Stochastic bifurcation of a three-dimensional stochastic Kolmogorov system

The paper proves a stochastic decomposition: x(t) = g(t) Ψ(∫ g^2, x/g0), where g solves a one-dimensional stochastic logistic equation (3.7). This yields: (i) a scalar radial factor g (hence Z=g^2) with threshold σ^2=2α for extinction vs persistence, and (ii) deterministic angular dynamics via a time change. From this they classify all ergodic stationary measures (rays and invariant cones) and Lyapunov exponents at δO, exactly matching the candidate’s reduction to a logistic diffusion for Z and a time-changed copy of (1.3). The paper’s Theorem 1.1 and Theorem 1.3 coincide with the model’s items (1)–(3a)–(3c), and Lemma 6.1 gives λi(δO)=α−σ^2/2 as in the model. The two arguments are essentially the same structure (radial logistic + deterministic direction via time-change), with the paper using an explicit decomposition formula and strong Feller/irreducibility on cones, and the model using an Ito/log-ratio and generator-splitting viewpoint. No substantive conflict detected. Citations: decomposition and logistic eq (3.7) with ug(ω) and threshold at 2α ; main classification Theorem 1.1 (rays vs cones) and pull-back Ω-limits Theorem 4.1/1.3 ; ray/cone stationary measures and uniqueness on cones (Props. 5.1, 5.3, 5.4) ; Lyapunov exponents Lemma 6.1 .