Lake Regime Shift: Nutrient Loading & Hysteresis (2D)

Nutrient Loading

External loading L 0.00

Internal recycling r 0.95

Flushing rate s 1.00

Lake State

Clear (oligotrophic)

Phosphorus P

0.00

dP/dt

0.00

Forward tip (L↑)

—

Backward tip (L↓)

—

L (loading) →↑ P (phosphorus)

How it works

Shallow-lake phosphorus P follows Carpenter–Ludwig–Brock's eutrophication model: external loading L feeds P, outflow/sedimentation removes it at rate s·P, and P recycled from lake-bed sediment adds a sigmoid feedback term:

dP/dt = L − s·P + r · P^q / (m^q + P^q)

q = 8, m = 1  (steep sigmoid: recycling only "switches on" near P ≈ m)

Because that recycling term saturates, the equilibrium relation L(P) = s·P − r·P^q/(m^q+P^q) folds back on itself for a wide range of r and s — the S-curve on the plot. The upper and lower branches are stable (clear and turbid states); the middle, backward-sloping branch is unstable and is never actually observed.

L slider — nutrient input from agriculture, sewage, runoff. Raise it slowly and P stays low until L crosses the upper fold, then P jumps abruptly to the turbid branch — a regime shift, not a gradual response.
Lower L back down and the lake does not clear up at the same loading it collapsed at — it stays turbid until L drops past the much lower lower fold. That gap is hysteresis: restoration needs far deeper cuts than prevention would have.
r (recycling) — how strongly bed sediment releases stored phosphorus back into the water once algae die and settle; raising it widens the hysteresis loop.
s (flushing) — how fast water (and dissolved P) leaves the lake; raising it narrows the loop and pushes both folds higher.
Auto ramp sweeps L up to the top and back down by itself so the jump-up / jump-down asymmetry is easy to see traced live on the plot — the panel above also logs the measured L at each jump.

This is the mechanism behind real documented lake collapses (e.g. Lake Mendota, the Norfolk Broads): why cutting phosphorus back to the level that once kept a lake clear often fails to restore it.

Clear, vegetated state Turbid, algae-bloom state