NEW STABILITY CRITERIA FOR PERIODIC PLANAR LOTKA-VOLTERRA SYSTEMS

Part (a): The candidate’s bounds and use of periodic identities, Hölder/Littlewood, and the L∞ caps U=(a/b)M, V=(d/f)M+(e/f)M U match the paper’s derivation of (up,vp) ∈ Cp; this is essentially the same argument as in the paper’s proof of Theorem 1(a) via (10)–(20) and Lemma 1 (U,V) . Part (b): The paper reduces the claim to the Lp-condition of Ortega–Rebelo (Theorem 2) and then upper-bounds its left-hand side by T(√(cMeM up vp) + 1/2(bM u1 + fM v1)); imposing this for all (xp,yp)∈Cp and (x1,y1)∈C1 proves uniqueness and asymptotic stability (Theorem 1(b)) . By contrast, the candidate attempts a direct “q-angle” proof but makes several critical mistakes: (i) the claimed angle dynamics θ′≥gq(θ) is asserted without justification; (ii) the factor gq(θ)≥1 is incorrectly handled (they effectively drop it when bounding ∫√(ξη) gq, which reverses the inequality), and they use bM,fM where bL,fL are needed to obtain lower bounds; (iii) the final inequality mixes a term with units T×(…) and a dimensionless J(q), yielding 0 ≤ T(…)-J(q), which is inconsistent with the structure of the known Lp-criterion. These gaps contradict the careful route taken in the paper (via Theorem 2 and its constant J(q)/2^{2-1/q}) and invalidate the candidate’s Part (b) proof .