Simulate Lotka-Volterra predator-prey dynamics. Visualize population cycles, phase portraits, and equilibrium stability.
This tool integrates the classic Lotka-Volterra pair of differential equations - prey grow at rate alpha, are removed at rate beta per predator, predators grow from eaten prey at rate delta and die at rate gamma - stepping both populations forward from the initial counts you set. It plots the two time series, the predator-versus-prey phase portrait, and the coexistence equilibrium. The surprise: the standard model is neutrally stable. Every starting point traces its own closed loop; nothing damps the swings, and any disturbance moves the system onto a different cycle.
A concrete input and expected output from the current implementation.
Input
alpha = 1.5 (prey growth), beta = 0.5 (attack rate), delta = 0.5 (conversion), gamma = 0.75 (predator death); initial prey 2.0, initial predators 2.0; run 30 time units.
Expected output
Both populations rise at first - prey at 1.0 per time unit, predators at 0.5 - then cycle around the equilibrium at prey = gamma/delta = 1.5, predators = alpha/beta = 3.0. Prey crest first, as predators pass 3.0; predators crest about a quarter cycle later, as prey fall back through 1.5. Near equilibrium the period is about 2*pi/sqrt(alpha*gamma) = 5.9 time units and lengthens with amplitude, so the run shows several full loops in the phase portrait.
The coexistence point (gamma/delta, alpha/beta) is a neutrally stable center, so the orbit is closed and the small-amplitude period follows from the linearized frequency sqrt(alpha*gamma). The quarter-cycle lag between peaks comes from the sign structure of the rates: prey stop rising exactly when predators cross alpha/beta, and predators stop rising exactly when prey cross gamma/delta.
From the starting prey and predator numbers, the tool advances both populations through time in small steps, recomputing each step from the current rates: prey change at N(alpha - beta*P) and predators at P(delta*N - gamma). Every step is recorded, so a run yields the two time-series curves plus the predator-versus-prey trace in the phase plane, and rerunning with changed parameters lets you compare runs side by side.
Setting both rates to zero gives two fixed points: the trivial one at (0, 0) and the coexistence point at prey = gamma/delta, predators = alpha/beta. Around that point the linearized motion is a rotation with frequency sqrt(alpha*gamma), so small cycles complete in about 2*pi/sqrt(alpha*gamma) time units, and larger loops take longer. Nothing damps the motion: the equilibrium is a neutrally stable center, not an attractor.
Because the classic equations contain no damping. Every orbit is a closed loop around (gamma/delta, alpha/beta), and its size is fixed by the starting counts, not the parameters. Add prey self-limitation (a logistic term) or a saturating predator response and the center usually becomes a damped spiral or a true limit cycle - so no, eternal cycles are an idealization of the bare model.
Not reliably. The model ignores carrying capacity, seasonality, refuges, age structure, and every other species in the food web, and it treats populations as smooth continuous numbers. Its honest use is qualitative: showing why two-species systems oscillate out of phase and which parameters shift the equilibrium. Treat the plotted values as illustrations, not forecasts.
The example and behavioral notes were checked against the browser implementation. Standards and primary references below define the relevant format, formula, or platform behavior.
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