A functional approach to stochastic Lotka-Volterra equations

Abstract

In this work, a stochastic two species Lotka-Volterra model with additive noise is studied, using a field-theoretical formalism. This nonlinear model describes the evolution of predator and prey populations, which are affected by stochastic external environmental effects described by the noise. The functional approach applied to this model is the Martin-Siggia-Rose (MSR) formalism. This method describes the evolution of observables of a stochastic system as a stochastic field theory. This allows an analytical description of noise effects. First, stochastic systems and methods are studied in general, before turning to functional descriptions of these systems. After a derivation of the MSR formalism and an analytical and numerical study of the deterministic Lotka-Volterra model, the stochastic case is investigated. Simulations of this model for different noise realizations are performed, for which an ensemble average is computed. For large simulation time, the average stochastic trajectory shows an inward spiraling motion towards the fixed point, starting from the periodic solution of the deterministic system. This inward motion increases for larger noise amplitudes or larger simulation time, which is caused by the diffusion of the individual simulations due to noise effects. Using the MSR formalism, the general behavior of the populations under the influence of noise is studied. This is done by deriving expressions for the one- and two-point correlation functions of the fluctuations in terms of Feynman diagrams. These are calculated and compared with the simulated results. For small simulation time, the one-point functions and the resulting total average populations show similar behavior. For larger simulation time, the analytical results obtained with the MSR formalism grow exponentially compared to the simulated results, which causes large deviations in all correlation functions.