Equilibria of large random Lotka–Volterra systems with vanishing species: a mathematical approach

Ecosystems with a large number of species are often modelled as Lotka–Volterra dynamical systems built around a large interaction matrix with random part. Under some known conditions, a global equilibrium exists and is unique. In this article, we rigorously study its statistical properties in the large dimensional regime. Such an equilibrium vector is known to be the solution of a so-called Linear Complementarity Problem. We describe its statistical properties by designing an Approximate Message Passing (AMP) algorithm, a technique that has recently aroused an intense research effort in the fields of statistical physics, machine learning, or communication theory. Interaction matrices based on the Gaussian Orthogonal Ensemble, or following a Wishart distribution are considered. Beyond these models, the AMP approach developed in this article has the potential to describe the statistical properties of equilibria associated to more involved interaction matrix models.