  It is well-established that including spatial structure and stochastic noise
in models for predator-prey interactions invalidates the classical
deterministic Lotka-Volterra picture of neutral population cycles. In contrast,
stochastic models yield long-lived, but ultimately decaying erratic population
oscillations, which can be understood through a resonant amplification
mechanism for density fluctuations. In Monte Carlo simulations of spatial
stochastic predator-prey systems, one observes striking complex spatio-temporal
structures. These spreading activity fronts induce persistent correlations
between predators and prey. In the presence of local particle density
restrictions (finite prey carrying capacity), there exists an extinction
threshold for the predator population. The accompanying continuous
non-equilibrium phase transition is governed by the directed-percolation
universality class. We employ field-theoretic methods based on the Doi-Peliti
representation of the master equation for stochastic particle interaction
models to (i) map the ensuing action in the vicinity of the absorbing state
phase transition to Reggeon field theory, and (ii) to quantitatively address
fluctuation-induced renormalizations of the population oscillation frequency,
damping, and diffusion coefficients in the species coexistence phase.

Populations and Evolution;Statistical Mechanics;Biological Physics