Dynamical systems on large networks with predator-prey interactions are stable and exhibit oscillations

TL;DR

This paper develops an exact spectral theory for large, sparse networks with predator-prey interactions, revealing conditions for stability and oscillations, and highlighting the unique stability of antagonistic systems in large networks.

Contribution

It introduces a precise spectral analysis of Jacobian matrices on random graphs, showing predator-prey systems can be stable and oscillatory, unlike generic systems.

Findings

Large systems with unbounded degree distributions are generally unstable.

Predator-prey interactions can stabilize large networks, allowing for stability in the infinite size limit.

Antagonistic systems exhibit oscillations and a phase transition related to mean degree.

Abstract

We analyse the stability of linear dynamical systems defined on sparse, random graphs with predator-prey, competitive, and mutualistic interactions. These systems are aimed at modelling the stability of fixed points in large systems defined on complex networks, such as, ecosystems consisting of a large number of species that interact through a food-web. We develop an exact theory for the spectral distribution and the leading eigenvalue of the corresponding sparse Jacobian matrices. This theory reveals that the nature of local interactions have a strong influence on system's stability. We show that, in general, linear dynamical systems defined on random graphs with a prescribed degree distribution of unbounded support are unstable if they are large enough, implying a tradeoff between stability and diversity. Remarkably, in contrast to the generic case, antagonistic systems that only contain interactions of the predator-prey type can be stable in the infinite size limit. This qualitatively feature for antagonistic systems is accompanied by a peculiar oscillatory behaviour of the dynamical response of the system after a perturbation, when the mean degree of the graph is small enough. Moreover, for antagonistic systems we also find that there exist a dynamical phase transition and critical mean degree above which the response becomes non-oscillatory.