  Field theory tools are applied to analytically study fluctuation and
correlation effects in spatially extended stochastic predator-prey systems. In
the mean-field rate equation approximation, the classic Lotka-Volterra model is
characterized by neutral cycles in phase space, describing undamped
oscillations for both predator and prey populations. In contrast, Monte Carlo
simulations for stochastic two-species predator-prey reaction systems on
regular lattices display complex spatio-temporal structures associated with
persistent erratic population oscillations. The Doi-Peliti path integral
representation of the master equation for stochastic particle interaction
models is utilized to arrive at a field theory action for spatial
Lotka-Volterra models in the continuum limit. In the species coexistence phase,
a perturbation expansion with respect to the nonlinear predation rate is
employed to demonstrate that spatial degrees of freedom and stochastic noise
induce instabilities toward structure formation, and to compute the fluctuation
corrections for the oscillation frequency and diffusion coefficient. The
drastic downward renormalization of the frequency and the enhanced diffusivity
are in excellent qualitative agreement with Monte Carlo simulation data.
