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

• Published in: Population ecology Vol. 57; no. 1; pp. 63 - 75
• Main Authors: Allesina, Stefano, Tang, Si
• Format: Journal Article
• Language: English
• Published: Tokyo Springer Japan 01.01.2015; Blackwell Publishing Ltd
• Subjects: Behavioral Sciences; Biomedical and Life Sciences; Complexity; Ecology; Ecosystem studies; Ecosystems; Eigenvalue; Eigenvalues; Equilibrium; Evolutionary Biology; Food chains; Food web; Forestry; Inequality; Life Sciences; mathematics and statistics; Ordinary differential equations; Plant Sciences; Population; population ecology; Random matrix; Special Feature: Review; Stability; Zoology; Eigenvalue; Food web; Random matrix; Stability; Complexity
• ISSN: 1438-3896; 1438-390X
• DOI: 10.1007/s10144-014-0471-0

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.