On Stability of Two Kinds of Delayed Chemical Reaction Networks

The paper proves three main points: (A) existence of a `cDCB realization for any delayed complex-balanced network and any positive diagonal Q (via an explicit realization that may require adding zero-delay reactions when Q is not scalar), (B) stability and then local asymptotic stability relative to a new invariant set H_θ, and (C) uniqueness and local asymptotic stability in each delayed stoichiometric class for a special family ¯`cDCB1, with a Lyapunov split argument. These are all supported in the text (Definition 2, Theorem 3, Lemmas 6–7, Theorem 8, Lemma 11, Theorem 12) and their displayed equations (especially (7)–(13), (16)–(18), and the A+B decomposition) . However, (i) the paper’s existence proof for non-scalar Q tacitly permits complexes with non-integer stoichiometric coefficients (or else requires additional hypotheses on Q) when realizing (10)–(13), which is not reconciled with its earlier integer-complex convention; and (ii) the derivation of local asymptotic stability repeatedly relies on LaSalle’s invariance for RFDEs but does not state the needed regularity assumptions on the vector field and solution semiflow. The candidate model’s Phase A likewise overlooks the non-scalar Q realization subtlety by proposing y′_i = Q ~ y′_i directly, which, under the paper’s standard CRN convention, can violate integrality; the paper explicitly avoids this by introducing additional (zero-delay) reactions to realize the vector field . Phases B and C of the model match the paper closely—mapping invariants and transporting stability for `cDCB via conjugacy (cf. (16)–(18)) and executing the A+B Lyapunov split for ¯`cDCB1 with the same key inequality—but the model leaves some assumptions implicit (uniqueness-in-class and LaSalle conditions). Hence both are substantially correct in spirit but incomplete in hypotheses and construction details.