A STUDY ON A CLASS OF PREDATOR-PREY MODELS WITH ALLEE EFFECT

Jianhang Xie, Changrong Zhu

incompletemedium confidence

Category: Not specified
Journal tier: Specialist/Solid
Processed: Sep 28, 2025, 12:56 AM
arXiv Links: Abstract ↗PDF ↗

Audit review

The paper asserts a codimension-two Bogdanov–Takens (BT) bifurcation near E8 for (s,h) near (s1,h3), but its proof does not state or verify the BT nondegeneracy conditions and implicitly assumes lim h11 = −(s1+2+q) < 0; this is not always true (it fails at m = 1/(q+2)), where the uv-coefficient in the normal form vanishes, so the BT degenerates unless such values are excluded. The reduction to a BT normal form u′=v, v′=l00 + l01 v + u^2 + u v is presented without checking that the Jacobian ∂(l00,l01)/∂(η1,η2) is nonzero via an explicit computation, nor are the special cases m ∈ {1/(2(q+1)), 1/(q+2)} treated, though s1 is singular at the former and b=0 at the latter (undermining nondegeneracy) . By contrast, the model computes the BT invariants explicitly, obtaining a=−q/2≠0 and b=2(q+1)(1−m(q+2))/(2m(q+1)−1), and states the necessary exclusions m≠1/(2(q+1)), m≠1/(q+2), together with a clear parameter unfolding, thereby supplying the missing hypotheses and a complete argument.

Referee report (LaTeX)

\textbf{Recommendation:} major revisions

\textbf{Journal Tier:} specialist/solid

\textbf{Justification:}

The manuscript presents a coherent bifurcation analysis of a Leslie–Gower predator–prey model and plausibly identifies a codimension-two Bogdanov–Takens point organizing local dynamics. However, the BT claim is stated without explicit nondegeneracy hypotheses, and a sign assertion used to normalize the quadratic terms fails at m = 1/(q+2), where the uv coefficient vanishes. These omissions affect correctness on a codimension-one set and should be remedied. With added conditions and explicit rank/nondegeneracy checks, the result would be rigorous and useful.