by Dirk Brockmann

This explorable of a pattern forming system is derived from a model that was designed to understand co-existance of cyclicly interacting species in a spatially extended model ecosystem. Despite its simplicity, it can generate a rich set of complex spatio-temporal patterns depending on the choice of parameters and initial conditions.

The Model:

The foundation of the model is a set of 3 species A, B and C that are distributed in space and locally interact in a cyclic way: When species A (red) encounters species B (green), A “eats” B and replicates. When B encounters C (blue), B wins and when C encounters A, C wins. Just like in the rock-paper-scissors game. This cyclic predation can be schematically represented by the reactions

[ A+B\rightarrow 2A,\quad B+C\rightarrow 2B, \quad C+A\rightarrow 2C ]

Every location $(x,y)$ in the simulation has a concentration of A,B, and C individuals. In addition to the cyclic interaction, individuals can move in space diffusively. Finally, all species compete symmetrically and self-regulate so the population does not grow indefinitely.

In the simulation the relative concentration of each type of species is encoded in rgb-color. When a location is brightly green, for example, this means that predominantly C exists at this location, etc. When a region is black, the concentration of all species is low, and white means that all species are present at a high concentration.

Try this:

When you start the simulation by pressing the play button, you may need to wait a little for a pattern to emerge. Eventually you should see a dynamic spiral wave pattern. Especially when predation and competition is high you have to wait a bit.

You can vary predation and competition and observe the impact on the pattern geometry.

When you change the parameters, it’s also a good idea to press the reset button (central button with left arrow) so the system starts again with a randomized initial condition.

Slow blobs can be observed when predation is maximal and competition is about 20% lower. When you decrease competition slowly to very low values the spirals become smaller and smaller and the pattern becomes flashy with white regions.


  • T, Reichenbach, M. Mobilia, E. Frey, Mobility promotes and jeopardizes biodiversity in rock–paper–scissors games, Nature, 448, 1046 (2007)

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