  We investigate the impact of cyclic competition on pattern formation in the
rock-paper-scissors game. By separately considering random and prepared initial
conditions, we observe a critical influence of the competition rate $p$ on the
stability of spiral waves and on the emergence of biodiversity. In particular,
while increasing values of $p$ promote biodiversity, they may act detrimental
on spatial pattern formation. For random initial conditions, we observe a phase
transition from biodiversity to an absorbing phase, whereby the critical value
of mobility grows linearly with increasing values of $p$ on a log-log scale,
but then saturates as $p$ becomes large. For prepared initial conditions, we
observe the formation of single-armed spirals, but only for values of $p$ that
are below a critical value. Once above, the spirals break up and form
disordered spatial structures, mainly because of the percolation of vacant
sites. Thus, there exists a critical value of the competition rate $p_{c}$ for
stable single-armed spirals in finite populations. Importantly though, $p_{c}$
increases with increasing system size, because noise reinforces the
disintegration of ordered patterns. In addition, we also find that $p_{c}$
increases with the mobility. These phenomena are reproduced by a deterministic
model that is based on nonlinear partial differential equations. Our findings
indicate that competition is vital for the sustenance of biodiversity and
emergence of pattern formation in ecosystems governed by cyclical interactions.
