Predator-Prey Quasi-cycles from a Path Integral Formalism

The existence of beyond mean field quasi-cycle oscillations in a simple spatial model of predator prey interactions is derived from a path integral formalism. The results agree substantially with those obtained from analysis of similar models using system size expansions of the master equation. In all of these analyses, the discrete nature of predator prey populations and finite size effects lead to persistent oscillations in time, but spatial patterns fail to form. The path integral formalism goes beyond mean field theory and provides a focus on individual realizations of the stochastic time evolution of population not captured in the standard master equation approach.