Marginally Stable Equilibria in Critical Ecosystems

TL;DR

This paper investigates the stability of large ecosystems modeled by symmetric random interactions, revealing a regime of marginally stable multiple equilibria that resemble a critical spin-glass phase, with implications for understanding systems at the edge of stability.

Contribution

It demonstrates that ecosystems with strong, heterogeneous interactions have multiple marginally stable equilibria and connects this to spin-glass physics, extending May's stability bounds.

Findings

Multiple equilibria are marginally stable in large ecosystems.

A relation between diversity and species response is established.

The multiple equilibria regime is analogous to a critical spin-glass phase.

Abstract

In this work we study the stability of the equilibria reached by ecosystems formed by a large number of species. The model we focus on are Lotka-Volterra equations with symmetric random interactions. Our theoretical analysis, confirmed by our numerical studies, shows that for strong and heterogeneous interactions the system displays multiple equilibria which are all marginally stable. This property allows us to obtain general identities between diversity and single species responses, which generalize and saturate May's bound. By connecting the model to systems studied in condensed matter physics, we show that the multiple equilibria regime is analogous to a critical spin-glass phase. This relation provides a new perspective as to why many systems in several different fields appear to be poised at the edge of stability and also suggests new experimental ways to probe marginal stability.