Figure 4
The image contains three panels of graphs related to stochastic impulse response and ecological resilience. Panel (a) features a main graph with individual trajectories in grey, 75% and 90% confidence intervals in blue, sample mean in red, sample median in green, and analytical mean response in dash-dotted line. The upper inset shows the empirical density of ΔI at t = 10, and the lower inset displays the cumulative critical probability πτ by horizon. Panel (b) includes two graphs mapping recovery time and critical probability across a grid of persistence and volatility values. Panel (c) presents five histograms showing the empirical distribution of ΔI(t) at different horizon snapshots (t = 2, 5, 10, 15, 20 years), with kernel density estimates and snapshot means marked. All values are approximated.Stochastic impulse response and ecological resilience. Panel (a) presents the full distributional dynamics following an acute environmental shock, together with inset diagnostics for the mid-horizon density and the cumulative probability of breaching the critical threshold. Panel (b) maps recovery time and critical probability across a two-dimensional grid of persistence and volatility values, identifying the combinations under which KPI measurement may be contaminated by natural recovery dynamics. Panel (c) tracks the evolution of the ΔI distribution from early recovery through long-run stationarity, documenting the gradual re-centring of ecological outcomes after the shock