Cyclic dominance

Setup

Three strategies, each beating one of the others and losing to the third. Dominance runs in a loop, so no ordering of the three exists and no single strategy is the right answer. In the population version the three types compete for space: a type kills the one it beats, the vacated space is recolonised, and individuals move around.

What makes this worth simulating is that the payoffs alone do not determine the outcome. The same three types with the same competitive relationships either coexist forever or collapse to one survivor, depending entirely on how far individuals move between interactions.

The model

In a well-mixed population, which is the limit of unbounded mobility, the deterministic replicator equations give closed orbits around the interior fixed point. Any finite population then drifts along those orbits until it touches a boundary, and the touch is absorbing.

Results

In a well-mixed population the interior fixed point is neutrally stable at best, so demographic noise carries the system outward until a type hits zero. Diversity is lost, and which type survives is arbitrary.

Add spatial structure and the picture inverts. Local interaction produces travelling waves and rotating spirals, and a type being wiped out in one region is invading in another. Reichenbach, Mobilia and Frey (2007) showed the boundary between the two regimes is sharp rather than gradual: there is a critical mobility above which the spiral wavelength exceeds the system size, the patterns merge, and coexistence fails. The transition is a genuine critical point.

Frean and Abraham (2001) added a result that is easy to state and hard to accept: strengthening one type’s competitive ability lowers its own equilibrium abundance and raises the abundance of the type it consumes. In a cycle, being better at winning does not translate into doing well.

Evidence

Sinervo and Lively (1996) documented the cycle in male side-blotched lizards (Uta stansburiana) in the inner Coast Range of California across six years of field observation. Males carry one of three throat colours with three mating strategies. Orange-throated males are aggressive and hold large territories, overrunning the smaller territories of blue-throated males. Blue males guard a single female closely, which defeats the yellow-throated sneakers who mimic females. Yellow sneakers exploit the large, thinly defended territories of orange males. Morph frequencies oscillated with roughly a six-year period, in the predicted order.

Kerr, Riley, Feldman and Bohannan (2002) built the bacterial version using three strains of Escherichia coli: a colicin-producing strain that kills the sensitive strain, a resistant strain that outgrows the producer, and the sensitive strain that outgrows the resistant one. In a flask, where everything mixes, diversity collapsed within a few transfers. On a static agar plate, where colonies interact only with neighbours, all three persisted. Same organisms, same competitive relationships, opposite outcome, with spatial structure the only difference. That is the experiment the simulator above reproduces.

Kirkup and Riley (2004) then showed the same three-way antagonism maintaining coexistence in a living host, in the mouse colon.

Limitations

The lattice model is a caricature of space. Real populations have habitat heterogeneity, non-uniform dispersal kernels and demographic structure, none of which appear here, and the specific value of the critical mobility is a property of this particular model rather than a constant of nature.

The lizard result is the most cited example, and the original paper covered six years at one site. It has since been followed by mark-recapture work at the same system running from the 1990s into the 2020s, so the cycle is much better attested than a single study would be. What that longer record also shows is that the three-strategy game is not the whole story: female morphs follow their own density-dependent strategies, and population density and year-to-year conditions modulate the male dynamics in ways the simple asexual model Sinervo and Lively wrote does not contain.

The bacterial work is cleaner in a different sense, because the competitive relationships were engineered rather than inferred, but it buys that cleanliness by studying a system built to order. Read cyclic dominance as well demonstrated in principle, decisively demonstrated in bacteria, and well supported but not exhaustive as an account of the lizards.

Related This model shares its mechanism with The war of attrition (Paper 05) and Colonel Blotto (Paper 06) and Professionals play minimax (Paper 08). In each of them, no fixed choice is stable, so the solution is a distribution or a cycle rather than a move.

References

1. Frean, M., & Abraham, E. R. (2001). Rock–scissors–paper and the survival of the weakest. Proceedings of the Royal Society B: Biological Sciences, 268(1474), 1323–1327. https://doi.org/10.1098/rspb.2001.1670
2. 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(6894), 171–174. https://doi.org/10.1038/nature00823
3. Kirkup, B. C., & Riley, M. A. (2004). Antibiotic-mediated antagonism leads to a bacterial game of rock–paper–scissors in vivo. Nature, 428(6981), 412–414. https://doi.org/10.1038/nature02429
4. Maynard Smith, J. (1982). Evolution and the theory of games. Cambridge University Press.
5. Reichenbach, T., Mobilia, M., & Frey, E. (2007). Mobility promotes and jeopardizes biodiversity in rock–paper–scissors games. Nature, 448(7157), 1046–1049. https://doi.org/10.1038/nature06095
6. Sinervo, B., & Lively, C. M. (1996). The rock–paper–scissors game and the evolution of alternative male strategies. Nature, 380(6571), 240–243. https://doi.org/10.1038/380240a0

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