Generalised correlations in disordered dynamical systems: Insights from the many-species Lotka-Volterra model

TL;DR

This paper investigates how single-index correlations in interaction matrices influence the dynamics and stability of disordered systems, using the generalized Lotka-Volterra model as a case study, revealing effects on noise levels, feedback, and species survival.

Contribution

It introduces a dynamic mean-field analysis of correlations in interaction matrices, extending understanding of disordered systems beyond independent entries, with specific focus on ecological models.

Findings

In-row correlations increase mean field noise levels.

Transpose pair correlations can either enhance or suppress feedback effects.

Correlations impact species survival and ecological stability.

Abstract

In the study of disordered systems, one often chooses a matrix of independent identically distributed interaction coefficients to represent the quenched random couplings between components, perhaps with some symmetry constraint or correlations between diagonally opposite pairs of elements. However, a more general set of couplings, which still preserves the statistical interchangeability of the components, could involve correlations between interaction coefficients sharing only a single row or column index. These correlations have been shown to arise naturally in systems such as the generalised Lotka-Volterra equations (gLVEs). In this work, we perform a dynamic mean-field analysis to understand how single-index correlations affect the dynamics and stability of disordered systems, taking the gLVEs as our example. We show that in-row correlations raise the level of noise in the mean field process, even when the overall variance of the interaction coefficients is held constant. We also see that correlations between transpose pairs of rows and columns can either enhance or suppress feedback effects, depending on the sign of the correlation coefficient. In the context of the gLVEs, in-row and transpose row/column correlations thus affect both the species survival rate and the stability of ecological equilibria.