Large system population dynamics with non-Gaussian interactions

TL;DR

This paper extends the dynamical mean field theory for ecological models to include non-Gaussian interactions, revealing that species abundance distributions depend on all cumulants of interactions, thus breaking universality and enabling inference of microscopic details from macroscopic data.

Contribution

It introduces a generalized mean field theory for non-Gaussian interactions in ecological models, showing the dependence on all cumulants and enabling microscopic inference from macroscopic distributions.

Findings

Mean field equations depend on all cumulants of interactions.

Breakdown of universality in species abundance distributions.

Analytical relationship between interaction and population distributions.

Abstract

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.