Investigating the different phases of the Generalized Lotka-Volterra equations in the large species limit

This thesis analyzes the Generalized Lotka-Volterra (GLV) equation with random Gaussian noise using the Dynamical Mean-Field Theory (DMFT) approach in the limit of a large number of species. Deriving a self-consistent effective equation we describe the system as a single specie interacting only with global order parameters. We analyze the global order parameter, focusing in particular on the correlation function, to get a deeper understanding of the system’s macroscopic behavior. The numerical simulations we performed focus on the multiple-attractors phase, where species’ populations exhibit persistent oscillations without converging to a fixed point. To derive an analytical expression of the species abundance distribution, we apply the Unified Colored-Noise Approximation (UCNA) approximating the system’s noise with an Ornstein-Uhlenbeck process with a quenched term. However, the method shows good results only in a specific case—excluding the memory term—and does not provide a general solution.