Self-organizing patterns in an evolutionary rock-paper-scissors game for stochastic synchronized strategy updates

We study a spatial evolutionary rock-paper-scissors game with synchronized strategy updating. Players gain their payoff from games with their four neighbors on a square lattice and can update their strategies simultaneously according to the logit rule, which is the noisy version of the best-response dynamics. For the synchronized strategy update two types of global oscillations (with an ordered strategy arrangement and periods of three and six generations) can occur in this system in the zero noise limit. At low noise values, all nine oscillating phases are present in the system by forming a self-organizing spatial pattern due to the comprising invasion and speciation processes along the interfaces separating the different domains.

Figure 1

Typical snapshot of the spatial distribution of strategies in a block of $90\times 90$ sites on a square lattice at $K=0.4$. Black stands for the $\mathbf{R}$ strategy, red (dark gray) for the $\mathbf{P}$ strategy, and green (light gray) for the $\mathbf{S}$ strategy.

Figure 2

Flow diagram for the cyclic behavior of domains. Solid (black) lines show how the domain composition changes in every generation due to the logit strategy adoption rule. The color code is the same as in Fig. 1. Domain types are denoted by the appropriate strategy names. Note the different names for the chessboard and antichessboard structures. Pure domains return to the same state in three generations; chessboardlike associations accomplish this in six generations. Interestingly, the nine available strategy associations form two disjoint cycles instead of a full cycle.

Figure 3

Flow diagram for the invasion and speciation processes between the different strategy associations (domains). Arrows on the solid (black) lines show the winning association when the two associations connected by the black line meet each other. Dashed (blue) lines display the cases when the encounter of two associations results in the appearance of a third species. The arrow points toward the newly established association. The meaning of the strategy colors is the same as in Fig. 2.

Figure 4

Propagation of a steplike irregularity at the boundary of two domains. The first step of the invasion (indicated by a star) happens with a lower noise-dependent probability. The propagation of the irregularity displayed on the subsequent panels is, however, the result of the most probable strategy adoptions. The six-generation-long cyclic behavior of the domains can be observed as well.

Figure 5

Snapshots showing the course of a speciation event at $K=0.3$. The new species $\mathbf{SR}$ emerges at the boundary of the $\mathbf{RR}$ and $\mathbf{SP}$ domains.

Figure 6

Autocorrelation function versus distance of even integers for three different noise levels. Dashed line shows the zero level. The system size used in the simulations was $400\times 400$.