The development of deep-ocean anoxia in a comprehensive ocean phosphorus model

The paper derives the fast–slow reduction for the oxygen–reductant subsystem (eqs. (3.1)–(3.4)), identifies the critical manifold r + ε19 ν = ν/g − λ6 s3/(λ11 + g), simplifies the r-equation using the identity δ25/λ6 − δ3 = 0, and states the threshold λ6 s3 ≷ ν with time scales ≈ ε37 (about 30 y) for anoxic approach and ≈ ε32/ε19 (about 3000 y) for oxic recovery. The candidate solution reproduces these steps essentially verbatim and adds standard normal-hyperbolicity (Tikhonov–Fenichel) justification and a monotonicity argument for uniqueness/stability of the anoxic equilibrium. See the paper’s equations and discussion around (3.1)–(3.3) for the slow manifold and reduced r-dynamics, the simplification to (3.4) and the 30 y estimate, and the oxic rescaling and 3000 y estimate in (3.5)–(3.6) .