Population oscillations in spatial stochastic Lotka-Volterra models: A field-theoretic perturbational analysis

TL;DR

This paper uses field-theoretic methods to analyze how stochastic noise and spatial structure cause population oscillations and pattern formation in predator-prey models, extending classical mean-field results.

Contribution

It introduces a field theory approach to spatial stochastic Lotka-Volterra models and computes fluctuation corrections to oscillation frequency and diffusion, revealing noise-induced instabilities.

Findings

Fluctuations lower the oscillation frequency significantly.

Spatial noise enhances diffusion and structure formation.

Results agree qualitatively with Monte Carlo simulations.

Abstract

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.