Seeding the Spatial Prisoner’s Dilemma with Ulam’s Spiral

Figure 2

Cooperation emerges, following collapse, from a small number of clusters. Figure 2 displays snapshots of the simulation across (a) Generation = 1, (b) Generation = 2, (c) Generation = 3, and (d) Generation = 4 when and . From an initial distribution at the prime-numbered values of Ulam’s spiral (panel (a), Generation = 1), cooperators’ share of the population collapses (panel (b), Generation = 2), stabilizes at 7 clusters (panel (c), Generation = 3), and those clusters begins to grow (panel (d), Generation = 4). This dynamic of collapse-then-growth, which Figure 2 shows in panels (a)–(d), appears across runs in which cooperators overtake the population, thus indicating that the configuration of cooperators at the locations where cooperators survive in Generations 2 and 3 are responsible for those instances in which cooperation evolves from an initial distribution of cooperators at the prime-number locations in Ulam’s spiral.