Evolutionary games and spatial periodicity

Spatial interactions are considered an important factor influencing a variety of evolutionary processes that take place in structured populations. It still remains an open problem to fully understand evolutionary game dynamics on networks except for certain limiting scenarios such as weak selection. Here we study the evolutionary dynamics of spatial games under strong selection where strategy evolution of individuals becomes deterministic in a fashion of winners taking all. We show that the long term behavior of the evolutionary process eventually converges to a particular basin of attraction, which is either a periodic cycle or a single fixed state depending on specific initial conditions and model parameters. In particular, we find that symmetric starting configurations can induce an exceedingly long transient phase encompassing a large number of aesthetic spatial patterns including the prominent kaleidoscopic cooperation. Our finding holds for any population structure and a broad class of finite games beyond the Prisoner’s Dilemma. Our work offers insights into understanding evolutionary dynamics of spatially extended systems ubiquitous in biology and ecology.