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

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.

arXiv commentUpdated version accepted for publication in Europhysics Letters. New titel, additional References and expanded supplementary material. 5 pages, 3 figures

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Cited by in corpus (4)

• Constrained optimization as ecological dynamics with applications to random quadratic programming in high dimensions
• A new spin on optimal portfolios and ecological equilibria
• On non-negative solutions to large systems of random linear equations
• Gradient descent dynamics and the jamming transition in infinite dimensions