Long time behavior of a degenerate stochastic system modeling the response of a population face to environmental impacts

The paper proves existence/uniqueness of πε and identifies its ε→0 limit using Hörmander-accessibility plus a Freidlin–Wentzell-style action functional adapted to the degenerate 2D setting, culminating in Theorem 2.4 (δ(1,0) if σ1<0; 1/2-mixture if σ1=0; δ(−1,0) if σ1>0) . The candidate solution obtains the same limit law by a different route: solving the 1D stationary Fokker–Planck for X, applying a Laplace principle to the resulting density, and then controlling Y via a stationarity computation. The model’s proof matches the paper’s conclusion but omits technical justification around zero flux and the interior degeneracy at x=−σ0/σ1; still, the selection rule and limiting measures coincide with the paper’s main result .