Fragile vs robust Multiple Equilibria phases in generalized Lotka-Volterra model with non-reciprocal interactions

We investigate the Multiple Equilibria phase of generalized Lotka-Volterra dynamics with random, non-reciprocal interactions. We compute the topological complexity of equilibria, which quantifies how rapidly the number of equilibria of the dynamical equations grows with the total number of species. We perform the calculation for arbitrary degree of non-reciprocity in the interactions, distinguishing between configurations that are dynamically stable to invasions by species absent from the equilibrium, and those that are not. We characterize the properties of typical (i.e., most numerous) equilibria at a given diversity, including their average abundance, mutual similarity, and internal stability. This analysis reveals the existence of two distinct ME phases, which differ in how internally stable equilibria behave under invasions by absent species. We discuss the implications of this finding for the system's dynamical behavior.