SADDLE-NODE BIFURCATIONS IN CHEMICAL REACTION NETWORKS

The paper’s main theorem (Theorem 4.2) correctly shows that if a network contains an SN‑pair of Child Selections, then one can choose monotone chemical kinetics so that ẋ = S f(x, λ) undergoes a saddle‑node bifurcation, with λ affecting a single reaction η = J1(m*) ≠ J2(m*) (definition and statement in the paper; see Definition 6 and Theorem 4.2 ). The proof uses an ε‑rescaling to isolate two dominant Child Selections, ensures a simple zero eigenvalue via nonzero Partial Child Selection contributions to tr Adj G, and verifies the nondegeneracy conditions SN2 and SN3 (Lemmas 7.1 and 7.2) by aligning ∂λ g with Sη and by exploiting freedom in second derivatives (Hill’s kinetics) . Crucially, the paper assumes from the outset that the stoichiometric matrix admits a positive right kernel vector r>0 (equilibrium constraints Sr=0 with rj>0 for all j; eq. (2) and surrounding text), rather than deriving this from the SN‑pair hypothesis . The model’s Step (1) attempts to deduce r>0 ∈ ker S from “opposite‑orientation adjacent bases” via a ‘positive circuit’, and then sets f( x̄, λ*) = r. This is incorrect in general: (i) the paper explicitly requires a separate structural assumption that S admits a strictly positive kernel vector (not implied by an SN‑pair), and (ii) the model’s construction would force some reaction rates to be zero if E>M+1, contradicting the positivity of chemical kinetics at positive concentrations (fj(x)>0) required in Definition 1 . The remaining steps of the model mirror the paper’s ε‑dominance (Lemma 5.1), simple eigenvalue (Lemma 6.3), and SN2/SN3 nondegeneracy arguments, but they rest on the flawed Step (1).