Modelling Population-Level Hes1 Dynamics: Insights from a Multi-Framework Approach

Across A–C and E the candidate solution matches the paper’s statements and arguments (unique homogeneous fixed point; two‑cell instability criterion (3.2); existence and stability of two non‑homogeneous equilibria via the γ-map; 2D hexagonal lattice instability (3.19)) as given in Propositions 3.1–3.4 and 3.8, with the same criteria and proof structure . For the full model (Part D), the candidate reproduces the instability threshold (3.15) and its equivalence to (3.2) (paper’s Proposition 3.6) , but then asserts that “all remaining eigenvalues equal −μi” to conclude stability of the patterned state. This spectral claim contradicts the paper’s characteristic polynomial structure for the full Jacobian, where factors (λ+μi) appear multiplicatively in a product minus a nonzero constant term, so λ=−μi are not eigenvalues in general; the paper instead proves stability by sign arguments on the full characteristic polynomial and a reduction lemma (Lemma 3.5; Proposition 3.7) . Hence the model’s proof of stability in D is flawed, though the conclusion matches the paper.