  We study several variants of the stochastic four-state rock-paper-scissors
game or, equivalently, cyclic three-species predator-prey models with conserved
total particle density, by means of Monte Carlo simulations on one- and
two-dimensional lattices. Specifically, we investigate the influence of spatial
variability of the reaction rates and site occupancy restrictions on the
transient oscillations of the species densities and on spatial correlation
functions in the quasi-stationary coexistence state. For small systems, we also
numerically determine the dependence of typical extinction times on the number
of lattice sites. In stark contrast with two-species stochastic Lotka-Volterra
systems, we find that for our three-species models with cyclic competition
quenched disorder in the reaction rates has very little effect on the dynamics
and the long-time properties of the coexistence state. Similarly, we observe
that site restriction only has a minor influence on the system's dynamical
properties. Our results therefore demonstrate that the features of the spatial
rock-paper-scissors system are remarkably robust with respect to model
variations, and stochastic fluctuations as well as spatial correlations play a
comparatively minor role.
