What you're seeing
A grid of cells, each occupied by one of three species — call them teal, amber, and orchid. They stand in a cycle of dominance, exactly like the children's game: teal beats amber, amber beats orchid, orchid beats teal. There is no strongest species; each one is both a predator and prey. At every microstep a cell looks at a random neighbor, and if that neighbor's species dominates it, the cell is invaded and converts.
Locally the rule is brutally simple and one-sided: a stronger neighbor always wins that encounter. But step back and the local invasions don't sum to a winner. They organize into rotating spiral waves — fronts of teal chasing amber chasing orchid chasing teal, curled around pivot points. Because each species is forever pursuing the one it beats while fleeing the one that beats it, the chase closes on itself and all three coexist indefinitely. No species is ever driven extinct. That coexistence is the emergent fact; no cell intends it.
The rule
The dynamics are about as bare as a spatial model gets:
- Cyclic dominance. With three species numbered 0, 1, 2, species i beats species (i+1) mod 3 and loses to (i+2) mod 3. The relation is non-transitive: there is no ordering from weakest to strongest.
- Local invasion. Pick a random cell and a random neighbor (von Neumann's four or Moore's eight). If the neighbor dominates the cell, the cell takes the neighbor's species. Many such microsteps happen per frame. The grid wraps at the edges (a torus).
Contrast this with non-cyclic competition. If one species simply beat both others, there would be no game: the strongest would sweep the board and the other two would vanish — a winner-take-all monoculture. Cyclic dominance removes the top of the food chain. Strength is relative and circular, so victory is never final.
The optional 5 species rule is rock-paper-scissors-lizard-Spock: each species beats the next two in the cycle and loses to the other two. It is still non-transitive, just finer-grained, and produces smaller, busier spirals.
Why it matters
Cyclic dominance is a general mechanism for coexistence — for how competition that would be winner-take-all can instead maintain diversity. The crucial ingredient is that the competition is spatial. In a well-mixed soup, where everyone interacts with everyone, a rock-paper-scissors system drifts: random fluctuations eventually knock one species out, and once one is gone the cycle collapses (with two species left, one simply beats the other). But on a lattice, a species can only fight its immediate neighbors. Local structure shelters the loser of any encounter elsewhere on the grid, the spirals keep the three species spatially separated, and the whole system becomes stable. Space turns a fragile cycle into a durable coexistence — at least while mixing stays limited. Reichenbach and colleagues showed there is a critical mobility: let the species wander and intermix too freely and the spirals grow larger than the arena itself, one species wins, and diversity collapses again. Coexistence lives in a window.
The spirals are the visible signature of this stabilization. They are the same kind of traveling wave seen in other excitable media — the Belousov–Zhabotinsky reaction, cardiac tissue — here driven by ecological invasion rather than chemistry.
In the wild
Unlike many toy models, cyclic dominance has been documented in real living systems:
- Side-blotched lizards (Uta stansburiana). Sinervo & Lively (1996) found that males come in three throat colors tied to three mating strategies — orange (aggressive, large territories), blue (mate-guarding), and yellow (sneaker mimics of females). Over years the common type is beaten by the next: orange out-competes blue, blue out-competes yellow, yellow out-competes orange. The frequencies oscillate in a rock-paper-scissors cycle.
- Colicin-producing E. coli. Kerr et al. (2002) built a real bacterial rock-paper-scissors from three strains: a toxin (colicin) producer, a resistant strain, and a sensitive strain, which cycle in dominance. The key result is spatial: grown on a static plate (local dispersal), all three strains coexisted; grown in a well-mixed flask, diversity collapsed. This is the experimental demonstration that spatial structure promotes coexistence.
- Sessile reef organisms. Competition among corals, algae, and sponges for space can be non-transitive (A overgrows B, B overgrows C, C overgrows A), which helps explain how many species share a crowded reef rather than one overgrowing all. The evidence here is more case-by-case than the two systems above, so treat it as suggestive rather than settled.
The lattice model on this page is a caricature of these systems, not a literal simulation of any one of them. Its value is to show that the qualitative outcome — coexistence through spatial, cyclic competition — falls out of almost nothing.
Try this
- Raise the speed to watch the spirals turn — the rotation becomes obvious once the fronts move quickly.
- Switch the rule to 5 species and watch the texture get finer and busier, with more, smaller spiral cores.
- Hit randomize and watch the ordered spirals re-form from pure noise within a few seconds — the pattern is an attractor, not a fluke of the seed.
- Toggle the neighborhood between Moore (8) and von Neumann (4) to see how the local interaction range changes the spiral grain.
References
- Sinervo, B. & Lively, C. M. (1996). "The rock–paper–scissors game and the evolution of alternative male strategies." Nature 380, 240–243.
- Kerr, B., Riley, M. A., Feldman, M. W., Bohannan, B. J. M. (2002). "Local dispersal promotes biodiversity in a real-life game of rock–paper–scissors." Nature 418, 171–174.
- Reichenbach, T., Mobilia, M., Frey, E. (2007). "Mobility promotes and jeopardizes biodiversity in rock–paper–scissors games." Nature 448, 1046–1049.