  Noise and spatial degrees of freedom characterize most ecosystems. Some
aspects of their influence on the coevolution of populations with cyclic
interspecies competition have been demonstrated in recent experiments [e.g. B.
Kerr et al., Nature {\bf 418}, 171 (2002)]. To reach a better theoretical
understanding of these phenomena, we consider a paradigmatic spatial model
where three species exhibit cyclic dominance. Using an individual-based
description, as well as stochastic partial differential and deterministic
reaction-diffusion equations, we account for stochastic fluctuations and
spatial diffusion at different levels, and show how fascinating patterns of
entangled spirals emerge. We rationalize our analysis by computing the
spatio-temporal correlation functions and provide analytical expressions for
the front velocity and the wavelength of the propagating spiral waves.
