Heterogeneous mean-field analysis of the generalized Lotka-Volterra model on a network

TL;DR

The paper develops a heterogeneous mean-field theory for the generalized Lotka-Volterra model on configuration-model networks, capturing degree heterogeneity in the high-connectivity limit. It introduces an effective single-species process with self-consistent kernels and derives explicit fixed-point distributions for both homogeneous and heterogeneous interaction strengths, including regimes where abundances diverge and map to replicator dynamics. Central to the analysis is the critical degree $g_c$, which governs extinction and survival patterns in competitive and cooperative settings, and the results quantify how network structure shapes inequality via the Gini coefficient and survival probabilities. The framework yields tractable predictions for fixed points and stability, enabling insights into how finite-connectivity corrections and network heterogeneity affect dynamics with potential applications in ecology and econophysics.

Abstract

We study the dynamics of the generalized Lotka-Volterra model with a network structure. Performing a high connectivity expansion for graphs, we write down a mean-field dynamical theory that incorporates degree heterogeneity. This allows us to describe the fixed points of the model in terms of a few simple order parameters. We extend the analysis even for diverging abundances, using a mapping to the replicator model. With this we present a unified approach for both cooperative and competitive systems that display complementary behaviors. In particular we show the central role of an order parameter called the critical degree, $g_c$. In the competitive regime $g_c$ serves to distinguish high degree nodes that are more likely to go extinct, while in the cooperative regime it has the reverse role, it will determine the low degree nodes that tend to go relatively extinct.

Figures (6)

• Figure 1: Left: Average abundance, $\left \langle x \right \rangle = \int \mathop{}\!\mathrm{d} x Q(x|t) x$, and conditional one, $\left \langle x \right \rangle_{|g} =\int \mathop{}\!\mathrm{d} x P_g(x|t) x$, as a function of time for bimodal graph \ref{['eq:bimodal-distribution']}, with $N = 3000$, $C = 300$, and $\mu = -7$. Right: $P_g(x|t)$ for $g = 4/3$ for different values of $C$. Symbols represent numerical integration of a single instance of \ref{['eq:model-competition']} and solid lines of the solution of the theory \ref{['eq:cavity-M-competitive']}.
• Figure 2: Left: Solution for the critical degree $g_c$ from \ref{['eq:g_c-equation']} for exponential degree distribution, $\nu(g)=\rme^{-g}$. Right: Abundance distribution at the fixed point for homogeneous model, \ref{['eq:model-competition']}, with $N=3000$ and $C=50$ for $\mu = -0.5$ (competitive) and $\mu = 0.25$ (cooperative). Symbols are averages over 50 instances of numerical simulations. Dashed and solid lines correspond to the theory \ref{['eq:equilibrium-minimal-model']}. Error bars are of the order of magnitude of the symbols and omitted for clarity.
• Figure 3: Gini coefficient of the abundance distributions $Q(x^*)$ and $Q(y^*)$ at the fixed point for different values of $\mu$. The solid line corresponds to the Gini coefficient predicted by the theory, \ref{['eq:equilibrium-minimal-model']} and \ref{['eq:y-minimal-equilibrium']}, and the solid circles corresponds to the Gini coefficient measured directly for a single realization of the model, \ref{['eq:model-competition']}, with $N = 3000$ and an exponential degree distribution with $C = 50$. Dashed lines corresponds to the fraction of extinct species. For values $\mu>1/2$, it corresponds to relative extinctions. It is interesting that the rise of the Gini coefficient is sharper in the cooperative side, $\mu>0$, than in the competitive one $\mu<0$.
• Figure 4: Left: Abundance distribution for the bimodal case, \ref{['eq:bimodal-distribution']}, with $N=1000$, $C = 50$, $\mu=1$, $\sigma=0.5$, and $\gamma =0.9$. Results are shown for averaging over 3000 instances. The dotted and the dashed line correspond to the theoretical distributions of each of the two degrees. We can see the full distribution corresponds to the mixture of them, solid line \ref{['eq:equilibrium-distribution']}. Right: Equilibirum distribution for an exponential degree distribution. Average over 300 instances. Error bars are of the order of magnitude of the symbols and omitted for clarity.
• Figure 5: Left: Survival probability, $\Phi$, for $\gamma=0$ and exponential distribution as a function of $\sigma$, from \ref{['eq:survival-probability']}. From top to bottom different values $\mu=$$-.2$, $0.33$, $0.4$, $0.5$, $0.7$, and $1$. Notice for $\mu = 0.7$ it is non-monotonic and for $\mu = 1$ it is an increasing function. Right: Phase boundary for the linear stability described by the HDMFT. Dashed line corresponds to the theoretical values, from \ref{['eq:critical-line-sigma']}, for non-heterogeneoud DMFT, $\textrm{VAR}[g] = 0$, and solid line for the bimodal degree distribution, \ref{['eq:bimodal-distribution']}. The phase boundary made by the solid diamonds was determined numerically from \ref{['eq:Jacobian']} for $N=2000,C = 66$. Error bars are of the order of magnitude of the symbols and omitted for clarity.