  We employ Monte Carlo simulations to numerically study the temporal evolution
and transient oscillations of the population densities, the associated
frequency power spectra, and the spatial correlation functions in the
(quasi-)steady state in two-dimensional stochastic May--Leonard models of
mobile individuals, allowing for particle exchanges with nearest-neighbors and
hopping onto empty sites. We therefore consider a class of four-state
three-species cyclic predator-prey models whose total particle number is not
conserved. We demonstrate that quenched disorder in either the reaction or in
the mobility rates hardly impacts the dynamical evolution, the emergence and
structure of spiral patterns, or the mean extinction time in this system. We
also show that direct particle pair exchange processes promote the formation of
regular spiral structures. Moreover, upon increasing the rates of mobility, we
observe a remarkable change in the extinction properties in the May--Leonard
system (for small system sizes): (1) As the mobility rate exceeds a threshold
that separates a species coexistence (quasi-)steady state from an absorbing
state, the mean extinction time as function of system size N crosses over from
a functional form ~ e^{cN} / N (where c is a constant) to a linear dependence;
(2) the measured histogram of extinction times displays a corresponding
crossover from an (approximately) exponential to a Gaussian distribution. The
latter results are found to hold true also when the mobility rates are randomly
distributed.
