PREDATOR & PREY
A meadow that keeps its own books. Hares breed, lynx hunt, and neither ever wins — the two counts chase each other forever.
Two lines of arithmetic, run forever.
Every hare added is a hare that might be eaten. Every lynx fed is a lynx that lives to hunt again. Alfred Lotka and Vito Volterra wrote that bookkeeping as a pair of coupled rates — the oldest working model in mathematical ecology.
Nothing here is scripted. The meadow above integrates these equations with a fourth-order Runge–Kutta step, sixty times a second. Change a count and the arithmetic does the rest.
Plot hares against lynx instead of against time and the oscillation becomes a single closed loop, travelled anticlockwise. The dashed crosshair in the instrument marks the equilibrium — the one pair of counts where nothing changes. The meadow circles it but never lands.
Cull a species and you throw the state onto a different loop. Watch both curves lurch: knock the hares down and the lynx, suddenly hungry, fall behind them; thin the lynx and the hares boom before the predators catch up.
Equilibrium — hares
20H* = γ ⁄ δ. Below it hares recover; above it, lynx multiply faster than the meadow can feed them.
Equilibrium — lynx
10L* = α ⁄ β. The predator count the prey birth-rate can sustain, if the loop ever stood still.
The orbit leaves a faint engraving of every loop it has already traced — a second-read detail. Leave the meadow running and the closed curve slowly darkens into the panel.