Rabbits in a Meadow
Every year, the number of rabbits depends on two things: how fast they breed, and how crowded the meadow already is. That's one simple rule — but turn up the breeding rate and watch a calm, steady population turn into total chaos.
The meadow
Try a setting
Watching the meadow…
Let a few years go by and I'll tell you what kind of pattern the population is making.
One knob, three worlds
Low breeding rate: the population settles to a steady number and stays there — easy to predict.
Turn it up: it starts to boom and bust — too many rabbits one year, a crash the next. Turn it higher and the cycle splits again: 4 years, then 8…
Highest: the swings stop repeating at all. Same rule, but now the future is a mystery.
Where does calm end?
Slowly raise the breeding rate from 2.5. What's the highest rate that still gives one steady number? (Hint: it's right around 3.0.)
And does the starting number of rabbits change the final pattern? Try the same rate with 20 rabbits, then 180. In the calm and cycle worlds it doesn't matter — but in the wild world, it changes everything.
For teachers & grown-ups
This is the logistic map, xₙ₊₁ = r·xₙ·(1−xₙ), where x is the population as a fraction of the meadow's carrying capacity and r is the breeding rate. It's the same equation as the Scientist-level Logistic Map tool, here told as a population story and capped at 200 rabbits for concreteness. As r rises it passes through the period-doubling cascade — steady (r < 3), period-2 (3 < r < 3.45), period-4, period-8, … — accumulating into chaos near r ≈ 3.57. The "pattern detector" simply checks the settled tail of the series for the smallest repeat period and reports chaos when none is found. The hypothesis prompts target two big ideas: a single control parameter can move a system between order and chaos, and sensitive dependence on initial conditions appears only in the chaotic regime (changing the starting count leaves the periodic attractors unchanged but completely reshuffles a chaotic run).