Quenched complexity of equilibria for asymmetric Generalized Lotka-Volterra equations

TL;DR

This paper analyzes the number and distribution of equilibria in high-dimensional asymmetric Lotka-Volterra ecosystems, using advanced statistical physics methods to compute the quenched complexity and compare it with other approximations.

Contribution

It introduces a detailed computation of the quenched complexity for asymmetric Lotka-Volterra systems using the replicated Kac-Rice formalism, extending methods from glass theory.

Findings

Computed the quenched complexity of equilibria

Compared quenched and annealed complexity results

Analyzed the distribution of equilibria based on diversity and stability

Abstract

We consider the Generalized Lotka-Volterra system of equations with all-to-all, random asymmetric interactions describing high-dimensional, very diverse and well-mixed ecosystems. We analyze the multiple equilibria phase of the model and compute its quenched complexity, i.e., the expected value of the logarithm of the number of equilibria of the dynamical equations. We discuss the resulting distribution of equilibria as a function of their diversity, stability and average abundance. We obtain the quenched complexity by means of the replicated Kac-Rice formalism, and compare the results with the same quantity obtained within the annealed approximation, as well as with the results of the cavity calculation and, in the limit of symmetric interactions, of standard methods to compute the complexity developed in the context of glasses.