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## A new proof of the competitive exclusion principle in the chemostat

 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.

Citation: Alain Rapaport, Mario Veruete. A new proof of the competitive exclusion principle in the chemostat. Discrete & Continuous Dynamical Systems - B, 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. [2] R. A. Armstrong and R. McGehee, Competitive exclusion, American Naturalist, 115 (1980), 151-170. doi: 10.1086/283553. [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. [4] P. de Leenheer, B. Li and H. L. Smith, Competition in the chemostat: Some remarks, Canadian Applied Mathematics Quarterly, 11 (2003), 229-248. [5] G. F. Gause, Experimental studies on the struggle for existence, Journal of Experimental Biology, 9 (1932), 389-402. [6] G. F. Gause, The Struggle for Existence, Dover, 1934. translated from Russian in 1971. [7] G. Hardin, The competitive exclusion principle, Science, 131 (1960), 1292-1297. [8] J. Harmand, C. Lobry, A. Rapaport and T. Sari, The Chemostat: Mathematical Theory of Microorganism Cultures, ISTE-Wiley, 2017. [9] S. Hsu, Limiting behavior for competing species, SIAM Journal on Applied Mathematics, 34 (1978), 760-763. doi: 10.1137/0134064. [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. [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. [12] W. Murdoch, C. Briggs and R. Nisbet, Consumer-Resource Dynamics, Princeton University Press, 2003. [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. [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. [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. [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. [17] H. Smith and P. Waltman, The Theory of the Chemostat: Dynamics of Microbial Competition, Cambridge, 1995. doi: 10.1017/CBO9780511530043. [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.

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##### References:
 [1] A. Ajbar and K. Alhumaizi, Dynamics of the Chemostat: A Bifurcation Theory Approach, CRC Press, Boca Raton, FL, 2012. [2] R. A. Armstrong and R. McGehee, Competitive exclusion, American Naturalist, 115 (1980), 151-170. doi: 10.1086/283553. [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. [4] P. de Leenheer, B. Li and H. L. Smith, Competition in the chemostat: Some remarks, Canadian Applied Mathematics Quarterly, 11 (2003), 229-248. [5] G. F. Gause, Experimental studies on the struggle for existence, Journal of Experimental Biology, 9 (1932), 389-402. [6] G. F. Gause, The Struggle for Existence, Dover, 1934. translated from Russian in 1971. [7] G. Hardin, The competitive exclusion principle, Science, 131 (1960), 1292-1297. [8] J. Harmand, C. Lobry, A. Rapaport and T. Sari, The Chemostat: Mathematical Theory of Microorganism Cultures, ISTE-Wiley, 2017. [9] S. Hsu, Limiting behavior for competing species, SIAM Journal on Applied Mathematics, 34 (1978), 760-763. doi: 10.1137/0134064. [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. [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. [12] W. Murdoch, C. Briggs and R. Nisbet, Consumer-Resource Dynamics, Princeton University Press, 2003. [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. [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. [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. [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. [17] H. Smith and P. Waltman, The Theory of the Chemostat: Dynamics of Microbial Competition, Cambridge, 1995. doi: 10.1017/CBO9780511530043. [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.
Growth functions and their break-even concentrations
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 $I_{i}$)
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