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Generalized Lotka-Volterra Model with Sparse Interactions: Non-Gaussian Effects and Topological Multiple-Equilibria Phase

Tommaso Tonolo1,2,*, Maria Chiara Angelini3, Sandro Azaele4, Amos Maritan4, and Giacomo Gradenigo1,2

  • *Contact author: tommaso.tonolo@gssi.it
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PRX Life 4, 013017 – Published 6 February, 2026

DOI: https://doi.org/10.1103/8hrf-twxb

Abstract

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 nontrivial 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 multiple-attractor 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.

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PRX Life 4, 013017– Published 6 February, 2026
  • Received 13 May 2025
  • Accepted 23 December 2025
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Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

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