Global Hopf bifurcation and connected components in a delayed predator-prey model

The paper proves the two claims (boundedness of all Hopf components for K<K0 and, when χ(n)=2, coincidence/nestedness of the two components connecting τ−n and τ+n) as Theorem 4.6, using explicit a priori bounds (Lemma 4.2), exclusion of period-1 (and hence 1/n) solutions in the scaled system (Lemma 4.3), sharp period estimates T(i)n ∈ (1/(n+1), 1/n) (Lemma 4.4), and the global Hopf alternative (Theorem 4.1) with index sum zero, together with the nonexistence of periodic orbits at τ=0 and for τ≥τmax (Theorem 4.5). This yields both boundedness and, when χ(n)=2, a single connected component that starts at τ−n and ends at τ+n, with components across n ordered/nested as described . By contrast, the model’s solution invokes a “virtual period/strip invariance” with strips In=(2/(2n+1), 2/(2n−1)) and claims In+1⊂In; this is incorrect (these intervals are disjoint and not nested), and the justification via virtual-period theory is not established for the RFDE setting at hand. The model also relies on that strip boundary never being met without a valid argument; the paper instead rules out T=1 and 1/n and directly bounds the T-projection, which suffices for compactness of components in Fuller’s space .