Slow-fast dynamics in stochastic Lotka-Volterra systems

TL;DR

This paper studies the long-term behavior of stochastic Lotka-Volterra systems with cyclic interactions, showing how rapid oscillations average out to yield a simplified stochastic differential equation describing the population dynamics.

Contribution

It introduces a novel averaging approach for stochastic Lotka-Volterra models with cyclic states, linking short-term Hamiltonian dynamics to long-term stochastic behavior.

Findings

Large population limit leads to explicit SDEs

Stationary measures converge under averaging

Short-term dynamics are approximated by Hamiltonian systems

Abstract

We investigate the large population dynamics of a family of stochastic particle systems with three-state cyclic individual behaviour and parameter-dependent transition rates. On short time scales, the dynamics turns out to be approximated by an integrable Hamiltonian system whose phase space is foliated by periodic trajectories. This feature suggests to consider the effective dynamics of the long-term process that results from averaging over the rapid oscillations. We establish the convergence of this process in the large population limit to the solutions of an explicit stochastic differential equation. Remarkably, this averaging phenomenon is complemented by the convergence of stationary measures. The proof of averaging follows the Stroock-Varadhan approach to martingale problems and relies on a fine analysis of the system's dynamical features.