Biological Sciences

Ecology

1925

Advanced

Lotka-Volterra Equations

\frac{dx}{dt} = \alpha x - \beta xy, \quad \frac{dy}{dt} = \delta xy - \gamma y

Predator-prey populations oscillate—prey growth minus predation, predators grow from eating prey.

By Alfred Lotka, Vito Volterra

Biological Sciences

1925 · Alfred Lotka

Human Reviewed

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Story PortalWho discovered this?Discover how Alfred Lotka and others shaped this equation.Visual PortalWhat does it look like?See the equation come alive in the Visual Studio.Machine PortalWhere is it used?Explore machines powered by this equation — wildlife management.Math PortalWhat does it derive from?Trace the mathematical lineage from logistic growth.Future PortalWhere is it going?Ecosystem restoration

Why it matters: Foundation of theoretical ecology and dynamical systems biology.

Discoverers: Alfred Lotka, Vito Volterra (1925)

What does it mean?

Predator-prey populations oscillate—prey growth minus predation, predators grow from eating prey.

Why should I care?

Foundation of theoretical ecology and dynamical systems biology.

Equation Compass

North — Prerequisites

West — History

East — Applications

South — Derivations

Derivation

Variables & Units

Symbol	Name	Unit	Meaning
$x$	Prey population	—	Prey count
$y$	Predator population	—	Predator count
$α$	Prey growth	—	Intrinsic growth rate
$β$	Predation rate	—	Encounter rate

Worked Example

Classic lynx-hare cycles in Canadian fur trade data.

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Sources & further reading

Search Wikipedia Search Wolfram MathWorld Open in Wolfram|Alpha Watch explainers on YouTube

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