Stability Analysis of Biochemical Reaction Networks Linearly Conjugated to complex balanced Systems with Time Delays Added

Xiaoyu Zhang, Shibo He, Chuanhou Gao, Denis Dochain

incompletemedium confidence

Category: math.DS
Journal tier: Specialist/Solid
Processed: Sep 28, 2025, 12:56 AM
arXiv Links: Abstract ↗PDF ↗

Audit review

The paper establishes the three structural classes (A)–(C), defines new invariant sets Dnψ, proves their invariance, and shows existence/uniqueness of a positive equilibrium per Dnψ. It then asserts local asymptotic stability relative to Dnψ by appealing back to Section IV’s Lyapunov functionals and uniqueness, but the argument is terse and does not explicitly invoke a LaSalle/Hale invariance principle for delay systems or verify its hypotheses. The candidate solution matches the paper’s structure (A)–(C) and the invariants (including the weighted δji construction), but key steps—strict negativity of the Lyapunov–Krasovskii derivative near equilibrium, domination of delay remainders, and injectivity of the invariant map via a Hessian argument—are stated at a high level without full justification. Thus both are plausible but omit essential details for a complete proof.

Referee report (LaTeX)

\textbf{Recommendation:} minor revisions

\textbf{Journal Tier:} specialist/solid

\textbf{Justification:}

The manuscript extends stability analysis for delayed networks related by linear conjugacy/dynamical equivalence to CBMAS by introducing structured classes and decompositional Lyapunov constructions. The redefinition of invariant sets that eliminates degeneracy is well-motivated and practically relevant. The final step to local asymptotic stability relative to these sets is correct in spirit but too terse; explicitly invoking a LaSalle-type invariance principle for delay equations and checking its hypotheses would complete the proof. Examples are useful and computations of the new invariants are illustrative.