat with multiple species, J. Theoretical Biology, 257 (2009), 252-259.  doi: 10.1016/j.jtbi.2008.11.015.  Google Scholar

[14]

A. Rapaport and J. Harmand, Biological control of the chemostat with nonmonotonic response and different removal rates, Mathematical Biosciences & Engineering, 5 (2008), 539-547.  doi: 10.3934/mbe.2008.5.539.  Google Scholar

[15]

T. Sari, Competitive exclusion for chemostat equations with variable yields, Acta Applicandae Mathematicae, 123 (2013), 201-219.  doi: 10.1007/s10440-012-9761-8.  Google Scholar

[16]

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.  Google Scholar

[17]

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

[18]

G. Wolkowicz and Z. Lu, Global dynamics of a mathematical model of competition in the chemostat: General response functions and differential death rates, SIAM Journal on Applied Mathematics, 52 (1992), 222-233.  doi: 10.1137/0152012.  Google Scholar

show all references

References:
[1]

A. Ajbar and K. Alhumaizi, Dynamics of the Chemostat: A Bifurcation Theory Approach, CRC Press, Boca Raton, FL, 2012.  Google Scholar

[2]

R. A. Armstrong and R. McGehee, Competitive exclusion, American Naturalist, 115 (1980), 151-170.  doi: 10.1086/283553.  Google Scholar

[3]

G. J. Butler and G. Wolkowicz, A mathematical model of the chemostat with a general class of functions describing nutrient uptake, SIAM Journal on Applied Mathematics, 45 (1985), 138-151.  doi: 10.1137/0145006.  Google Scholar

[4]

P. de Leenheer, B. Li and H. L. Smith, Competition in the chemostat: Some remarks, Canadian Applied Mathematics Quarterly, 11 (2003), 229-248.   Google Scholar

[5]

G. F. Gause, Experimental studies on the struggle for existence, Journal of Experimental Biology, 9 (1932), 389-402.   Google Scholar

[6]

G. F. Gause, The Struggle for Existence, Dover, 1934. translated from Russian in 1971. Google Scholar

[7]

G. Hardin, The competitive exclusion principle, Science, 131 (1960), 1292-1297.   Google Scholar

[8]

J. Harmand, C. Lobry, A. Rapaport and T. Sari, The Chemostat: Mathematical Theory of Microorganism Cultures, ISTE-Wiley, 2017.  Google Scholar

[9]

S. Hsu, Limiting behavior for competing species, SIAM Journal on Applied Mathematics, 34 (1978), 760-763.  doi: 10.1137/0134064.  Google Scholar

[10]

S. Hsu, S. Hubbell and P. Waltman, A mathematical theory for single-nutrient competition in continuous cultures of microorganisms, SIAM Journal on Applied Mathematics, 32 (1977), 366-383.  doi: 10.1137/0132030.  Google Scholar

[11]

B. Li, Global asymptotic behavior of the chemostat: General response functions and differential removal rates, SIAM Journal on Applied Mathematics, 59 (1998), 411-422.  doi: 10.1137/S003613999631100X.  Google Scholar

[12]

W. Murdoch, C. Briggs and R. Nisbet, Consumer-Resource Dynamics, Princeton University Press, 2003. Google Scholar

[13]

A. Rapaport, D. Dochain and J. Harmand, Long run coexistence in the chemostat with multiple species, J. Theoretical Biology, 257 (2009), 252-259.  doi: 10.1016/j.jtbi.2008.11.015.  Google Scholar

[14]

A. Rapaport and J. Harmand, Biological control of the chemostat with nonmonotonic response and different removal rates, Mathematical Biosciences & Engineering, 5 (2008), 539-547.  doi: 10.3934/mbe.2008.5.539.  Google Scholar

[15]

T. Sari, Competitive exclusion for chemostat equations with variable yields, Acta Applicandae Mathematicae, 123 (2013), 201-219.  doi: 10.1007/s10440-012-9761-8.  Google Scholar

[16]

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.  Google Scholar

[17]

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

[18]

G. Wolkowicz and Z. Lu, Global dynamics of a mathematical model of competition in the chemostat: General response functions and differential death rates, SIAM Journal on Applied Mathematics, 52 (1992), 222-233.  doi: 10.1137/0152012.  Google Scholar

Figure 1.  Growth functions and their break-even concentrations
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 2.  Illustration of the intervals $I_{i} = [s_1^-, s_i^+]$ for $i\in\{1, \cdots, N-1\}$ (in green, the values of the break-even concentrations $\lambda_{i}$, in orange, the nested intervals $I_{i}$)
Figure Options

Download full-size image

Download as PowerPoint slide

[1]

