Generalized Lotka-Volterra Systems with Time Correlated Stochastic Interactions

In this Letter, we explore the dynamics of species abundances within ecological communities using the generalized Lotka-Volterra (GLV) model. At variance with previous approaches, we present an analysis of GLV dynamics with temporal stochastic fluctuations in interaction strengths between species. We develop a dynamical mean field theory (DMFT) tailored for scenarios with colored noise interactions, which we term annealed disorder, and simple functional responses. Our DMFT framework enables us to show that annealed disorder acts as an effective environmental noise; i.e., every species experiences a time-dependent environment shaped by the collective presence of all other species. We then derive analytical predictions for the species abundance distribution that well match empirical observations. Our results suggest that annealed disorder in interaction strengths favors species coexistence and leads to a large pool of very rare species in the systems, supporting the insurance theory of biodiversity. This Letter offers new insights not only into the modeling of large ecosystem dynamics but also proposes novel methodologies for examining ecological systems.