Catastrophic shifts, hysteresis, and early-warning signals near tipping points.
Nutrient P1.50
Recovery Rate r0.50
Disturbance σ0.05
Initial State x₀3.50
Scheffer's lake eutrophication model: dx/dt = r·x(1−x/K) − α·x²/(h²+x²)·P. Below critical P, clear water is stable;
above it, turbid water dominates. Hysteresis means restoration requires P far below the tipping point.
Increasing variance and autocorrelation are early-warning signals of impending collapse.