Existence of periodic solution of a non-autonomous allelopathic phytoplankton model with fear effect

The paper proves existence of a positive T-periodic solution for the non-autonomous allelopathic phytoplankton model using Mawhin’s coincidence degree theory, after a log change of variables and operator setup, see the model statement (1.3)–(1.4) and the use of L, N, P, Q in Section 3 . However, in verifying the degree condition on Ker(L), the homotopy Ψ is written with sign inconsistencies, and the text claims Ψ(z,0)=0 has a unique solution even though the displayed equations at μ=0 cannot have a positive solution; the boundary non-vanishing argument is also logically flawed (the implication is reversed) . These issues render the current proof incomplete, despite the plausibility of the overall strategy and the standard lemmas employed. The candidate model’s alternative Poincaré-map-in-a-rectangle argument correctly handles three faces but relies on a top-edge inward-pointing inequality that does not follow from the paper’s (A3); after substituting m0 := k2 r2^M/(1 + w1 k1), the required inequality becomes independent of r2^M, so (A3) cannot guarantee it. Hence the candidate proof also has a gap. Given these gaps on both sides, the most accurate judgment is that both are incomplete.