  If rock beats scissors and scissors beat paper, one might assume that rock
beats paper too. But this is not the case for intransitive relationships that
make up the famous rock-paper-scissors game. However, the sole presence of
paper might prevent rock from beating scissors, simply because paper beats
rock. This is the blueprint for the rock-paper-scissors game with protection
spillovers, which has recently been introduced as a new paradigm for
biodiversity in well-mixed microbial populations. Here we study the game in
structured populations, demonstrating that protection spillovers give rise to
spatial patterns that are impossible to observe in the classical
rock-paper-scissors game. We show that the spatiotemporal dynamics of the
system is determined by the density of stable vortices, which may ultimately
transform to frozen states, to propagating waves, or to target waves with
reversed propagation direction, depending further on the degree and type of
randomness in the interactions among the species. If vortices are rare, the
fixation to waves and complex oscillatory solutions is likelier. Moreover,
annealed randomness in interactions favors the emergence of target waves, while
quenched randomness favors collective synchronization. Our results demonstrate
that protection spillovers may fundamentally change the dynamics of cyclic
dominance in structured populations, and they outline the possibility of
programming pattern formation in microbial populations.
