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Bistability with equal and independent removal in small bimolecular networks

  • Bo Peng
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Abstract

We classify two-species mass-action networks with at most three internal directed reactions on {X,Y,2X,X+Y,2Y}, positive inflows, and linear removal of both species. Among the 681 networks up to species exchange, exactly 27 admit two distinct positive hyperbolic attracting equilibria. The same 27 admit bistability with equal removal rates and with independently chosen positive removal rates. Each is minimal under reaction deletion. We give rational equal-removal witnesses for all positive cases and all-parameter exclusions for the remaining 654. The proof combines nullcline geometry, polynomial sign tests, resultant root bounds, and exact flux-cone decompositions. A separate rational-arithmetic implementation verifies the complete classification and its equilibrium certificates.

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Cite

BibTeXRIS
BibTeX
@misc{peng2026bistability,
  title = {{Bistability with equal and independent removal in small bimolecular networks}},
  author = {Bo Peng},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.32739},
  eprint = {2610.32739},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.32739},
}
RIS
TY  - UNPB
TI  - Bistability with equal and independent removal in small bimolecular networks
AU  - Bo Peng
PY  - 2026
DA  - 2026/10/23
PB  - arXiv
DO  - 10.48550/arxiv.2610.32739
UR  - https://arxiv.org/abs/2610.32739
AB  - We classify two-species mass-action networks with at most three internal directed reactions on {X,Y,2X,X+Y,2Y}, positive inflows, and linear removal of both species. Among the 681 networks up to species exchange, exactly 27 admit two distinct positive hyperbolic attracting equilibria. The same 27 admit bistability with equal removal rates and with independently chosen positive removal rates. Each is minimal under reaction deletion. We give rational equal-removal witnesses for all positive cases and all-parameter exclusions for the remaining 654. The proof combines nullcline geometry, polynomial sign tests, resultant root bounds, and exact flux-cone decompositions. A separate rational-arithmetic implementation verifies the complete classification and its equilibrium certificates.
ER  -