Sparse Interactions Reshape Stability in Random Lotka-Volterra Dynamics

TL;DR

This paper develops an exact stability phase diagram for sparse ecological networks modeled by Lotka-Volterra equations, revealing how network structure influences ecosystem resilience and multi-stability.

Contribution

It introduces a thermodynamically exact framework using the dynamic cavity method to analyze stability in sparse, structured ecological networks, a significant advance over traditional fully connected models.

Findings

Finite connectivity induces a topological phase transition leading to multi-stability.

Sparse networks exhibit different stability properties compared to fully connected models.

The method provides a scalable way to analyze large ecological systems.

Abstract

Classical approaches to ecological stability rely on fully connected interaction models, yet real ecosystems are sparse and structured--a feature that qualitatively reshapes their collective dynamics. Here, we establish a thermodynamically exact stability phase diagram for generalized Lotka-Volterra dynamics on sparse random graphs, resolving how finite connectivity and interaction heterogeneity jointly govern ecosystem resilience. Using a small-coupling expansion of the dynamic cavity method, we derive an effective single-site stochastic process that is solvable via population dynamics. Our approach uncovers a topological phase transition--driven purely by the finite connectivity structure of the network--that leads to multi-stability. This instability is fundamentally distinct from the disorder-driven transitions induced by quenched randomness of the couplings. Our framework overcomes the considerable computational cost of direct simulations, offering a scalable and versatile analysis of stability, biodiversity, and alternative stable states in realistic, large-scale ecological ecosystems.