COMPLEX DYNAMICS OF A PREDATOR–PREY MODEL WITH CONSTANT-YIELD PREY HARVESTING AND ALLEE EFFECT IN PREDATOR

I read the uploaded paper and the candidate solution. Both analyze the same nondimensional system (4), dx/dt = x(1−x) − qxy − h, dy/dt = s y (1 − y/x)(y − m) with 0 < m < 1. The paper proves: (A) a codimension‑1 saddle‑node at E1 = (1/2, 0) when h = h2 = 1/4 by verifying the Sotomayor conditions with explicit left/right eigenvectors and nonzero second‑derivative and parameter transversality terms, matching the candidate’s computation of J(E1), v = (1,0), w = (2sm, −q), w·f_h = −2sm ≠ 0 and w·D^2f[v,v] = −4sm ≠ 0 (paper Theorem 15/Section 3.1; see the displayed calculation in the PDF) . (B) For the interior equilibrium E8 = (x8, x8) (Δ2 > 0), the paper derives tr J(E8) = 1 − (2+q)x8 + s(m − x8), det J(E8) = s(m − x8)(1 − 2(q+1)x8), defines s2 = (2x8 + qx8 − 1)/(m − x8), shows d(tr)/ds|_{s2} = m − x8 ≠ 0 when m < x8, and establishes a Hopf bifurcation near s = s2; it then uses Perko’s formula for the first Lyapunov coefficient σ to classify super/subcriticality by the sign of σ. The candidate reproduces these formulas and invokes the same Perko-based σ computation without adding new sign conditions, in line with the paper (paper Theorems 13 and 16; Lemma 4; derivations for d(tr)/ds and σ) . (C) On Δ2 = 0 (h = h3 = 1/[4(q+1)]) and s = s1 = (4h−1)/(2(m−2h)), E7 = (2h,2h) has a double zero eigenvalue with a single Jordan block; the paper proves E7 is a codimension‑2 cusp and, using a near-identity normal form reduction with parameters (η1,η2) for (h,s), obtains the BT normal form and concludes a codimension‑2 Bogdanov–Takens bifurcation near E8 for (s,h) near (s1,h3). The candidate states the same cusp nondegeneracy and transversality conditions and the same BT unfolding conclusion, consistent with the paper’s detailed normal‑form construction (paper Theorem 12 for the cusp and Theorem 17 for BT) . Overall, the candidate follows the paper’s route closely; there are no substantive conflicts. Minor presentational gaps in the candidate (e.g., a quoted value for a cusp coefficient without derivation) do not affect correctness.