Conditional Random Matrix Ensembles and the Stability of Dynamical Systems

There has been a long-standing and at times fractious debate whether complex and large systems can be stable. In ecology, the so-called `diversity-stability debate arose because mathematical analyses of ecosystem stability were either specific to a particular model (leading to results that were not general), or chosen for mathematical convenience, yielding results unlikely to be meaningful for any interesting realistic system. Mays work, and its subsequent elaborations, relied upon results from random matrix theory, particularly the circular law and its extensions, which only apply when the strengths of interactions between entities in the system are assumed to be independent and identically distributed (i.i.d.). Other studies have optimistically generalised from the analysis of very specific systems, in a way that does not hold up to closer scrutiny. We show here that this debate can be put to rest, once these two contrasting views have been reconciled --- which is possible in the statistical framework developed here. Here we use a range of illustrative examples of dynamical systems to demonstrate that (i) stability probability cannot be summarily deduced from any single property of the system (e.g. its diversity), and (ii) our assessment of stability depends on adequately capturing the details of the systems analysed. Failing to condition on the structure of dynamical systems will skew our analysis and can, even for very small systems, result in an unnecessarily pessimistic diagnosis of their stability.