# Predators and Prey

> The Lotka–Volterra model in the plane of the two populations: a direction field, the closed orbit of a solution, and a state that an action moves around it.

A population of prey $x$ grows when it is left alone, and is eaten at a rate proportional to the number of meetings between prey and predators. A population of predators $y$ dies out when it is left alone, and grows with the same meetings. Alfred Lotka (1925) and Vito Volterra (1926) wrote this as $x' = ax - bxy$ and $y' = dxy - cy$.

The picture is the plane of the two populations, not a plot against time. At each point the arrow shows where the populations go next. Many prey feed more predators; many predators leave few prey; few prey starve the predators; few predators let the prey recover. So the state turns around the equilibrium $\left(\frac{c}{d}, \frac{a}{b}\right)$, the grey dot inside the orbit, where both populations stay constant.

In this model the state returns exactly to its start: the quantity $V(x, y) = dx - c\ln x + by - a\ln y$ does not change along a solution, so each solution is a closed level curve of $V$. The blue curve is the level curve through the start $(p, q)$. An action moves the state along it with the fourth-order Runge–Kutta method, $h = 0.03$ time units every 30 milliseconds. Edit the rates $a$, $b$, $c$, $d$ or the start $p$, $q$ (all six must stay positive), then run the second action to put the state back at the start. Near the equilibrium one turn takes the time $\frac{2\pi}{\sqrt{ac}}$; a larger orbit takes longer.

After a JSXGraph example by The Center of Mobile Learning with Digital Technology.

- [Open the interactive version](https://graph-paper.io/showcase/en/predator-prey)
- [Learn more on Wikipedia](https://en.wikipedia.org/wiki/Lotka%E2%80%93Volterra_equations)