Sparse Interactions Reshape Stability in Random Lotka-Volterra Dynamics

TL;DR

The paper tackles how finite connectivity and interaction heterogeneity reshape stability in generalized Lotka–Volterra ecosystems on sparse networks. It develops a thermodynamically exact framework based on the Gaussian Expansion Cavity Method to map the many-species dynamics onto an effective single-species stochastic process, solvable via population dynamics. A topology-driven transition from a globally stable single equilibrium to a fragmented multi-equilibrium landscape emerges purely from network structure, persisting under disorder and extending beyond fully connected models. The resulting phase diagrams provide a scalable analytic tool to assess stability, biodiversity, and regime shifts in large ecological networks and potentially other sparse non-equilibrium systems.

Abstract

Classical approaches to ecological stability rely on fully connected interaction models, yet real ecosystems are sparse and structured--a feature that qualitatively reshapes their collective dynamics. Here, we establish a thermodynamically exact stability phase diagram for generalized Lotka-Volterra dynamics on sparse random graphs, resolving how finite connectivity and interaction heterogeneity jointly govern ecosystem resilience. Using a small-coupling expansion of the dynamic cavity method, we derive an effective single-site stochastic process that is solvable via population dynamics. Our approach uncovers a topological phase transition--driven purely by the finite connectivity structure of the network--that leads to multi-stability. This instability is fundamentally distinct from the disorder-driven transitions induced by quenched randomness of the couplings. Our framework overcomes the considerable computational cost of direct simulations, offering a scalable and versatile analysis of stability, biodiversity, and alternative stable states in realistic, large-scale ecological ecosystems.

Figures (7)

• Figure 1: Species abundance distributions on sparse random graphs. Validation of the effective theory against microscopic simulations. The plots compare stationary abundances from direct Monte Carlo integration (MC, markers) with the fixed-point population-dynamics solution (PopDyn, lines). Left: Random Regular Graphs (RRG) with connectivity $K=10$, exhibiting near-Gaussian statistics. Right: Low connectivity $K=3$, showing strong skewness. Parameter sets $(m,\sigma,\gamma)$ are listed in the legends. MC data aggregate 20 graph realizations (20 initial conditions each); error bars denote standard errors. PopDyn population size is $P\in[10^5,10^7]$.
• Figure 2: Topological transition at $\sigma=0$ on Random Regular Graphs. Phase boundary in the $(m,K)$ plane for homogeneous couplings ($J_{ij}=m/K, \sigma=0$). Markers indicate the critical point $m_c(K)$ where the Population Dynamics (PopDyn) develops a non-vanishing variance $q$, signaling the transition from a globally stable Single Equilibrium (SE) to a fragmented Multiple Equilibria (ME) phase. The solid line has been obtained by interpolation. The dashed line plots the analytical stability boundary derived from the Kesten-McKay spectral edge, $m_c(K) = -K/(2\sqrt{K-1})$, marking the loss of linear stability for the fully feasible equilibrium. Right panels: Representative microscopic trajectories of species abundances $x_i(t)$ illustrating the dynamical regimes. Top ($m=-1.2$, SE): All species converge to a unique, homogeneous abundance value. Middle ($m=-1.7$, ME): Just below the transition, species settle into distinct fixed points depending on initial conditions, breaking the homogeneity. Bottom ($m=-4$, deep ME): Deep in the fragmented phase, the dependence on initial conditions persists and the fraction of extinct species ($x_i=0$) increases significantly.
• Figure 3: Stability phase diagrams in the $(m, \sigma)$ plane. Results for Random Regular Graphs with $K=10$ (left) and $K=3$ (right) obtained via Population Dynamics. Dashed lines mark the transition from the Single Equilibrium (SE) to the Multiple Equilibria (ME) phase; solid lines mark the transition to the Unbounded Growth (UG) phase. Different colors correspond to interaction symmetry values $\gamma \in [-1, 1]$ (see legend). Note the shrinking of the stable SE region as $\gamma \to 1$ and the severe compression of stability at low connectivity ($K=3$). Dynamical regimes are identified by the PopDyn behavior: SE (convergent to $q=0$), ME (non-convergent/oscillatory, $q>0$), UG (divergent $\mu$).
• Figure S1: Numerical determination of the topological phase transition ($\sigma=0$). Each row corresponds to a different connectivity $K$. The columns display the population averages of the mean abundance $\overline{\langle\mu\rangle}$ (left), the cavity variance $\overline{\langle q\rangle}$ (center), and the susceptibility $\overline{\langle\chi\rangle}$ (right). The errors represent the standard deviation of the population distribution. In the central column ($q$), the blue horizontal line indicates the baseline noise floor (mean $\pm$ std), while the vertical red line marks the detected critical point $m_c$ (with its estimated error width). Note the sharp discontinuity in $q$ characteristic of the topological transition.
• Figure S2: Detection of the Single Equilibrium (SE) to Multiple Equilibria (ME) transition. Parameters: $K=10, \gamma=-1$. Each row corresponds to a different disorder strength $\sigma$. The columns display the population averages of the mean abundance $\overline{\langle\mu\rangle}$ (left), the cavity variance $\overline{\langle q\rangle}$ (center), and the susceptibility $\overline{\langle\chi\rangle}$ (right). The errors represent the standard deviation of the population distribution. In the central column ($q$), the blue horizontal line indicates the baseline noise floor (mean $\pm$ std), while the vertical red line marks the detected critical point $m_c$ (with its estimated error width). The transition is identified by the departure of the cavity variance $q$ from zero, while the mean abundance $\mu$ remains finite. Unlike the topological case, the order parameter grows continuously from zero, resulting in larger uncertainties for $m_c$ as the distribution slowly broadens.