HOPF BIFURCATIONS OF REACTION NETWORKS WITH ZERO-ONE STOICHIOMETRIC COEFFICIENTS

The paper’s main theorem states that any zero–one reaction network admitting a Hopf bifurcation must have rank at least 4, and proves it by transforming the Jacobian via extreme rays J(h,λ), establishing sign patterns for diagonals and principal minors, and showing the relevant Hurwitz determinant(s) are strictly positive when rank ≤ 3; see the statement of Theorem 3.1 and the proof strategy around the Jacobian factorization (their Eq. (4.1)), Lemma 4.2, Lemma 6.5, and the concluding argument that rules out Hopf for ranks 1–3 . The candidate model’s solution reaches the same conclusion using a different route: it works directly with the mass–action Jacobian factorization, analyzes signs of principal minors via Cauchy–Binet, and invokes Routh–Hurwitz in dimensions 1–3 to preclude Hopf. The key logical milestones (nonpositive diagonal entries; positivity of the 2×2 minors and of the 3D Hurwitz determinant) align with the paper’s conclusions, even though the model sketches some steps (e.g., the mixed Gram expansion) rather than deriving them in full. Hence both are correct, with substantively different proof organizations.