Ecological Regime Shift — Fold Catastrophe

Lake ecosystem hysteresis: clear ↔ turbid states with nutrient loading

Turbidity: low
State: clear
The lake model: dx/dt = r·x(1−x/K) − h·x²/(x²+x₀²) + P, where x = algae biomass. Two stable equilibria (clear/turbid) coexist in the hysteresis zone. Once the tipping point is crossed, the system jumps catastrophically and requires much lower P to recover — irreversibility is the hallmark of fold catastrophes (cusp catastrophe in higher dimensions).