Long-time behaviour of a model for p62-ubiquitin aggregation in cellular autophagy

The paper rigorously proves the dichotomy for the zero steady state via a time-regularization and a q-chart blow-up, establishing local asymptotic stability for ᾱ > 1 and instability for ᾱ < 1 (Theorem 1 and supporting lemmas) . By contrast, the candidate solution’s Phase-2 proof contains a critical flaw: it asserts that the reduced y-nullcline satisfies ẏ0 = 0 ⇒ y = 1, but the explicit factorization shows ẏ0 = (1 − y)[κ2 x − κ− y((n − 2) − 2y)/(n − 2)], so there is a second branch for y ≠ 1 on D (nonempty for n ≥ 5). This undermines the claimed uniqueness of the reduced equilibrium and the ensuing uniform attraction argument. In addition, the Dulac–Bendixson calculation and positive invariance of the reduced domain D are stated without rigorous boundary verification. The final conclusion matches the paper’s theorem, but the provided proof is not sound as written.