Finite-amplitude establishment beyond local invasion in cooperative chemical networks
Abstract
Local kinetic exponents and next-generation radii diagnose infinitesimal invasion, but they need not determine whether a nonlinear organized state is accessible. Motivated by the kinetic component of Gate~7 in a twelve-gate nonequilibrium framework, we study cooperative networks whose productive linearization contains no closed cycle. We prove that an acyclic productive Jacobian has R=0 and λkin=−mini di<0 for every drive, while a nonlinear return path can create a stable finite-amplitude branch through a saddle-node. In a canonical catalyst--assembly model, the saddle's stable manifold---not a threshold in either concentration---is the deterministic establishment boundary, so equal-abundance seeds of different composition can have opposite outcomes. At finite copy number, establishment becomes a committor and first-passage problem. For a benchmark protocol with Γ=2, Ω=16, η=0.02, and horizon T=1000, density-Hill and pair-count generators give first-entrance probabilities $0.633$ and $0.339$; their interpolated half-success drives are $1.900$ and $2.148$, an approximately 13.1% shift. The associated constant-source deterministic system retains a stable low state, so these transitions are fluctuation- and discrete-arrival-enabled. The finite-amplitude regime is not confined to exact acyclicity: a weak linear return leaves a finite interval with 0<R<1, λkin<0, and bistability. A general cooperative-return criterion separates the superlinear local condition from the global return-profile structure required for a fold. The resulting hierarchy of branch existence, basin accessibility, and finite-time stochastic establishment replaces a single onset threshold by a protocol-dependent kinetic criterion.
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BibTeX
@misc{martinezreyes2026finiteamplitude,
title = {{Finite-amplitude establishment beyond local invasion in cooperative chemical networks}},
author = {Ferran Martínez-Reyes and Slava G. Turyshev},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.05458},
eprint = {2611.05458},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.05458},
}
RIS
TY - UNPB
TI - Finite-amplitude establishment beyond local invasion in cooperative chemical networks
AU - Ferran Martínez-Reyes
AU - Slava G. Turyshev
PY - 2026
DA - 2026/11/03
PB - arXiv
DO - 10.48550/arxiv.2611.05458
UR - https://arxiv.org/abs/2611.05458
AB - Local kinetic exponents and next-generation radii diagnose infinitesimal invasion, but they need not determine whether a nonlinear organized state is accessible. Motivated by the kinetic component of Gate~7 in a twelve-gate nonequilibrium framework, we study cooperative networks whose productive linearization contains no closed cycle. We prove that an acyclic productive Jacobian has R=0 and λ_kin=−min_i d_i<0 for every drive, while a nonlinear return path can create a stable finite-amplitude branch through a saddle-node. In a canonical catalyst--assembly model, the saddle's stable manifold---not a threshold in either concentration---is the deterministic establishment boundary, so equal-abundance seeds of different composition can have opposite outcomes. At finite copy number, establishment becomes a committor and first-passage problem. For a benchmark protocol with Γ=2, Ω=16, η=0.02, and horizon T=1000, density-Hill and pair-count generators give first-entrance probabilities $0.633$ and $0.339$; their interpolated half-success drives are $1.900$ and $2.148$, an approximately 13.1% shift. The associated constant-source deterministic system retains a stable low state, so these transitions are fluctuation- and discrete-arrival-enabled. The finite-amplitude regime is not confined to exact acyclicity: a weak linear return leaves a finite interval with 0<R<1, λ_kin<0, and bistability. A general cooperative-return criterion separates the superlinear local condition from the global return-profile structure required for a fold. The resulting hierarchy of branch existence, basin accessibility, and finite-time stochastic establishment replaces a single onset threshold by a protocol-dependent kinetic criterion.
ER -