Rock-Paper-Scissors Spatial Ecology — Spiral Waves

Cyclic Dominance and Spatial Biodiversity

Rock-Paper-Scissors (RPS) dynamics model cyclic predator-prey interactions where species A kills B, B kills C, and C kills A. In well-mixed populations, two species always go extinct. But on a spatial lattice, local neighborhood effects allow all three species to coexist indefinitely through self-organized spiral waves — a striking example of how space changes the rules of ecological competition.

The model (Reichenbach, Mobilia & Frey 2007, Nature) uses three rates: selection σ (predator kills prey), reproduction μ (empty site colonization), and mobility m (random swap with neighbor). When mobility is low, spirals form and coexistence is maintained. Above a critical mobility m_c, spirals break apart and two of three species go extinct. The spiral wavelength scales as λ ~ 1/σ. This mechanism is thought to explain biodiversity in microbial communities (e.g., Escherichia coli strains).