Systems of random linear equations and the phase transition in MacArthur's resource-competition model

TL;DR

This paper investigates the phase transition in high-dimensional resource-competition models, linking it to the existence of solutions to random linear equations, and employs statistical mechanics techniques to analyze this phenomenon.

Contribution

It reveals that the phase transition in MacArthur's model is a general feature related to solutions of random linear systems, extending previous findings.

Findings

Identifies the phase transition as a property of solutions to random linear equations.

Maps the problem to high-dimensional fractional volume analysis.

Uses statistical mechanics methods to characterize the transition.

Abstract

Complex ecosystems generally consist of a large number of different species utilizing a large number of different resources. Several of their features cannot be captured by models comprising just a few species and resources. Recently, Tikhonov and Monasson have shown that a high-dimensional version of MacArthur's resource competition model exhibits a phase transition from a 'vulnerable' to a 'shielded' phase in which the species collectively protect themselves against an inhomogeneous resource influx from the outside. Here we point out that this transition is more general and may be traced back to the existence of non-negative solutions to large systems of random linear equations. Employing Farkas' Lemma we map this problem to the properties of a fractional volume in high dimensions which we determine using methods from the statistical mechanics of disordered systems.