Generalized Lotka-Volterra model with sparse interactions: non-Gaussian effects and topological multiple-equilibria phase

We study the equilibrium phases of a generalized Lotka-Volterra model characterized by a species interaction matrix which is random, sparse and symmetric. Dynamical fluctuations are modeled by a demographic noise with amplitude proportional to the effective temperature T. The equilibrium distribution of species abundances is obtained by means of the cavity method and the Belief Propagation equations, which allow for an exact solution on sparse networks. Our results reveal a rich and non-trivial phenomenology that deviates significantly from the predictions of fully connected models. Consistently with data from real ecosystems, which are characterized by sparse rather than dense interaction networks, we find strong deviations from Gaussianity in the distribution of abundances. In addition to the study of these deviations from Gaussianity, which are not related to multiple-equilibria, we also identified a novel topological glass phase, present at both finite temperature, as shown here, and at T=0, as previously suggested in the literature. The peculiarity of this phase, which differs from the multiple-equilibria phase of fully-connected networks, is its strong dependence on the presence of extinctions. These findings provide new insights into how network topology and disorder influence ecological networks, particularly emphasizing that sparsity is a crucial feature for accurately modeling real-world ecological phenomena.

💡 Research Summary

The paper investigates the equilibrium properties of a generalized Lotka‑Volterra (gLV) ecosystem model in which the interaction matrix is random, sparse, and symmetric. Unlike the traditional fully‑connected (dense) models that have dominated theoretical ecology, the authors focus on a random regular graph with fixed degree k = 3, reflecting the sparsity observed in real ecological networks. Species abundances n_i evolve according to a stochastic differential equation that includes intrinsic growth, logistic self‑limitation, symmetric pairwise interactions α_{ij}, and demographic noise of amplitude T. The interaction strengths are drawn from a Gaussian distribution with mean μ/k and variance σ²/k, while a reflecting wall at a small λ prevents negative abundances.

Because a random regular graph is locally tree‑like (loop length ∼log N), the cavity method can be applied exactly in the thermodynamic limit. The authors derive cavity marginals η_{i→j}(n_i) and, using belief‑propagation (BP) equations, obtain the exact single‑site marginal distributions η_i(n_i) for any given disorder realization. Two types of averages are considered: a sample average over species for a fixed disorder, and a disorder average over many realizations of the random graph and couplings. This framework allows them to compute the full Gibbs‑Boltzmann distribution P(n) ∝ exp(−H_eff/T) without resorting to replica symmetry breaking or mean‑field approximations.

The first set of results explores the effect of the interaction variance σ at fixed temperature T = 1 and mean interaction μ̂ = 0.1. For a small σ̂ = 0.02 the averaged abundance distribution η(n) is well described by a truncated Gaussian, mirroring dense‑model predictions. However, increasing σ̂ to 0.20 drives a dramatic crossover: η(n) becomes highly skewed, with a long tail that fits a Gamma distribution. Importantly, this non‑Gaussian behavior occurs in a regime where the BP algorithm converges to a single equilibrium state, demonstrating that sparsity alone can generate fat‑tailed abundance distributions even when the underlying couplings remain Gaussian.

The second major finding concerns the case σ = 0 (all non‑zero interactions are identical). By varying μ and T, the authors map out a phase diagram using both BP and direct Langevin simulations, finding excellent agreement. For sufficiently large μ̂ and low T, the system enters a “topological multiple‑attractor” phase, which the authors term a topological glass. Unlike the multiple‑equilibria phase of dense models—where many metastable states arise from disorder in the couplings—here the multiplicity stems from different surviving sub‑networks (different topologies of extinct versus persisting species). The presence of the extinction wall λ is crucial: without it the glassy phase collapses. This phase persists at finite temperature, indicating that thermal fluctuations do not erase the topological origin of the multiple basins.

The paper situates its contributions within the literature. Prior works on sparse ecological stability (e.g., spectral analyses) did not address abundance distributions or phase transitions. Studies that linked non‑Gaussian abundances to heterogeneous degree distributions or non‑Gaussian couplings are challenged: the present work shows that even with homogeneous degree (regular graph) and Gaussian couplings, sparsity alone suffices to produce the observed deviations. Moreover, the identification of a topological glass phase extends earlier zero‑temperature results (e.g., Ref.