  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
ecological timeseries data of plankton abundances.
