# Lotka–Volterra Predator-Prey Dynamics Simulator

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

> Canonical page: https://elysiatools.com/en/tools/lotka-volterra-predator-prey

- **Category:** Health

- **Keywords:** lotka volterra, predator prey, lynx hare, population cycles, population dynamics, phase plane, limit cycle, coexistence equilibrium, ecology simulator, predation model, oscillation, rk4

## Overview

The Lotka–Volterra Predator-Prey Dynamics Simulator runs an RK4 integration of the classic oscillator dN/dt = rN − aNP, dP/dt = bNP − mP. Enter prey growth, predation, conversion, and mortality rates plus starting populations and duration to get time-series cycles, a closed phase-plane orbit, the coexistence equilibrium, and period analysis.

## Inputs

- **Prey growth rate r (per time unit)** (number): Per-capita prey growth with no predators present.
- **Predation rate a (per predator)** (number): Removal of prey per encounter — multiplies N·P.
- **Conversion efficiency b (prey → predator)** (number): New predators produced per prey consumed — sets the prey level needed to sustain predators (N* = m/b).
- **Predator mortality m (per time unit)** (number): Per-capita predator death rate with no prey.
- **Initial prey population N₀** (number): Starting prey population — sets the orbit amplitude together with P₀.
- **Initial predator population P₀** (number): Starting predator population.
- **Simulation duration (time units)** (number): How long to integrate — a few cycle periods (T ≈ 2π/√(r·m)) make the orbit readable.

## When to use

- When you need closed predator–prey orbits, time series, and the coexistence point N* = m/b, P* = r/a without coding an integrator.
- When you want to compare the linearized period 2π/√(r m) with the cycle length measured from the RK4 trajectory.
- When you are checking how N₀ and P₀ set orbit amplitude around the equilibrium for given rates.

## How it works

- Enter prey growth r, predation a, conversion b, predator mortality m, initial N₀ and P₀, and simulation length tMax.
- RK4 steps the Lotka–Volterra ODEs; Hamiltonian drift is monitored so the orbit stays closed.
- The HTML result plots N(t) and P(t), the N–P phase plane with the closed orbit and equilibrium, and reports cycle count, measured period, and amplitude ranges.
- Linearized period T ≈ 2π/√(r m) is shown beside the measured period from the integrated trajectory.

## Use cases

- Demonstrate lynx–hare style oscillations, predator lag, and a closed orbit around (m/b, r/a).
- Explore how raising r, a, b, or m shortens the period and widens population swings.
- Verify that a chosen (N₀, P₀) produces a conserved closed orbit with reported amplitudes and period.

## Frequently asked questions

### 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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