The stability–complexity relationship at age 40: a random matrix perspective

TLDR

Since Robert May’s 1972 work, the local asymptotic stability of large ecological systems has been a central focus of theoretical ecology. The authors review May’s work through random matrix theory and aim to present the technical material accessibly while pointing to primary literature. They analyze May’s stability criterion using random matrix theory, which studies large matrices with randomly sampled coefficients. They derive May’s stability criterion via random matrix theory, extend the circular law to better understand ecological systems, and outline challenges whose resolution will improve predictions of large ecological network stability.

Abstract

Abstract Since the work of Robert May in 1972, the local asymptotic stability of large ecological systems has been a focus of theoretical ecology. Here we review May's work in the light of random matrix theory, the field of mathematics devoted to the study of large matrices whose coefficients are randomly sampled from distributions with given characteristics. We show how May's celebrated “stability criterion” can be derived using random matrix theory, and how extensions of the so‐called circular law for the limiting distribution of the eigenvalues of large random matrix can further our understanding of ecological systems. Our goal is to present the more technical material in an accessible way, and to provide pointers to the primary mathematical literature on this subject. We conclude by enumerating a number of challenges, whose solution is going to greatly improve our ability to predict the stability of large ecological networks.