H. L. Smith, X. Q. Zhao. Competitive exclusion in a discrete-time, size-structured chemostat model. Discrete & Continuous Dynamical Systems - B, 2001, 1 (2) : 183-191. doi: 10.3934/dcdsb.2001.1.183
[2]

Zhiqi Lu. Global stability for a chemostat-type model with delayed nutrient recycling. Discrete & Continuous Dynamical Systems - B, 2004, 4 (3) : 663-670. doi: 10.3934/dcdsb.2004.4.663
[3]

Yixiang Wu, Necibe Tuncer, Maia Martcheva. Coexistence and competitive exclusion in an SIS model with standard incidence and diffusion. Discrete & Continuous Dynamical Systems - B, 2017, 22 (3) : 1167-1187. doi: 10.3934/dcdsb.2017057
[4]

Azmy S. Ackleh, Youssef M. Dib, S. R.-J. Jang. Competitive exclusion and coexistence in a nonlinear refuge-mediated selection model. Discrete & Continuous Dynamical Systems - B, 2007, 7 (4) : 683-698. doi: 10.3934/dcdsb.2007.7.683
[5]

Azmy S. Ackleh, Keng Deng, Yixiang Wu. Competitive exclusion and coexistence in a two-strain pathogen model with diffusion. Mathematical Biosciences & Engineering, 2016, 13 (1) : 1-18. doi: 10.3934/mbe.2016.13.1
[6]

Yanxia Dang, Zhipeng Qiu, Xuezhi Li. Competitive exclusion in an infection-age structured vector-host epidemic model. Mathematical Biosciences & Engineering, 2017, 14 (4) : 901-931. doi: 10.3934/mbe.2017048
[7]

Chunqing Wu, Patricia J.Y. Wong. Global asymptotical stability of the coexistence fixed point of a Ricker-type competitive model. Discrete & Continuous Dynamical Systems - B, 2015, 20 (9) : 3255-3266. doi: 10.3934/dcdsb.2015.20.3255
[8]

Hao Wang, Katherine Dunning, James J. Elser, Yang Kuang. Daphnia species invasion, competitive exclusion, and chaotic coexistence. Discrete & Continuous Dynamical Systems - B, 2009, 12 (2) : 481-493. doi: 10.3934/dcdsb.2009.12.481
[9]

M. R. S. Kulenović, Orlando Merino. Competitive-exclusion versus competitive-coexistence for systems in the plane. Discrete & Continuous Dynamical Systems - B, 2006, 6 (5) : 1141-1156. doi: 10.3934/dcdsb.2006.6.1141
[10]

Frederic Mazenc, Gonzalo Robledo, Michael Malisoff. Stability and robustness analysis for a multispecies chemostat model with delays in the growth rates and uncertainties. Discrete & Continuous Dynamical Systems - B, 2018, 23 (4) : 1851-1872. doi: 10.3934/dcdsb.2018098
[11]

Timothy Blass, Rafael De La Llave, Enrico Valdinoci. A comparison principle for a Sobolev gradient semi-flow. Communications on Pure & Applied Analysis, 2011, 10 (1) : 69-91. doi: 10.3934/cpaa.2011.10.69
[12]

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
[13]

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
[14]

Tobias Black. Global existence and asymptotic stability in a competitive two-species chemotaxis system with two signals. Discrete & Continuous Dynamical Systems - B, 2017, 22 (4) : 1253-1272. doi: 10.3934/dcdsb.2017061
[15]

Azmy S. Ackleh, Linda J. S. Allen. Competitive exclusion in SIS and SIR epidemic models with total cross immunity and density-dependent host mortality. Discrete & Continuous Dynamical Systems - B, 2005, 5 (2) : 175-188. doi: 10.3934/dcdsb.2005.5.175
[16]

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
[17]

Lin Niu, Yi Wang. Non-oscillation principle for eventually competitive and cooperative systems. Discrete & Continuous Dynamical Systems - B, 2019, 24 (12) : 6481-6494. doi: 10.3934/dcdsb.2019148
[18]

Shigeaki Koike, Takahiro Kosugi. Remarks on the comparison principle for quasilinear PDE with no zeroth order terms. Communications on Pure & Applied Analysis, 2015, 14 (1) : 133-142. doi: 10.3934/cpaa.2015.14.133
[19]

Xiaowei Tang, Xilin Fu. New comparison principle with Razumikhin condition for impulsive infinite delay differential systems. Conference Publications, 2009, 2009 (Special) : 739-743. doi: 10.3934/proc.2009.2009.739
[20]

Thomas Leroy. Relativistic transfer equations: Comparison principle and convergence to the non-equilibrium regime. Kinetic & Related Models, 2015, 8 (4) : 725-763. doi: 10.3934/krm.2015.8.725

2018 Impact Factor: 1.008

Tools

Metrics

  • PDF downloads (102)
  • HTML views (548)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences