Thought Toys · Exhibit 04
Give rabbits a meadow and foxes to hunt them, and neither side ever wins. The numbers swing up and down forever, locked in a chase where the foxes always crest a beat after the rabbits. Two simple rules; one eternal, lopsided loop.
Rabbit and fox populations oscillate in a never-settling cycle, with foxes peaking shortly after rabbits, tracing a closed loop in the phase plot.
Rabbits, left alone, would multiply without end. Foxes, with nothing to eat, would starve. Put them together and each becomes the other's brake. When rabbits are plentiful the foxes feast and breed — so the fox line climbs after the rabbit line, never with it. Then the swollen fox population eats the meadow bare, the rabbits crash, and soon the hungry foxes crash too. With the foxes gone, the rabbits rebound, and the whole chase starts over.
Watch the right-hand plot, where foxes are drawn against rabbits instead of against time. The state doesn't drift to a cosy middle and doesn't spiral away — it rides the same closed loop around and around, forever. That pale dot in the middle is the one balance point where births and deaths cancel exactly; press Go to balance and everything freezes on it. But the balance is a knife-edge: nudge it with Release rabbits and the populations don't return, they just orbit it on a wider ring.
Turn the foxes deadlier or the rabbits friskier and the loop changes shape and pace — yet the lag never disappears. Predators forever peak a quarter-turn behind their prey. It's the oldest rhythm in ecology, and it falls out of nothing but two lines of arithmetic.
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