Fluctuation spectra of large random dynamical systems reveal hidden structure in ecological networks

Yvonne Krumbeck,Tim Rogers,Qian Yang,George W A Constable

doi:10.1038/s41467-021-23757-x

Yvonne Krumbeck, Tim Rogers + Show 2 more

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https://doi.org/10.1038/s41467-021-23757-x

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Abstract

Understanding the relationship between complexity and stability in large dynamical systems—such as ecosystems—remains a key open question in complexity theory which has inspired a rich body of work developed over more than fifty years. The vast majority of this theory addresses asymptotic linear stability around equilibrium points, but the idea of ‘stability’ in fact has other uses in the empirical ecological literature. The important notion of ‘temporal stability’ describes the character of fluctuations in population dynamics, driven by intrinsic or extrinsic noise. Here we apply tools from random matrix theory to the problem of temporal stability, deriving analytical predictions for the fluctuation spectra of complex ecological networks. We show that different network structures leave distinct signatures in the spectrum of fluctuations, and demonstrate the application of our theory to the analysis of ecological time-series data of plankton abundances.

Highlights

Understanding the relationship between complexity and stability in large dynamical systems —such as ecosystems—remains a key open question in complexity theory which has inspired a rich body of work developed over more than fifty years
Our approach enables the computation of the power spectral density of fluctuations in large random systems of a very general class; a full and detailed derivation is given in the Methods
In the case of ecosystem stability, the dynamical system in question is that describing the interactions of different species

Summary

Introduction

Understanding the relationship between complexity and stability in large dynamical systems —such as ecosystems—remains a key open question in complexity theory which has inspired a rich body of work developed over more than fifty years. The notion of stability referred to by May and these later works is that of asymptotic linear stability of an equilibrium point While this definition is a natural mathematical choice, it belies the rich array of interpretations of ‘ecological stability’ present in the empirical ecological literature[11]. Recently Arnoldi et al.[22] employed the term ‘variability’ to describe the inverse of temporal stability in a random community matrix model. In that work, they consider the scale of response to persistent external (environmental) noise applied to an ecosystem

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