Global stability of Michaelis-Menten networks
Abstract
In a very nice paper, Blanchini, Breda, Giordano and Liessi studied networks of species coupled through Michaelis-Menten interactions. Under standard assumptions of strong connectedness and non-degeneracy, they show that there is a unique equilibrium, which is (locally) asymptotically stable. This note shows that global stability, which was stated there as an open problem, holds as well. In fact, whenever a positive equilibrium exists, it attracts every solution with positive initial condition, with no connectedness or non-degeneracy hypothesis at all; those assumptions are needed only in order to guarantee existence and to cover solutions starting on the boundary of the orthant. The key is to note that the system is contractive with respect to the Thompson part metric.
Related
- Mood stability and change map onto distinct insula-centered networks
- Global assemblage stability is driven by asynchrony shaped by environmental heterogeneity
- Correlated traits generate the structure and stability of multiplex ecological networks
- Operator-valued free probability applied to recurrent neural networks with heterogeneity in local and global circuit structures
- Global density and biomass of arbuscular mycorrhizal fungal networks
BibTeX
@misc{sontag2026global,
title = {{Global stability of Michaelis-Menten networks}},
author = {Eduardo D. Sontag},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.24351},
eprint = {2610.24351},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.24351},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Global stability of Michaelis-Menten networks
AU - Eduardo D. Sontag
PY - 2026
DA - 2026/10/17
PB - arXiv
DO - 10.48550/arxiv.2610.24351
UR - https://arxiv.org/abs/2610.24351
AB - In a very nice paper, Blanchini, Breda, Giordano and Liessi studied networks of species coupled through Michaelis-Menten interactions. Under standard assumptions of strong connectedness and non-degeneracy, they show that there is a unique equilibrium, which is (locally) asymptotically stable. This note shows that global stability, which was stated there as an open problem, holds as well. In fact, whenever a positive equilibrium exists, it attracts every solution with positive initial condition, with no connectedness or non-degeneracy hypothesis at all; those assumptions are needed only in order to guarantee existence and to cover solutions starting on the boundary of the orthant. The key is to note that the system is contractive with respect to the Thompson part metric.
ER -