Equilibria of large random Lotka-Volterra systems with vanishing species: a mathematical approach

TL;DR

This paper rigorously analyzes the statistical properties of equilibrium states in large random Lotka-Volterra ecosystems using an Approximate Message Passing algorithm, focusing on models with Gaussian and Wishart interaction matrices.

Contribution

It introduces an AMP-based method to characterize equilibria in high-dimensional random ecological models, extending to complex interaction matrices.

Findings

Describes statistical properties of equilibria in large ecosystems.

Develops an AMP algorithm for analyzing equilibrium distributions.

Applicable to Gaussian and Wishart interaction matrices.

Abstract

Ecosystems with a large number of species are often modelled as Lotka-Volterra dynamical systems built around a large random interaction matrix. Under some known conditions, a global equilibrium exists and is unique. In this article, we rigorously study its statistical properties in the large dimensional regime. Such an equilibrium vector is known to be the solution of a so-called Linear Complementarity Problem (LCP). We describe its statistical properties by designing an Approximate Message Passing (AMP) algorithm, a technique that has recently aroused an intense research effort in the fields of statistical physics, Machine Learning, or communication theory. Interaction matrices taken from the Gaussian Orthogonal Ensemble, or following a Wishart distribution are considered. Beyond these models, the AMP approach developed in this article has the potential to describe the statistical properties of equilibria associated to more involved interaction matrix models.