STABILITY ANALYSIS OF A MODIFIED LESLIE–GOWER PREDATION MODEL WITH WEAK ALLEE EFFECT ON THE PREY

The paper correctly identifies the loci for saddle-node (along Δ=0) and the Bogdanov–Takens degeneracy (det J=tr J=0), but its BT claim includes a sign error in the formula for Q (it states Q = S(1 + A + M − u1)/(−1 + A − M + u1), which is negative of the correct value) and explicitly omits the nondegeneracy/unfolding checks, citing intractability; see Theorem 3.4 and surrounding discussion, as well as the trace condition tr(J(u,u)) = u(Qu − S(A+u)) implying Q = S(A+u)/u at trace zero . The saddle-node theorem application (Theorem 3.5) is outlined using a rescaled vector field f(u,v;Q) and provides ker(J), U·f_Q ≠ 0, and U·D^2f(V,V) ≠ 0, albeit with a questionable simplification of the second derivative term; still, the intended conclusion (a fold along Δ=0) is sound . By contrast, the model solution computes with the original system, finds the precise left/right kernels, verifies Sotomayor’s conditions, and performs full BT nondegeneracy and transversality checks in (Q,S), obtaining the correct BT condition Q = S(ud + A)/ud = S(1 + A + M − u1)/(1 − A + M − u1), consistent with tr J = 0 and det J = 0 at E1 .