  The classical game of rock-paper-scissors have inspired experiments and
spatial model systems that address robustness of biological diversity. In
particular the game nicely illustrates that cyclic interactions allow multiple
strategies to coexist for long time intervals. When formulated in terms of a
one-dimensional cellular automata, the spatial distribution of strategies
exhibits coarsening with algebraically growing domain size over time, while the
two-dimensional version allows domains to break and thereby opens for long-time
coexistence. We here consider a quasi-one-dimensional implementation of the
cyclic competition, and study the long-term dynamics as a function of rare
invasions between parallel linear ecosystems. We find that increasing the
complexity from two to three parallel subsystems allows a transition from
complete coarsening to an active steady state where the domain size stays
finite. We further find that this transition happens irrespective of whether
the update is done in parallel for all sites simultaneously, or done randomly
in sequential order. In both cases the active state is characterized by
localized bursts of dislocations, followed by longer periods of coarsening. In
the case of the parallel dynamics, we find that there is another phase
transition between the active steady state and the coarsening state within the
three-line system when the invasion rate between the subsystems is varied. We
identify the critical parameter for this transition, and show that the density
of active boundaries have critical exponents that are consistent with the
directed percolation universality class. On the other hand, numerical
simulations with the random sequential dynamics suggest that the system may
exhibit an active steady state as long as the invasion rate is finite.
