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Abstract Many reaction networks arising in applications are multistationary, that is, they have the capacity for more than one steady state, while some networks exhibit absolute concentration robustness (ACR), which means that some species concentration is the same at all steady states. Both multistationarity and ACR are significant in biological settings, but only recently has attention focused on the possibility for these properties to coexist. Our main result states that such coexistence in at-most-bimolecular networks (which encompass most networks arising in biology) requires at least three species, five complexes and three reactions. We prove additional bounds on the number of reactions for general networks based on the number of linear conservation laws. Finally, we prove that, outside of a few exceptional cases, ACR is equivalent to non-multistationarity for bimolecular networks that are small (more precisely, one-dimensional or up to two species). Our proofs involve analyses of systems of sparse polynomials, and we also use classical results from chemical reaction network theory.
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This content will become publicly available on April 30, 2026
On pattern formation in the thermodynamically-consistent variational Gray-Scott model
In this paper, we explore pattern formation in a four-species variational Gary-Scott model, which includes all reverse reactions and introduces a virtual species to describe the birth–death process in the classical Gray-Scott model. This modification transforms the classical Gray-Scott model into a thermodynamically consistent closed system. The classical two-species Gray-Scott model can be viewed as a subsystem of the variational model in the limiting case when the small parameter ε, related to the reaction rate of the reverse reactions, approaches zero. We numerically explore pattern formation in this physically more complete Gray-Scott model in one spatial dimension, using non-uniform steady states of the classical model as initial conditions. By decreasing ε, we observed that the stationary patterns in the classical Gray-Scott model can be stabilized as the transient states in the variational model for a significantly small ε. Additionally, the variational model admits oscillating and traveling wave-like patterns for small ε. The persistent time of these patterns is on the order of O(1/ε). We also analyze the energy stability of two uniform steady states in the variational Gary-Scott model for fixed. Although both states are stable in a certain sense, the gradient flow type dynamics of the variational model exhibit a selection effect based on the initial conditions, with pattern formation occurring only if the initial condition does not converge to the boundary steady state, which corresponds to the trivial uniform steady state in the classical Gray-Scott model.
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- PAR ID:
- 10607908
- Publisher / Repository:
- Elsevier
- Date Published:
- Journal Name:
- Mathematical Biosciences
- Volume:
- 385
- Issue:
- C
- ISSN:
- 0025-5564
- Page Range / eLocation ID:
- 109453
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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