Population over time
Phase plane — predator vs. prey
Applies on the next reset
The Lotka-Volterra equations behind every predator-prey cycle
Two populations locked in a feedback loop: prey grow on their own, but predators cut them down — and predators, in turn, depend entirely on eating prey to grow. Change any of the four rates on the right (α prey growth, β predation, γ predator death, δ predator growth) and watch both populations settle into a new rhythm.
The top chart plots both populations against time. Watch how a rise in prey is always followed — a little later — by a rise in predators, who then overhunt the prey until food runs short and their own numbers fall too. That lag between cause and effect is what keeps the whole system oscillating instead of settling down.
The bottom chart tells the same story a different way: instead of time on the horizontal axis, it plots predators against prey directly, so each moment becomes a single point. As the two populations rise and fall together, that point traces a closed loop — a cycle — over and over, with the bright dot marking where the system is right now. A wider loop means bigger swings between boom and bust; a small loop close to the center means both populations are hovering near their natural balance.