Predator-Prey Dynamics

Differential Equations · Axiom Academy

The Lotka-Volterra equations: two coupled ODEs that turn a rabbit-and-fox island into a permanent, predictable cycle — never crashing to zero, never settling still. An island holds two species: rabbits (prey) and foxes (predators). Their populations never stop moving — up, down, up again, forever. Two equations explain exactly why. Set how good the foxes are at catching rabbits, hit Go, and watch both populations play out over 20 years. Same relationship every time: The phase plane: closed loops, not a spiral Same two populations, plotted against each other instead of time. Drag the starting point and watch its own loop trace out — every starting point gets its own permanent loop , nested around one center. Change a rate, move the equilibrium The center of every loop sits at one point: . Drag the fox death rate and watch that point — and every orbit around it — slide. Every loop you dragged in Beat 2 stayed its own permanent size — none of them shrank onto a shared cycle. That's the mathematical signature of a center : the coexistence equilibrium has purely imaginary eigenvalues, so nearby trajectories neither spiral in nor spiral out — they orbit forever at whatever amplitude they started with. Closed orbits, not a limit cycle — a distinction worth keeping straight as you classify critical points in this unit.

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