FROM HABITAT DECLINE TO COLLAPSE: A SPATIALLY EXPLICIT APPROACH CONNECTING HABITAT DEGRADATION TO DESTRUCTION

The paper proves uniform-in-time convergence uc → u∞ as c → ∞ under µ1,∞ ≠ 0 via a compactness/mollification argument for finite times (Lemma 3.8) and uniform asymptotic stability for the two spectral regimes (Cor. 3.7 for µ1,∞ > 0; Lemma 3.9 for µ1,∞ < 0), culminating in Theorem 1.3 . The candidate solution reaches the same conclusion using Mosco/Dirichlet-form convergence and semigroup methods for the finite-time part, plus spectral-gap arguments for the tail. While the candidate’s approach is valid in spirit, it omits some technical justification when upgrading L2 semigroup convergence to C(Ω) bounds in the mild formulation and when asserting a uniform spectral gap for the linearizations. Thus, both proofs are essentially correct, but the methods differ and the model’s proof needs a few additional details to be fully rigorous.