---
tag: 00Y5QZ82AA
kind: example
name: Lotka–Volterra equations
aliases: [predator-prey-equations, lotka-volterra-predator-prey-model]
msc: [92D25, 34C25]
instance_of: [0673D85QKZ]
uses: [0G4X36AE0X, 0SS1P1YC9A, 01DSNP0B4R, 09ZWYRQ41J]
sources:
  - url: https://en.wikipedia.org/wiki/Lotka%E2%80%93Volterra_equations
    locator: Population equilibrium
  - url: https://en.wikipedia.org/wiki/Lotka%E2%80%93Volterra_equations
    locator: Hamiltonian structure of the system
same_as:
  wikidata: Q150052
  wikipedia: Lotka–Volterra equations
---

## Example

Let $\alpha, \beta, \gamma, \delta > 0$ be constants. Let $x(t) > 0$ be the size of a prey population and $y(t) > 0$ the size of a predator population at time $t$.
The Lotka–Volterra equations are the [[autonomous system|07BNRFQ0PJ]]
$$x' = \alpha x - \beta x y, \qquad y' = \delta x y - \gamma y.$$
The [[equilibrium points|0G4X36AE0X]] in the closed quadrant $x, y \ge 0$ are $(0, 0)$ and $(\gamma / \delta, \alpha / \beta)$.
The function $V(x, y) = \delta x - \gamma \ln x + \beta y - \alpha \ln y$ is a [[first integral|0SS1P1YC9A]] on the open quadrant $x, y > 0$.

## Remarks

Along a solution, $\frac{d}{dt} V = (\delta x - \gamma)(\alpha - \beta y) + (\beta y - \alpha)(\delta x - \gamma) = 0$.
The [[linearisation|09ZWYRQ41J]] at $(0, 0)$ has the eigenvalues $\alpha$ and $-\gamma$, so the origin is a saddle. At $(\gamma/\delta, \alpha/\beta)$ the eigenvalues are $\pm i \sqrt{\alpha \gamma}$.
The level curves of $V$ in the open quadrant are closed curves around $(\gamma/\delta, \alpha/\beta)$. So each solution other than this equilibrium is [[periodic|01DSNP0B4R]]: the two populations oscillate.

## Explanations

### L1 human

Prey grow when there are few predators, and predators grow when there is much prey.
This feedback makes both populations rise and fall in cycles, with the predator peak after the prey peak.