  We investigate the Generalized Lotka-Volterra (GLV) equations, a central
model in theoretical ecology, where species interactions are assumed to be
fixed over time and heterogeneous (quenched noise). Recent studies have
suggested that the stability properties and abundance distributions of large
disordered GLV systems depend, in the simplest scenario, solely on the mean and
variance of the distribution of species interactions. However, empirical
communities deviate from this level of universality. In this article, we
present a generalized version of the dynamical mean field theory for
non-Gaussian interactions that can be applied to various models, including the
GLV equations. Our results show that the generalized mean field equations have
solutions which depend on all cumulants of the distribution of species
interactions, leading to a breakdown of universality. We leverage on this
informative breakdown to extract microscopic interaction details from the
macroscopic distribution of densities which are in agreement with empirical
data. Specifically, in the case of sparse interactions, which we analytically
investigate, we establish a simple relationship between the distribution of
interactions and the distribution of species population densities.
