Prospects for Acoustically Monitoring Ecosystem Tipping Points

The supporting information explicitly derives the OU linearization near a stable equilibrium and shows ρ(τ)=e^{λτ} and Var(η)=−σ^2/(2λ) in the stationary regime, with both trending toward 1 and ∞ respectively as λ→0−. The same file also gives the discrete-time AR(1) form with lag-1 autocorrelation ρ(1)=α and Var(η)=σ^2/(1−α^2), and notes α=e^{λΔt}, so ρ(1)→1 and the variance diverges as α→1 when approaching critical slowing down. These match the candidate solution exactly; the model further provides the exact discrete-time noise mapping from the continuous OU, which the paper summarizes but does not detail. See the SI summary and derivations for these formulas and limits , and the detailed continuous-time and discrete-time derivations .