  We apply a perturbative Doi--Peliti field-theoretical analysis to the
stochastic spatially extended symmetric Rock-Paper-Scissors (RPS) and
May--Leonard (ML) models, in which three species compete cyclically. Compared
to the two-species Lotka--Volterra predator-prey (LV) model, according to
numerical simulations, these cyclical models appear to be less affected by
intrinsic stochastic fluctuations. Indeed, we demonstrate that the qualitative
features of the ML model are insensitive to intrinsic reaction noise. In
contrast, and although not yet observed in numerical simulations, we find that
the RPS model acquires significant fluctuation-induced renormalizations in the
perturbative regime, similar to the LV model. We also study the formation of
spatio-temporal structures in the framework of stability analysis and provide a
clearcut explanation for the absence of spatial patterns in the RPS system,
whereas the spontaneous emergence of spatio-temporal structures features
prominently in the LV and the ML models.
