Predator-Prey (Lotka-Volterra) Calculator

Simulate populations of inteting predator and prey species over time using the differential Lotka-Volterra numerical model.

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Input Parameters

Enter starting populations and intetion coefficients.

Initial Prey (x₀)

Initial Predators (y₀)

Model Coefficients

Prey Growth Rate (α)

Predation Rate / Capture Efficiency (β)

Predator Mortality Rate (γ)

Reproduction Efficiency (δ)

Prey Range (Min - Max)

Predators Range (Min - Max)

Simulated Timeline (scroll)

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Phase Shift Dynamics

Predator population peaks lag behind prey population peaks by approximately one-quarter of a full cycle period.

Classic Ecological Predator-Prey Systems Reference

Prey Species	Predator Species	Observed Cycle Period	Primary Limiting Mechanism
Snowshoe Hare	Canada Lynx	9 – 11 Years	Winter food availability + predation mortality
Moose (Isle Royale)	Gray Wolf	12 – 15 Years	Isolation, canine parvovirus, severe winters
Paramecium caudatum	Didinium nasutum	3 – 5 Days	Microscopic lab culture clearance rate

Methodology & Equations

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Prey Differential Equation

dx/dt = α·x - β·x·y

Prey growth is exponential ($alpha x$) diminished by encounters with predators ($eta x y$).

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Predator Differential Equation

dy/dt = δ·x·y - γ·y

Predator growth depends on prey consumption ($delta x y$) minus natural mortality ($gamma y$).

Frequently Asked Questions

What are the Lotka-Volterra predator-prey differential equations? ▼

The Lotka-Volterra model consists of two coupled first-order differential equations: $(dx / dt) = α x - β x y$ (prey growth minus predation) and $(dy / dt) = δ x y - γ y$ (predator growth from prey minus natural mortality).

Why do predator and prey populations oscillate out of phase? ▼

An increase in prey ($x$) provides abundant food, driving predator ($y$) reproduction. As predators increase, predation pressure causes the prey population to crash, subsequently causing the predator population to starve and decrease, starting the cycle anew.

What are the equilibrium points in the Lotka-Volterra model? ▼

The non-trivial equilibrium point where both populations remain steady is $x^* = (γ / δ)$ (prey) and $y^* = (α / β)$ (predators).