Properties of equilibria and glassy phases of the random Lotka-Volterra model with demographic noise

TL;DR

This paper analyzes the equilibria and glassy phases of the disordered Lotka-Volterra model with demographic noise, revealing multiple stable states and a Gardner transition, with implications for ecological and other complex systems.

Contribution

It provides a comprehensive theoretical characterization of phase transitions and stability in a disordered ecological model with demographic noise, including new insights into the Gardner transition.

Findings

Exponential number of stable equilibria in certain phases

Identification of a Gardner transition to a marginally stable phase

Numerical simulations confirming analytical predictions

Abstract

In this letter we study a reference model in theoretical ecology, the disordered Lotka-Volterra model for ecological communities, in the presence of finite demographic noise. Our theoretical analysis, which takes advantage of a mapping to an equilibrium disordered system, proves that for sufficiently heterogeneous interactions and low demographic noise the system displays a multiple equilibria phase, which we fully characterize. In particular, we show that in this phase the number of stable equilibria is exponential in the number of species. Upon further decreasing the demographic noise, we unveil a "Gardner" transition to a marginally stable phase, similar to that observed in jamming of amorphous materials. We confirm and complement our analytical results by numerical simulations. Furthermore, we extend their relevance by showing that they hold for others interacting random dynamical systems, such as the Random Replicant Model. Finally, we discuss their extension to the case of asymmetric couplings.