Global Stability of the Periodic Solution of the Three Level Food Chain Model with Extinction of Top Predator

The paper’s main theorem (Theorem 3.13) proves that, under 0<a1<1, λ1<(1−a1)/2, the transverse average growth of z along the boundary cycle Γ is negative (condition (3.14)), and the differential-inequality condition (3.21) holds, then z(t)→0 and every trajectory converges to the unique x–y limit cycle Γ on H2={z=0} in R3+; the proof combines a Floquet-multiplier computation around Γ (Proposition 3.10) and a differential-inequality extinction lemma (Lemma 3.11), together with uniqueness of the boundary limit cycle when λ1<(1−a1)/2 (Lemma 3.9) and the absorbing box (2.1) for boundedness . The candidate solution reaches the same conclusion but via a different route: it frames the system as a 3D competitive flow, invokes Hirsch’s 3D Poincaré–Bendixson property, and applies food-chain persistence/extinction theory by checking that the top predator’s invasion exponent is negative on all boundary minimal sets (Γ and sets with y=0), deducing z→0 and convergence to Γ. It also shows that (3.21) forces λ2>(1+a1)2/4, hence no interior equilibrium, aligning with the paper’s classification (Table 1) where λ2>(1+a1 2)2 eliminates E* . Both arguments are logically consistent with the paper’s model (1.3) and hypotheses; the paper’s proof uses differential inequalities and explicit Floquet multipliers, whereas the model relies on monotone-system and persistence results. We did not find a substantive flaw in either approach under the stated assumptions.