Predator and Prey

More rabbits feed more foxes, and more foxes eat more rabbits. Two coupled rates of change make both populations rise and fall in an endless loop.

prey 10.0predators 5.0

t = 0.0

How it works

Systems of ODEs and phase planes

In one sentence

In the Lotka–Volterra model, prey $x$ grow at rate $\alpha$ and are eaten in proportion to encounters $xy$ , while predators $y$ grow with encounters ( $\delta xy$ ) and die at rate $\gamma$ ; solutions are closed orbits around the equilibrium $(\gamma/\delta,\ \alpha/\beta)$ .

General formula

{\htmlData{sym=0}{x}}' = {\htmlData{sym=2}{\alpha}} {\htmlData{sym=0}{x}} - {\htmlData{sym=3}{\beta}} {\htmlData{sym=0}{x}} {\htmlData{sym=1}{y}}, \qquad {\htmlData{sym=1}{y}}' = {\htmlData{sym=4}{\delta}} {\htmlData{sym=0}{x}} {\htmlData{sym=1}{y}} - {\htmlData{sym=5}{\gamma}} {\htmlData{sym=1}{y}}

$\alpha x$ : prey breed. $-\beta xy$ : predators meet prey and eat them. $\delta xy$ : well-fed predators raise more young. $-\gamma y$ : predators die off. Every term is a birth, a death or a meeting.

What the symbols mean

$x$
Prey population
How many rabbits there are right now.
$y$
Predator population
How many foxes there are right now.
$\alpha$
Prey birth rate
How fast rabbits multiply when there are no foxes around.
$\beta$
Predation rate
How often a fox–rabbit encounter ends with the rabbit eaten. Encounters happen in proportion to x·y.
$\delta$
Predator growth from food
How efficiently eaten rabbits turn into new foxes.
$\gamma$
Predator death rate
How fast foxes die off when there is nothing to eat.

Each point in the phase plane is a state: $x$ rabbits and $y$ foxes. Arrows show the rates $x'$ and $y'$ there.
Solutions follow the arrows. A closed loop means the cycle repeats forever.
The equilibrium sits where both rates vanish: $x^* = \gamma/\delta$ , $y^* = \alpha/\beta$ .

Learn the math

Complex EigenvaluesLin Alg · Lesson 29, Eigenvalues and Eigenvectors Euler's MethodDiff Eq · Lesson 12, First Order Differential Equations Systems of Differential EquationsDiff Eq · Lesson 30, Systems of Differential Equations The Phase PlaneDiff Eq · Lesson 32, Systems of Differential Equations Complex EigenvaluesDiff Eq · Lesson 34, Systems of Differential Equations Modeling with SystemsDiff Eq · Lesson 36, Systems of Differential Equations

Learn the math

Complex EigenvaluesLin Alg · Lesson 29, Eigenvalues and Eigenvectors

Euler's MethodDiff Eq · Lesson 12, First Order Differential Equations

Systems of Differential EquationsDiff Eq · Lesson 30, Systems of Differential Equations

The Phase PlaneDiff Eq · Lesson 32, Systems of Differential Equations

Complex EigenvaluesDiff Eq · Lesson 34, Systems of Differential Equations

Modeling with SystemsDiff Eq · Lesson 36, Systems of Differential Equations