What equations does the simulator integrate?

The classic Lotka–Volterra system dN/dt = rN − aNP and dP/dt = bNP − mP, using RK4.

Where is the coexistence equilibrium?

At N* = m/b and P* = r/a, plotted on the phase plane with the closed orbit.

How is period estimated?

A linearized value T ≈ 2π/√(r m) is given together with the period measured from the simulated cycles.

Do starting populations change the cycle?

Yes. Neutral closed orbits have amplitude fixed by N₀ and P₀; predators typically lag prey by about a quarter cycle.

What does the output show?

HTML time-series curves, the closed phase-plane orbit with equilibrium, cycle statistics, measured period, and min/max populations.

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Lotka–Volterra Predator-Prey Dynamics Simulator

RK4 simulation of the classic predator–prey oscillator with phase plane, closed orbits, equilibrium, and period analysis.

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2 examples

Classic lynx-and-hare cycle: equilibrium (40, 50), period ≈ 14–15

With r = 0.5, a = 0.01, b = 0.01, m = 0.4 the coexistence equilibrium sits at N* = m/b = 40, P* = r/a = 50, and orbits starting from (60, 30) cycle with a measured period near 15 (linearized estimate 2π/√(rm) ≈ 14.05). Prey swing 18.5–73.9 while predators lag a quarter cycle behind, swinging 25.3–87.0 — the Hamiltonian drift stays at the 1e-12 level, confirming the closed orbit.

{
  "type": "html"
}

Fast aggressive predator: dramatic swings N 2–100, P 10–143

Turn the rates up — r = 1.0, a = 0.02, b = 0.02, m = 0.5 with a small predator inoculum (25, 10): the system still orbits (25, 50) but with far larger amplitude (prey crash to ≈2 before booming past 100; predators overshoot to ≈143) and a shorter linearized period 2π/√(rm) ≈ 8.89. Higher turnover ⇒ wider, faster cycles.

{
  "type": "html"
}

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7 options

SettingsAdjust formats, ranges, numbers, and modes.7

Prey growth rate r (per time unit)numberRequiredPer-capita prey growth with no predators present.Predation rate a (per predator)numberRequiredRemoval of prey per encounter — multiplies N·P.Conversion efficiency b (prey → predator)numberRequiredNew predators produced per prey consumed — sets the prey level needed to sustain predators (N* = m/b).Predator mortality m (per time unit)numberRequiredPer-capita predator death rate with no prey.Initial prey population N₀numberRequiredStarting prey population — sets the orbit amplitude together with P₀.Initial predator population P₀numberRequiredStarting predator population.Simulation duration (time units)numberRequiredHow long to integrate — a few cycle periods (T ≈ 2π/√(r·m)) make the orbit readable.

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Lotka–Volterra Predator-Prey Dynamics Simulator

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Examples

1. Classic lynx-and-hare cycle

2. High-turnover predator, large-amplitude crashes

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