  Numerical studies of the May-Leonard model for cyclically competing species
exhibit spontaneous spatial structures in the form of spirals. It is desirable
to obtain a simple coarse-grained evolution equation describing spatio-temporal
pattern formation in such spatially extended stochastic population dynamics
models. Extending earlier work on the corresponding deterministic system, we
derive the complex Ginzburg-Landau equation as the effective representation of
the fully stochastic dynamics of this paradigmatic model for cyclic dominance
near its Hopf bifurcation, and for small fluctuations in the three-species
coexistence regime. The internal stochastic reaction noise is accounted for
through the Doi-Peliti coherent-state path integral formalism, and subsequent
mapping to three coupled non-linear Langevin equations. This analysis provides
constraints on the model parameters that allow time scale separation and in
consequence a further reduction to just two coarse-grained slow degrees of
freedom.
