Equilibrium and surviving species in a large Lotka-Volterra system of differential equations

TL;DR

This paper investigates the equilibrium states and the number of surviving species in large random Lotka-Volterra systems, combining theoretical conditions with heuristics and simulations to understand complex ecological dynamics.

Contribution

It introduces a heuristic method for estimating surviving species in large LV systems with random interactions, supported by theoretical conditions and numerical validation.

Findings

Conditions for unique equilibrium are established.

Heuristic accurately predicts the number of surviving species.

Numerical simulations confirm theoretical and heuristic results.

Abstract

Lotka-Volterra (LV) equations play a key role in the mathematical modeling of various ecological, biological and chemical systems. When the number of species (or, depending on the viewpoint, chemical components) becomes large, basic but fundamental questions such as computing the number of surviving species still lack theoretical answers. In this paper, we consider a large system of LV equations where the interactions between the various species are a realization of a random matrix. We provide conditions to have a unique equilibrium and present a heuristics to compute the number of surviving species. This heuristics combines arguments from Random Matrix Theory, mathematical optimization (LCP), and standard extreme value theory. Numerical simulations, together with an empirical study where the strength of interactions evolves with time, illustrate the accuracy and scope of the results.