Equilibria in a large Lotka-Volterra model for complex ecosystems: a random matrix perspective

The thesis investigates the properties of the equilibria of a system of Lotka-Volterra equations linearly coupled via a random matrix. Such a framework is often used to model complex ecosystems where the coupling term represents species interaction. The idea to describe species interactions by a random matrix goes back to the seminal work of Robert May that in the 70s initiated the debate complexity vs. stability in ecological studies. The present thesis, after a review of May’s work, focuses on the study of the equilibria of a large set of Lotka-Volterra equations. Our aim is to predict the properties of the equilibria of such systems from the statistics of the spectrum of the (random) interaction matrix. We focus on two key properties: stability and feasibility, that is the existence of an equilibrium where all species survive.