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

## TL;DR

This paper develops a numerical implementation of dynamical mean field theory (DMFT) for disordered systems, specifically applying it to the Lotka-Volterra ecosystem model to analyze complex dynamics like chaos and aging.

## Contribution

The paper derives DMFT via the dynamical cavity method and introduces a versatile numerical solution approach applicable to various systems.

## Key findings

- Successfully captures chaotic and aging dynamics in the Lotka-Volterra model
- Provides a general numerical framework for DMFT in disordered systems
- Demonstrates the method's effectiveness through simulations

## Abstract

Dynamical mean field theory (DMFT) is a tool that allows 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.