Global dynamics in a predator-prey model with cooperative hunting and Allee effect and bifurcation induced by diffusion and delays

The paper proves, under the “weak cooperation” hypothesis c < 1/(r(1−a)) and the assumptions a(pH) < 0 and that every periodic orbit is orbitally stable, that pH < p# and gives the four phase portraits; see Theorem 4 and its proof, which explicitly derives pH < p# and the basin separation by Γs(Ea) with uniqueness of the attracting cycle in pH < p < p# . The candidate solution reproduces this structure: forward invariance/boundedness (cf. Lemma 1 ), local equilibria and Hopf (Theorem 2 ), the role of Γs(Ea) and the heteroclinic at p# (Propositions 4–6 and Theorem 3 for the weak-cooperation regime ). The only gaps are that the model answer does not name the weak-cooperation condition explicitly (it is used implicitly via the “unique interior equilibrium” premise) and sketches, rather than cites, the rigorous monotonicity and basin-separation lemmas the paper provides. Net: both are correct and closely aligned.