Thought Toys · Cycles & change · Exhibit 95
A fish stock under a fixed quota looks fine right up until it isn't. But knock it and time how long it takes to come back, and you get a number that starts stretching long before anything visibly breaks.
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The amber line is a fish stock. It grows on its own, faster when there is room and slower as it fills the sea, and every season a fixed quota is taken out. Where those two balance is the dashed line: the healthy level the stock sits at and returns to.
Press knock it. Something takes a bite out of the population — a bad winter, a disease — and the line dips, then climbs back. The shaded stretch is how long the healing took. At a low quota it is quick.
Now raise the quota and knock it again. The stock still settles at a healthy level. It still recovers. It just takes longer. Keep going and the recovery time stretches out dramatically: at a quota of 14.4 the same small knock takes five times as long to heal as it did with no fishing at all.
That is the warning, and the reason it is useful is what the dashed line is doing meanwhile. It barely moves. Halfway to the cliff the healthy stock is still 87% of its unfished level, and at 96% of the way it is still three-fifths. If you only watch the population you see a fishery in decent shape. The recovery time is telling you something the population level is hiding: the system is losing its grip.
Push the quota past 15 and there is no healthy level left to return to. The dashed line vanishes because the balance point does not exist any more, and the stock falls all the way to nothing. Nothing was special about that last small increase. The ground had been going soft for a long time.
Real ecologists cannot walk up to a lake and knock it, so they read the same slowness out of ordinary wobble. Turn up how noisy the world is and watch: near the cliff the same weather makes bigger swings that take longer to settle. Rising variance and a rising tendency for each year to resemble the last are exactly the early-warning signals used on lakes, ice cores and climate records.
One honest check is built in. Press use a system with no cliff: a stock that has no tipping point at all, moved by the same dial. Its recovery time is dead flat across the whole range. So a rising recovery time is not what turning a knob looks like — it is what an approaching cliff looks like.
improve/verify/95-critical-slowing-down.js, 62 checks).
The recovery time is measured from the simulation by timing the decay of a knock, never assumed: across
seven quotas from 3 to 14.85 it matches theory to better than 0.3%, and a log-log fit returns the exponent
−0.500. Convergence tests confirm the number belongs to the system and not to the measurement — halving the
knock twice, and quartering the integrator step, leave it unchanged. With noise, the measured lag-1
autocorrelation matches exp(−λΔt) and the variance ratio tracks σ²/(2λ).
Five negative controls. The fold-free system returns at a fixed rate whatever the dial says,
giving a scaling exponent of 0 — and no rise in either variance or autocorrelation under the same noise. A
claimed exponent of −1 is refuted by the fit. Halfway to the cliff the recovery time is only √2 times its
unstressed value, so the alarm is genuinely quiet until late rather than always rising. And a wrong critical
quota of rK/2 is refuted directly: the stock is already gone at h = 20.
Also in Cycles & change: Predator & prey →
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