Thought Toys · Cycles & change · Exhibit 75
The same growth rule, applied to two populations that differ only in their starting size. One sits above a critical line and climbs to full health. The other sits below it and quietly goes extinct — no shock, no bad luck, just too few to begin with.
Nine populations, one rule, two outcomes recovers declines your population
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Every one of the nine thin background lines obeys the exact same growth rule. They differ only in where they start. The dashed amber line is the threshold, A: lines that start above it curve upward, settling at the dashed cyan carrying-capacity line above. Lines that start below it curve downward, toward zero — not because anything went wrong, but because a population that small can't find each other, defend itself, or reproduce fast enough to outrun its own losses. The thicker amber line is yours: drag the starting population slider and watch it join whichever family its starting point belongs to.
Now drag the starting population slowly toward the threshold from either side, and watch the time it takes to visibly move away. Far from the threshold, the outcome is obvious almost immediately. Close to it, the population seems to hang there, barely changing, for a long stretch before it finally commits to rising or falling. That hesitation isn't noise and it isn't a display glitch — it's the same threshold, showing up as slowness rather than as an outcome. In a real population you wouldn't get a warning label; you'd get exactly this: things looking fine, or at least stable, right up until they very slowly stop being so.
Drag the A slider and the threshold itself moves — a more fragile population (small A) needs less to recover; a more fragile-to-decline one (large A) needs more. The nine background lines recolour immediately, because whether a fixed starting population of, say, 40 recovers or declines depends entirely on which side of the current threshold it sits on — the rule never changed, only the line did.
improve/verify/75-allee.js, 20 checks): an independent
bisection search over starting populations — not the formula, a simulation of the actual differential equation
— locates the basin boundary and finds it lands on A itself to four decimal places, for three
different thresholds. The stability of all three equilibria is confirmed two separate ways: a numerical
derivative of the model, and an empirically measured exponential rate from simulation, both matching the
closed-form prediction. Near the threshold, the time to move a fixed distance away more than triples as the
starting point closes in on it — critical slowing down, measured, not assumed. Two negative controls confirm
the test has teeth: a plain logistic model with the threshold term removed grows to K from a population of
0.01, since it has no threshold at all; and a plausible-but-wrong guess of "half the carrying capacity"
(A=50) is directly contradicted by a population of 40, which survives under the real rule.
Also in Cycles & change: Predator & prey →
← the cabinet · Thought Toys — a cabinet of explorable explanations. Exhibit 75.