Beyond May: Complexity-stability relationships in disordered dynamical systems

Robert May famously used random matrix theory to predict that large, complex systems cannot admit stable fixed points. However, this general conclusion is not supported by empirical observation: from cells to biomes, biological systems are large, complex -- and, by and large, stable. In this paper, we revisit May's argument in the light of recent developments in both ecology and random matrix theory. Using a non-linear generalization of the Lotka-Volterra model, we show that there are in fact two kinds of complexity-stability relationships in disordered dynamical systems: if self-interactions grow faster with density than cross-interactions, complexity is destabilizing; but if cross-interactions grow faster than self-interactions, complexity is stabilizing. Our result shows that May's principle that "complexity begets instability" is not a general property of complex systems; instead, it is a property of a subclass of weakly cross-regulated disordered systems.