Local equations for the generalized Lotka-Volterra model on sparse asymmetric graphs

TL;DR

This work addresses the challenge of analyzing stochastic generalized Lotka-Volterra dynamics on sparse, often asymmetric graphs. It introduces local Fokker-Planck closures—IBMF and PBMF—to obtain tractable stationary distributions, with PBMF connecting to Belief Propagation in the symmetric limit. The authors derive and validate these closures across undirected/asymmetric, directed networks and finite temperatures, mapping phase diagrams and characterizing fluctuations. The approach offers a fast, versatile framework for predicting stable states in realistic ecological communities and can extend to economics and evolutionary game theory on sparse networks.

Abstract

Real ecosystems are characterized by sparse and asymmetric interactions, posing a major challenge to theoretical analysis. We introduce a new method to study the generalized Lotka-Volterra model with stochastic dynamics on sparse graphs. By deriving local Fokker-Planck equations and employing a mean-field closure, we can efficiently compute stationary states for both symmetric and asymmetric interactions. We validate our approach by comparing the results with the direct integration of the dynamical equations and by reproducing known results and, for the first time, we map the phase diagram for sparse asymmetric networks. Our framework provides a versatile tool for exploring stability in realistic ecological communities and can be generalized to applications in different contexts, such as economics and evolutionary game theory.

Figures (10)

• Figure 1: Comparing individual abundances from IBMF and simulations in a random regular graph at finite temperature $T=0.015$. The connectivity is $c=3$, and the immigration rate is $\lambda=10^{-6}$. Each $\alpha_{ij}$ is independently drawn from the Gaussian $\mathcal{N}(0, \sigma)$, with $\sigma=0.15$ (the interactions are asymmetric). Each point in the main graphic has coordinates $(n_i^{\text{SIM}}, n_i^{\text{IBMF}})$, where $n_i^{\text{SIM}}$ is the average stationary abundance of species $i$ obtained from 100 simulations of the dynamics, and $n_i^{\text{IBMF}}$ is the prediction of IBMF for the same species. The black dashed line is just the linear function $f(x)=x$. The system has $N=1024$ species, thus there are $1024$ points in the main graphic. The inserted graphic in the top-left corner shows the temporal evolution of four species with small stationary abundances. The corresponding points are marked with the same colors in the main graphic. The horizontal lines are the predictions of IBMF for the same species. The graphic is in semi-log scale. Analogously, the inserted graphic in the bottom-right is done with four species whose abundances are not small. Colored lines show the results of simulations, and the horizontal black lines show the predictions made with IBMF.
• Figure 2: Transitions obtained simulating the gLV model for $T=0$, asymmetric interactions ($\alpha_{ij}$ is chosen independently of $\alpha_{ji}$), and $\lambda=10^{-6}$. For several pairs $(\mu, \sigma)$, we run the dynamics for $10000$ different random regular graphs with connectivity $c=3$ and size $N=1024$. The interaction strengths are drawn from the Gaussian distribution: $\alpha_{ij}\sim\mathcal{N}(\mu, \sigma)$. By repeating the simulation $10$ times with different initial conditions for each graph, we identify one of three possible outcomes: i) all realizations converge to the same fixed point, ii) all the realizations converge but the fixed points are different, or iii) the abundances in at least one of the simulations grow and diverge for long times. The blue points mark, for each $\mu$, the maximum value of $\sigma$ at which more than $50\%$ of the samples are of type i). The red points mark, for each $\mu$, the minimum value of $\sigma$ at which more than $50\%$ of the samples are of type iii).
• Figure 3: Transitions of the gLV model for different system sizes at $T=0$. The interactions are asymmetric ($\alpha_{ij}$ is chosen independently of $\alpha_{ji}$) and defined on random regular graphs with connectivity $c=3$. Points represent the results of simulations with immigration rate $\lambda=10^{-6}$, and lines are the predictions made with IBMF for the same sizes. Each transition was determined using $10000$ graphs. Simulations are repeated for $10$ different initial conditions. IBMF was run with damping (see Appendix \ref{['app:damping']}) for $10$ different random initial conditions. (a) For each system size $N$ and average strength $\mu$, points (lines) mark the maximum value of $\sigma$ such that simulations (IBMF) converged to the same fixed point in more than $50\%$ of the interaction graphs. (b) Points (lines) mark the minimum value of $\sigma$ such that simulations (IBMF) displayed unbounded growth (not converged) more than $50\%$ of the interaction graphs.
• Figure 4: Probability that IBMF does not converge ($P_{\text{nc}}$) in directed graphs. IBMF is run over different realizations of the interaction graph with a given average connectivity $c$, size $N$, and interaction strength $\mu$. There is no unique function for all $\mu>1$, and dashed lines in the top panel are obtained exactly as in Ref. StavPRE2024 (see the text for clarification). (a) Toy model without noise in the interactions ($\sigma=0$). The colored lines in the main graphic represent the results of IBMF for $N=65536$ and different values of $\mu$. In the inserted graphic, IBMF (lines) is run instead for systems with $N=16384$ species, and the points represent the results of simulations of the dynamics for the same system size. The vertical line marks the value $c=\mkern1mu\mathrm{e}\mkern1mu$. The error bars for IBMF predictions are small and are not included in the graphics. (b) The interaction strengths are drawn from the distribution $\mathcal{N}(\mu, \sigma)$ with $\mu=0.7$ and two values of $\sigma$. The values of $P_{\text{nc}}$ for different values of $c$ are represented using points with their corresponding error bars. Lines are a guide to the eye.
• Figure 5: Predictions of IBMF and BP in the presence of thermal noise for graphs with symmetric and homogeneous interactions (drawn using $\sigma=0$). Both techniques are applied to random regular graphs with connectivity $c=3$. The immigration rate is $\lambda=10^{-6}$. Damping is not used in the iteration process (see Appendix \ref{['app:damping']}). (a), (b), and (c) Points represent the probability distributions $\tilde{P}_{BP}(n)$ obtained with BP (see Eq. \ref{['eq:PBP_tilde']}) for $\mu=0.04$, $\mu=0.06$, and $\mu=0.12$. The temperature is $T=0.03$ in all cases, and the system size is $N=128$. Black continuous lines are fits to the points via a truncated Gaussian. The inserted graphics show the relative deviation $\Delta_{BP-G}$ of the points with respect to the fits (see Eq. \ref{['eq:BP_G_dev']}), as a function of the abundance $n$. (d) For each temperature $T$, we mark the maximum value of $\mu$ where BP converges (green points). We also run IBMF on $10000$ graphs, each with $10$ times different initial conditions, and mark the maximum value of $\mu$ where it converges to the same fixed point in at least $50\%$ of the graphs (blue points). System sizes are $N=128$ and $N=1024$ for BP and IBMF, respectively. (e) Average abundance predicted by BP and IBMF as a function of $\mu$. At each temperature $T$, the black dashed lines are obtained with BP over the range of values of $\mu$ where this algorithm converges (to the left of the green points in panel (d)). Analogously, the continuous colored lines are obtained with IBMF where it converges to a single fixed point (to the left of the blue points in panel (a)). The colored points represent the average over many fixed points of IBMF where the fixed point is no longer unique (to the right of the blue points in the panel (a)), sampled by running IBMF with $10000$ distinct initial conditions. The inserted graphic shows the skewness $\gamma$ of the distribution $\hat{P}_{BP}(n)$ for the same values of $T$ and $\mu$ presented in the main graphic. We use colored lines in the region where IBMF finds a single equilibrium point, and continuous black lines where it finds different fixed points. BP and IBMF are run over the same graph with $N=1024$ species.