Numerical implementation of dynamical mean field theory for disordered systems: application to the Lotka–Volterra model of ecosystems

Dynamical mean field theory (DMFT) is a tool that allows one to analyze the stochastic dynamics of N interacting degrees of freedom in terms of a self-consistent 1-body problem. In this work, focusing on models of ecosystems, we present the derivation of DMFT through the dynamical cavity method, and we develop a method for solving it numerically. Our numerical procedure can be applied to a large variety of systems for which DMFT holds. We implement and test it for the generalized random Lotka–Volterra model, and show that complex dynamical regimes characterized by chaos and aging can be captured and studied by this framework.