Natural communities in sparse networks: a statistical mechanics approach

Ecological communities are high-dimensional interacting systems whose stability emerges from the structure of their interaction networks. This thesis applies tools from the statistical mechanics of disordered systems to study the equilibrium properties of large ecosystems modeled by generalized Lotka–Volterra (gLV) equations with sparse and symmetric interactions. Moving beyond the traditional fully-connected (mean-field) setting, we analyze systems on locally tree-like random graphs using the Cavity Method and Belief Propagation (BP). In the single-equilibrium regime where BP converges, we compute species abundance marginals and show that increasing interaction disorder leads to a robust crossover from Gaussian to Gamma-like distributions. This behaviour is consistent with empirical observations and differs from the results obtained in fully-connected models. Remarkably, unlike dense systems, sparse ecosystems do not develop a glassy phase at large interaction disorder. At zero disorder, however, a peculiar phenomenon emerges, specific to sparse topologies: increasing competition among species induces a topological multiple-equilibria phase, signalled by the lack of convergence of the BP equations. To characterize this regime, we develop a one-step replica symmetry breaking (1RSB) cavity framework. The persistent non-convergence of both RS and 1RSB iterative schemes across the multiple-equilibria regime suggests a more intricate organization of equilibria, compatible with a full replica symmetry breaking (FRSB) phase with hierarchically structured basins. Altogether, these results demonstrate how sparsity reshapes stability, abundance distributions, and phase structure in large ecological communities, showing distinctive results compared to fully-connected models.