Feasibility of sparse large Lotka-Volterra ecosystems

TL;DR

This paper investigates the conditions under which large, sparse ecological networks modeled by Lotka-Volterra equations have positive equilibrium solutions, using random matrix theory and concentration inequalities.

Contribution

It provides explicit thresholds for the existence of positive equilibria in large sparse ecosystems, extending previous results to the sparse case.

Findings

Existence of positive equilibrium depends on interaction strength and sparsity.

Thresholds for feasibility are derived based on n and d.

Stability of the equilibrium is established.

Abstract

Consider a large ecosystem (foodweb) with n species, where the abundances follow a Lotka-Volterra system of coupled differential equations. We assume that each species interacts with d other species and that their interaction coefficients are independent random variables. This parameter d reflects the connectance of the foodweb and the sparsity of its interactions especially if d is much smaller that n. We address the question of feasibility of the foodweb, that is the existence of an equilibrium solution of the Lotka-Volterra system with no vanishing species. We establish that for a given range of d with an extra condition on the sparsity structure, there exists an explicit threshold depending on n and d and reflecting the strength of the interactions, which guarantees the existence of a positive equilibrium as the number of species n gets large. From a mathematical point of view, the study of feasibility is equivalent to the existence of a positive solution (component-wise) to the equilibrium linear equation. The analysis of such positive solutions essentially relies on large random matrix theory for sparse matrices and Gaussian concentration of measure. The stability of the equilibrium is established. The results in this article extend to a sparse setting the results obtained by Bizeul and Najim in Proc. AMS 2021.