Fundamental clusters in spatial 2×2 games

The notion of fundamental clusters is introduced, serving as a rule of thumb to characterize the statistical properties of the complex behaviour of cellular automata such as spatial 2×2 games. They represent the smallest cluster size determining the fate of the entire system. Checking simple growth criteria allows us to decide whether the cluster–individuals, e.g. some mutant family, are capable of surviving and invading a resident population. In biology, spatial 2×2 games have a broad spectrum of applications ranging from the evolution of cooperation and intraspecies competition to disease spread. This methodological study allows simple classifications and long–term predictions in various biological and social models to be made. For minimal neighbourhood types, we show that the intuitive candidate, a 3×3 cluster, turns out to be fundamental with certain weak limitations for the Moore neighbourhood but not for the Von Neumann neighbourhood. However, in the latter case, 2×2 clusters generally serve as reliable indicators to whether a strategy survives. Stochasticity is added to investigate the effects of varying fractions of one strategy present at initialization time and to discuss the rich dynamic properties in greater detail. Finally, we derive Liapunov exponents for the system and show that chaos reigns in a small region where the two strategies coexist in dynamical equilibrium.