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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2026
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2605.16049 |
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| _version_ | 1866918504160034816 |
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| author | Conradi, Carsten Mincheva, Maya Uecker, Hannes |
| author_facet | Conradi, Carsten Mincheva, Maya Uecker, Hannes |
| contents | We study the onset of spatial instabilities in reaction networks where the spatially homogeneous system admits a steady state parameterization. We formulate a sufficient condition -- based on the signs of the constant and leading coefficients of the characteristic polynomial of the linearized Jacobian scaled by the diffusion coefficients -- that guarantees a Turing-like instability to spatially inhomogeneous solutions on appropriately chosen domains $Ω$. We also present a specific condition on the domain size $|Ω|$ required to trigger this instability. As a consequence of employing a monomial parameterization, these conditions take the form of algebraic polynomial inequalities involving only rate constants and diffusion coefficients. We apply these ideas to a network describing the sequential and distributive (de-)phosphorylation of a protein at two binding sites, ultimately deriving a condition involving only the four catalytic constants of the enzymes and the diffusion coefficients of the four enzyme-substrate complexes that guarantees a Turing-like instability. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_16049 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Conditions for spatial instabilities and pattern formation from monomial steady state parameterizations Conradi, Carsten Mincheva, Maya Uecker, Hannes Dynamical Systems Analysis of PDEs Molecular Networks 35B36, 35K57, 92B05, 92C42 We study the onset of spatial instabilities in reaction networks where the spatially homogeneous system admits a steady state parameterization. We formulate a sufficient condition -- based on the signs of the constant and leading coefficients of the characteristic polynomial of the linearized Jacobian scaled by the diffusion coefficients -- that guarantees a Turing-like instability to spatially inhomogeneous solutions on appropriately chosen domains $Ω$. We also present a specific condition on the domain size $|Ω|$ required to trigger this instability. As a consequence of employing a monomial parameterization, these conditions take the form of algebraic polynomial inequalities involving only rate constants and diffusion coefficients. We apply these ideas to a network describing the sequential and distributive (de-)phosphorylation of a protein at two binding sites, ultimately deriving a condition involving only the four catalytic constants of the enzymes and the diffusion coefficients of the four enzyme-substrate complexes that guarantees a Turing-like instability. |
| title | Conditions for spatial instabilities and pattern formation from monomial steady state parameterizations |
| topic | Dynamical Systems Analysis of PDEs Molecular Networks 35B36, 35K57, 92B05, 92C42 |
| url | https://arxiv.org/abs/2605.16049 |