Foxes and rabbits, forever chasing each other. Two coupled equations produce the endless boom-and-bust cycle at the heart of population ecology.
Lotka-Volterra Predator-PreyLive
coupled population cycles
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Two coupled equations produce the classic boom-and-bust cycle: prey multiply, predators feast and multiply, prey crash, predators starve, and the cycle repeats. The phase portrait shows the closed orbit — populations forever chasing each other, never settling.
Reading this result: These parameters trace a closed orbit: prey and predator populations cycle forever without settling, staying about a quarter-period out of phase — the predator peak lags the prey peak. The orbit encircles the fixed point at (prey 20.0, predator 10.0) = (c/d, a/b). Raising predation b (now 0.1) or predator death c (now 1.5) shifts and reshapes that orbit — a higher death rate c pushes the prey equilibrium up and tends to lengthen the boom-and-bust period, while stronger coupling widens the amplitude of each swing.
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How it works
The Lotka-Volterra equations model prey that reproduce and predators that eat them. Prey abundance feeds predator growth; predator pressure crashes the prey; predators then starve, and the cycle repeats. The phase portrait reveals closed orbits — the populations never settle to a steady state but circle a shared equilibrium indefinitely.
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