Predator–prey.
The Lotka–Volterra equations are the simplest model of two species locked in a feedback loop: rabbits breed, foxes eat them, foxes starve without food. Drag the sliders to change birth and death rates and watch the phase plane trace a closed orbit. The orbits are neutrally stable — flip on the noise slider and you see structural stability fail in real time: the trajectory drifts across orbits and, eventually, the meadow empties out.
What you're looking at.
Two coupled ODEs: ẋ = αx − βxy says prey grow exponentially on their own, but get eaten in proportion to how often a predator runs into one (the xy encounter term). And ẏ = δxy − γy says predators turn eaten prey into more predators, but die off without food. The non-trivial fixed point is at (γ/δ, α/β); small perturbations orbit around it with linear-theory period T = 2π/√(αγ).
Here's the big surprise: those orbits are neutrally stable. They neither spiral in to the fixed point nor spiral out — every closed orbit is a valid steady state, parametrised by initial conditions. Move a slider and the system jumps to a different closed orbit at a different radius. It does not relax back to the "right" one. That's a property of the model, not a numerical bug — which is why this lab uses RK4 rather than forward Euler (which would visibly spiral outward and fake an instability that isn't really there).
Now slide noise σ above zero. The trajectory drifts across orbits, a stochastic random walk in the conserved quantity. Given enough time it will brush against an axis and one species crashes to extinction — the model has no floor to catch it. That fragility is the original critique of Lotka–Volterra as a real ecological model: real populations have density-dependent terms (logistic growth, saturation in predation) that turn the neutral orbit into a proper attractor.