Equilibrium in a large Lotka-Volterra system with pairwise correlated interactions

TL;DR

This paper investigates the existence and stability of equilibria in large random Lotka-Volterra systems with correlated interactions, revealing phase transitions and conditions for positive and stable equilibria.

Contribution

It introduces a phase transition analysis for equilibria in large elliptic random matrices within Lotka-Volterra models, extending understanding of ecological stability.

Findings

Identifies conditions for positive equilibrium existence

Describes phase transition related to matrix normalization

Provides heuristics for counting positive components in equilibria

Abstract

We study the equilibria of a large Lokta-Volterra system of coupled differential equations in the case where the interaction coefficients form a large random matrix. In the case where this random matrix follows an elliptic model , we study the existence of a (componentwise) positive equilibrium and describe a phase transition for the matrix normalization.If there is no positive equilibrium, we provide conditions on the model parameters for the existence of a stable equilibrium (with vanishing components) and state heuristics to compute the number of positive components of the equilibrium. Lotka-Volterra systems are important in mathematical biology/ theoretical ecology.