Competitive exclusion in a chemostat with unequal removal rates
Let $n \ge 1$, let $D > 0$ be the dilution rate, $S_{\mathrm{in}} > 0$ the feed concentration of the nutrient and $d_i > 0$ the removal rate of species $i$. For each $i$ let $f_i : [0, \infty) \to [0, \infty)$ be $C^1$ with $f_i(0) = 0$ and $f_i'(s) > 0$ for $s > 0$, and suppose that each equation $f_i(\lambda_i) = d_i$ has a finite positive solution, where after relabelling $$0 < \lambda_1 < \lambda_2 \le \cdots \le \lambda_n, \qquad \lambda_1 < S_{\mathrm{in}}$$ (for $n = 1$ only the conditions on $\lambda_1$). With the yields scaled to one, the nutrient concentration $S(t)$ and the species concentrations $x_i(t)$ satisfy $$\dot S = D(S_{\mathrm{in}} - S) - \sum_{i=1}^n f_i(S)\, x_i, \qquad \dot x_i = \bigl(f_i(S) - d_i\bigr) x_i \quad (1 \le i \le n).$$ The $d_i$ may differ from each other and from $D$.
When every $d_i$ equals $D$, competitive exclusion was proved by Hsu, Hubbell and Waltman for Michaelis–Menten responses $f_i(S) = m_i S/(a_i + S)$ and by Armstrong and McGehee for general increasing responses. Hsu proved it for Michaelis–Menten responses with arbitrary removal rates, using a Lyapunov function, and Wolkowicz and Lu, and Li, proved it for general responses and different removal rates under additional hypotheses.
Problem. For every such system and every initial state with $S(0) \ge 0$ and $x_i(0) > 0$ for all $i$, does the solution satisfy $$\lim_{t \to \infty} S(t) = \lambda_1, \qquad \lim_{t \to \infty} x_1(t) = \frac{D(S_{\mathrm{in}} - \lambda_1)}{d_1}, \qquad \lim_{t \to \infty} x_i(t) = 0 \quad (2 \le i \le n)?$$
Source: Hsu (2005), Open problem 1, for model (2.9) under hypothesis (H); also raised by Sari and Mazenc (2011).
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- DS — Dynamical Systems
- Secondary
- CA
- Keywords
- AIM Open Applied Problems, chemostats, competitive exclusion, Lyapunov functions
References
@article{hsu2005,
author = {Hsu, Sze-Bi},
title = {A survey of constructing {L}yapunov functions for mathematical models in population biology},
journal = {Taiwanese J. Math.},
volume = {9},
number = {2},
year = {2005},
pages = {151--173},
doi = {10.11650/twjm/1500407791},
}
@article{sari2011,
author = {Sari, Tewfik and Mazenc, Fr{\'e}d{\'e}ric},
title = {Global dynamics of the chemostat with different removal rates and variable yields},
journal = {Math. Biosci. Eng.},
volume = {8},
number = {3},
year = {2011},
pages = {827--840},
doi = {10.3934/mbe.2011.8.827},
}
@article{hsu1977,
author = {Hsu, S. B. and Hubbell, S. and Waltman, P.},
title = {A mathematical theory for single-nutrient competition in continuous cultures of micro-organisms},
journal = {SIAM J. Appl. Math.},
volume = {32},
number = {2},
year = {1977},
pages = {366--383},
doi = {10.1137/0132030},
}
@article{armstrong1980,
author = {Armstrong, Robert A. and McGehee, Richard},
title = {Competitive exclusion},
journal = {Amer. Nat.},
volume = {115},
number = {2},
year = {1980},
pages = {151--170},
doi = {10.1086/283553},
}
@article{hsu1978,
author = {Hsu, S. B.},
title = {Limiting behavior for competing species},
journal = {SIAM J. Appl. Math.},
volume = {34},
number = {4},
year = {1978},
pages = {760--763},
doi = {10.1137/0134064},
}
@article{wolkowicz1992,
author = {Wolkowicz, Gail S. K. and Lu, Zhiqi},
title = {Global dynamics of a mathematical model of competition in the chemostat: general response functions and differential death rates},
journal = {SIAM J. Appl. Math.},
volume = {52},
number = {1},
year = {1992},
pages = {222--233},
doi = {10.1137/0152012},
}
@article{li1998,
author = {Li, Bingtuan},
title = {Global asymptotic behavior of the chemostat: general response functions and different removal rates},
journal = {SIAM J. Appl. Math.},
volume = {59},
number = {2},
year = {1998},
pages = {411--422},
doi = {10.1137/S003613999631100X},
}
@misc{aim-chemostat-unequal-removal-exclusion,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{Competitive exclusion in a chemostat with unequal removal rates}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/a153f469ee0ea64e876b2ad97c79d252c19a3e22/problems/334-chemostat-unequal-removal-exclusion.md},
note = {Entry 334 at commit a153f46, 2026-10-13},
}
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-00414, "Competitive exclusion in a chemostat with unequal removal rates", clopen.solutions.