Stability vs. complexity of ecosystems: a random matrix approach

In this thesis we investigate the complexity and stability of ecosystems using the random matrix theory. We start analyzing the approach of Robert May, whose work is considered one of the most influential in theoretical ecology. Then, motivated by recent studies which highlighted that living systems can be very sparse, we focus on large sparse ecosystems: we examine a Lotka-Volterra system using a large random matrix to reflect the interactions between species. We face the question of feasibility, that is the existence of an equilibrium solution with no vanishing species and we prove that a sharp phase transition occurs: above a certain threshold the feasible solution exists, below it does not. In the end we show some stability results and that feasibility and global stability occur simultaneously.