Impact of a block structure on the Lotka-Volterra model

TL;DR

This paper examines how a block-structured interaction matrix influences the equilibrium states of the Lotka-Volterra model, focusing on two communities and their intra- and inter-community dynamics.

Contribution

It introduces a block variance profile into the LV model and analyzes the effects on equilibrium feasibility and species attrition for two communities.

Findings

Intra- and inter-community interactions significantly affect equilibrium stability.

Feasibility of equilibrium depends on variance within blocks.

Species attrition can occur due to interaction structure.

Abstract

The Lotka-Volterra (LV) model is a simple, robust, and versatile model used to describe large interacting systems such as food webs or microbiomes. The model consists of $n$ coupled differential equations linking the abundances of $n$ different species. We consider a large random interaction matrix with independent entries and a block variance profile. The $i$th diagonal block represents the intra-community interaction in community $i$, while the off-diagonal blocks represent the inter-community interactions. The variance remains constant within each block, but may vary across blocks. We investigate the important case of two communities of interacting species, study how interactions affect their respective equilibrium. We also describe equilibrium with feasibility (i.e., whether there exists an equilibrium with all species at non-zero abundances) and the existence of an attrition phenomenon (some species may vanish) within each community. Information about the general case of $b$ communities ($b> 2$) is provided in the appendix.