  Cyclic (rock-paper-scissors-type) population models serve to mimic complex
species interactions. Focusing on a paradigmatic three-species model with
mutations in one dimension, we observe an interplay between equilibrium and
non-equilibrium processes in the stationary state. We exploit these insights to
obtain asymptotically exact descriptions of the emerging reactive steady state
in the regimes of high and low mutation rates. The results are compared to
stochastic lattice simulations. Our methods and findings are potentially
relevant for the spatio-temporal evolution of other non-equilibrium stochastic
processes.
