Ecological Regime Shifts — Fold Bifurcation & Hysteresis

Lake Model

Phosphorus loading L: 0.60 Loss rate s: 0.50 Recycling rate r: 0.50 Hill coefficient h: 0.50 Perturbation Δx: 0.5

dx/dt = L − sx + rx²/(x²+h²)

x: phosphorus in water
L: external loading
s: loss (sedimentation)
r·x²/(x²+h²): internal recycling
(sigmoidal, catastrophic above rc)

Regime: —

Fixed points: —

Shallow lakes can shift abruptly between clear (low P) and turbid (high P, algae bloom) states. Once past the tipping point, the system collapses to the turbid attractor. Recovery requires loading to be reduced far below the threshold — the hallmark of fold bifurcation hysteresis.