Pith Number

pith:564AX5AY

pith:2026:564AX5AYVRKONFGVHQEV6HBLGU

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Conditions for spatial instabilities and pattern formation from monomial steady state parameterizations

Carsten Conradi, Hannes Uecker, Maya Mincheva

For reaction networks admitting a monomial steady-state parameterization, the signs of the constant and leading coefficients in the diffusion-scaled characteristic polynomial supply a sufficient condition for Turing-like spatial instability

arxiv:2605.16049 v1 · 2026-05-15 · math.DS · math.AP · q-bio.MN

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Claims

C1strongest claim

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.

C2weakest assumption

The spatially homogeneous reaction network admits a monomial steady-state parameterization (explicitly invoked in the abstract and used to obtain algebraic inequalities); if this parameterization does not exist or is not monomial, the reduction to polynomial inequalities in rate constants and diffusion coefficients fails.

C3one line summary

Sufficient algebraic conditions on the signs of the constant and leading coefficients of the diffusion-scaled characteristic polynomial are derived to guarantee Turing instability on suitable domains, then applied to a two-site phosphorylation network to obtain a condition involving only four rate-1

References

38 extracted · 38 resolved · 0 Pith anchors

[1] Turing patterns, 70 years later , Nature Computational Science 2 (2022), no. 8, 463–464 2022

[2] Ashbaugh and Rafael D 1993

[3] C. Conradi, M. Mincheva, and H. Uecker, Supplementary information; Matlab sources and documentation , 2026, A vailable at [27], tab Applications 2026

[4] Carsten Conradi, Elisenda Feliu, and Maya Mincheva, On the existence of Hopf bifurcations in the sequential and distributive double phosphorylation cycle , Mathematical Biosciences and Engineering 17 2020

[5] Carsten Conradi, Dietrich Flockerzi, and Jörg Raisch, Multistationarity in the activation of an MAPK: parametriz- ing the relevant region in parameter space , Mathematical Biosciences 211 (2008), no. 2008

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Receipt and verification

First computed	2026-05-20T00:01:50.649253Z
Builder	pith-number-builder-2026-05-17-v1
Signature	Pith Ed25519 (`pith-v1-2026-05`) · public key
Schema	pith-number/v1.0

Canonical hash

efb80bf418ac54e694d53c095f1c2b35076974829f11358f81fe4d8627b31177

Aliases

arxiv: 2605.16049 · arxiv_version: 2605.16049v1 · doi: 10.48550/arxiv.2605.16049 · pith_short_12: 564AX5AYVRKO · pith_short_16: 564AX5AYVRKONFGV · pith_short_8: 564AX5AY

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Verify this Pith Number yourself

curl -sH 'Accept: application/ld+json' https://pith.science/pith/564AX5AYVRKONFGVHQEV6HBLGU \
  | jq -c '.canonical_record' \
  | python3 -c "import sys,json,hashlib; b=json.dumps(json.loads(sys.stdin.read()), sort_keys=True, separators=(',',':'), ensure_ascii=False).encode(); print(hashlib.sha256(b).hexdigest())"
# expect: efb80bf418ac54e694d53c095f1c2b35076974829f11358f81fe4d8627b31177

Canonical record JSON

{
  "metadata": {
    "abstract_canon_sha256": "3dd8bba57d8ab1ee31da41e6533f149ab2f64ae30c2115b0f41aa3eff842865c",
    "cross_cats_sorted": [
      "math.AP",
      "q-bio.MN"
    ],
    "license": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
    "primary_cat": "math.DS",
    "submitted_at": "2026-05-15T15:18:26Z",
    "title_canon_sha256": "8681dd537cd4ba9d5623a2929024a3ae6900ebecab215c20d470dd0c01b25bda"
  },
  "schema_version": "1.0",
  "source": {
    "id": "2605.16049",
    "kind": "arxiv",
    "version": 1
  }
}