Large LV systems of differential equations are widely used to model the evolution of ecological communities of species with interactions. In this talk, we will discuss the conditions of existence of a single equilibrium in the case where the interactions are random. We will in particular describe a phase transition from equilibria with no vanishing species to equilibria with vanishing species. As we will see, the nature of the mathematical problem changes with this transition as we move from a linear algebra problem to a non linear one. Parts of the talk rely on joint work with Pierre Bizeul and Maxime Clénet.
