Population Dynamics
Cyclic Dominance and Spiral Waves
In the Spatial Rock-Paper-Scissors model, three competing strategies are arranged on a 2D grid. Each strategy beats one and loses to another in a cyclic fashion: Rock beats Scissors, Scissors beats Paper, Paper beats Rock. When cells can invade beaten neighbors, the system spontaneously generates mesmerizing spiral wave patterns from random initial conditions.
Cyclic Dominance
Unlike competitive exclusion where one species drives others to extinction, cyclic dominance maintains biodiversity indefinitely. No single strategy can dominate because it is always vulnerable to another. This R > S > P > R cycle creates a perpetual chase across the spatial grid.
Spiral Wave Formation
The invasion process creates traveling wavefronts where each strategy chases the one it beats. Where wavefronts meet at angles, they curl into spirals. These spirals are self-sustaining: the tip rotates continuously, generating new arms. Similar spiral waves appear in chemical reactions (Belousov-Zhabotinsky) and cardiac tissue.
May-Leonard Model
The underlying mathematics follows the May-Leonard replicator equation. In the well-mixed (mean-field) case, trajectories spiral inward toward the coexistence fixed point. On a spatial lattice, local interactions break the mean-field assumption, stabilizing the spiral waves and preserving coexistence far longer than panmictic models predict.
Real-World Applications
Cyclic dominance appears in nature: side-blotched lizards (Uta stansburiana) have three male morphs (orange, blue, yellow) following RPS dynamics; marine sessile organisms compete cyclically for substrate; E. coli populations with toxin-antitoxin genes exhibit rock-paper-scissors interactions. In economics, multi-party market competition and technology adoption cycles show analogous patterns.
Role of Mutation
Small mutation rates introduce stochastic noise that prevents the system from settling into static domains. Mutation seeds new strategy clusters that can nucleate fresh spiral waves. Too much mutation destroys spatial structure entirely, producing a well-mixed noisy state. The balance between selection (invasion) and mutation controls pattern richness.
Biodiversity Maintenance
Spatial structure is a powerful mechanism for maintaining biodiversity. Without space, cyclic RPS games converge to a neutral equilibrium that is easily destabilized. On a lattice, local interactions create protective clusters that shield strategies from extinction. This insight helps explain why biodiversity is often higher in spatially structured habitats.