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The classical game of rock-paper-scissors have inspired experiments andspatial model systems that address robustness of biological diversity. Inparticular the game nicely illustrates that cyclic interactions allow multiplestrategies to coexist for long time intervals. When formulated in terms of aone-dimensional cellular automata, the spatial distribution of strategiesexhibits coarsening with algebraically growing domain size over time, while thetwo-dimensional version allows domains to break and thereby opens for long-timecoexistence. We here consider a quasi-one-dimensional implementation of thecyclic competition, and study the long-term dynamics as a function of rareinvasions between parallel linear ecosystems. We find that increasing thecomplexity from two to three parallel subsystems allows a transition fromcomplete coarsening to an active steady state where the domain size staysfinite. We further find that this transition happens irrespective of whetherthe update is done in parallel for all sites simultaneously, or done randomlyin sequential order. In both cases the active state is characterized bylocalized bursts of dislocations, followed by longer periods of coarsening. Inthe case of the parallel dynamics, we find that there is another phasetransition between the active steady state and the coarsening state within thethree-line system when the invasion rate between the subsystems is varied. Weidentify the critical parameter for this transition, and show that the densityof active boundaries have critical exponents that are consistent with thedirected percolation universality class. On the other hand, numericalsimulations with the random sequential dynamics suggest that the system mayexhibit an active steady state as long as the invasion rate is finite.

Three is much more than two in coarsening dynamics of cyclic competitions