# American Institute of Mathematical Sciences

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

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