Asymptotic stability of delayed complex balanced reaction networks with non-mass action kinetics

The paper establishes three claims for delayed complex-balanced kinetic systems with product-form kinetics: (i) invariance of delayed stoichiometric classes via conserved functionals c_v (Proposition 4.1), (ii) existence and uniqueness of a positive equilibrium in each delayed class (Proposition 4.2), and (iii) local asymptotic stability relative to the delayed class using a Lyapunov–Krasovskii functional (Theorem 4.3). These are proved rigorously by (a) showing c_v is constant along trajectories, (b) characterizing equilibria as the log-γ toric manifold and proving class-wise uniqueness through a strictly convex, coercive function g, and (c) constructing a specific Lyapunov–Krasovskii functional (equation (14)) whose derivative is nonpositive and vanishes only on constant, complex-balanced equilibria; LaSalle’s principle then yields local convergence . By contrast, the model’s Step 3 hinges on an incorrect monotonicity argument: it asserts that the negative term in DG(α) penalizes only components along span{y_k}, "which is contained in S," so positivity on S^⊥ follows. But span{y_k} need not be contained in S (which is spanned by reaction vectors y_k' − y_k), and components along S^⊥ can be penalized by (y_k·ξ)^2; consequently, the Jacobian need not be positive definite on S^⊥, invalidating the claimed strict monotonicity and the use of a global inverse theorem. The model’s Lyapunov–Krasovskii ansatz in Step 4 also differs materially from the paper’s and lacks a complete derivation of V̇ ≤ 0 using the delayed terms; the paper’s proof derives V̇ ≤ 0 via inequality (9) in terms of q_k(t) and q_k(t − τ_k), tied to the complex-balance identities, and identifies the largest invariant set via q_k'(t) = q_k(t − τ_k) (then shows it consists of constant equilibria) .