When predators and prey occupy space, their oscillations become traveling waves
Lotka-Volterra treats predators and prey as two numbers, and the populations oscillate forever. Real ecosystems are not well-mixed beakers. Give the species a landscape and the uniform oscillations break into traveling waves of pursuit and escape.
Prey bloom in one region; predators converge, eat through the local prey, and start to starve. Meanwhile distant prey patches multiply undisturbed. The cycle sweeps across the landscape as waves, spirals, or turbulence, depending on parameters.
The model below places predators and prey on a 2D grid. Each cell holds a prey density and a predator density. Three things happen at each cell on every time step:
From a random initial state, spatial waves emerge after a short transient. The pattern falls out of local growth, predation, and diffusion alone.
Click the canvas to drop a prey bloom and watch predators converge on it.
Figure 1. A 2D predator-prey landscape. Prey (green) grow and diffuse. Predators (red) chase, eat, and diffuse. Watch spatial waves of pursuit and escape form from random initial conditions. Click to inject a local prey bloom and see a chase-front form.
Early on the grid is noisy. Within a few dozen generations, patches form; within a few hundred, wavefronts sweep across the domain. Prey (green) leads, predators (red) follow, depleted zones (dark) trail.
With the right balance, the pursuit waves curl into rotating spirals — the same mechanism as in the Belousov-Zhabotinsky reaction: a broken wavefront has a free end that pivots and curls back on itself.
Raise the prey growth rate and the system tips into chaotic turbulence. Lower diffusion and the oscillations localize, approaching the well-mixed limit where everything pulses in unison.
Figure 2. Tune the parameters to see different spatial regimes. "Spirals" produces stable rotating spiral arms. "Chaos" creates turbulent, constantly breaking wavefronts. "Uniform" synchronizes the entire grid into a single global oscillation.
In the spiral regime, populations never go globally extinct: at any moment, some region is in a prey boom while another is in a predator boom. The spatial desynchronization stabilizes coexistence.
Real landscapes have rivers, ridgelines, fences. Walls block diffusion entirely.
A wall can shelter prey while predators on the other side starve. The sheltered pocket becomes a refugium, and prey spill out from it to recolonize depleted areas — a steady food source that sustains the broader ecosystem.
Click and drag to draw barriers (or erase them). Walls block diffusion and movement.
Figure 3. Draw barriers to reshape the landscape, or switch to Erase to carve openings. Notice how prey refugia form in sheltered pockets. The "Maze" preset creates a labyrinth that fragments the population into semi-isolated patches.
Spatial models explain why prey species persist when non-spatial Lotka-Volterra predicts extinction. Refugia, asynchronous oscillations, and recolonization stabilize coexistence — the same logic behind habitat corridors and reserves in conservation biology.