Enhancing Population Persistence by a Protection Zone in a Reaction-Diffusion Model with Strong Allee Effect

The paper’s Theorem 2.1 establishes the same threshold: λ1(α,ℓ)<0 implies uniform persistence (with a uniform positive constant under Robin, or relative to a fixed positive profile under Dirichlet), and λ1(α,ℓ)>0 yields small‑data extinction; see the statement and proof sketch around Theorem 2.1 and its supporting arguments via compactness/robust persistence and the perturbed eigenvalue problem (2.5)–(2.8) . The candidate solution proves the same claims using a gradient-flow Lyapunov functional and a minimal positive equilibrium argument. Aside from a minor sign error in the Robin boundary terms of the energy, the model’s argument follows standard semigroup and maximum‑principle techniques and reaches the same conclusions; consequently, both are correct, with different proofs.