Feasibility and Single Parameter Scaling of Extinctions in Large Ecological Communities

TL;DR

This paper uses random matrix theory to analyze conditions for species coexistence and extinctions in large ecological communities, revealing that feasibility breaks down before stability as species number grows.

Contribution

It derives an analytical expression for extinction probabilities and proposes a single-parameter scaling law for extinctions in generalized Lotka-Volterra models.

Findings

Feasibility is broken before stability in large communities.

Species abundances are Gaussian at equilibrium in the weakly interacting regime.

A conjectured single-parameter scaling law governs species extinctions.

Abstract

Multispecies ecosystems modelled by generalized Lotka-Volterra equations exhibit stationary population abundances, where large number of species often coexist. Understanding the precise conditions under which this is at all feasible and what triggers species extinctions is a key, outstanding problem in theoretical ecology. Using standard methods of random matrix theory, I show that distributions of species abundances are Gaussian at equilibrium, in the weakly interacting regime. One consequence is that feasibility is generically broken before stability, for large enough number of species. I further derive an analytical expression for the probability that $n=0,1,2,...$ species go extinct and conjecture that a single-parameter scaling law governs species extinctions. These results are corroborated by numerical simulations in a wide range of system parameters.