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

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 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.

Figures (25)

• Figure 1: Species abundance average distribution for two values of the coupling variance $\hat{\sigma}$ in the single equilibrium phase. For small $\hat{\sigma} = 0.02$ (blue points) the distribution follows a Gaussian distribution (black line), while for large $\hat{\sigma} = 0.20$ (orange points) the species abundance follows a Gamma distribution (red line) as in \ref{['eq:Gamma']}, with parameters $\alpha=7.5$ and $\beta=0.029$ plus a gaussian peaked at $n=0$. Inset: marginal at $\hat{\sigma} = 0.20$ computed using BP (orange points) and from the dynamics (green points). The two agree very well. In both plots the parameters are $T=1$, $\hat{\mu}=0.1$, $N=256$.
• Figure 2: Kurtosis $\kappa(\hat{\sigma},T)$ of the average species abundance in the unique fixed point phase. For high variance of the couplings $\hat{\sigma}$ it is evident that $\kappa> 0$ implying that the marginals distributions are developing a non-Gaussian tail, while the kurtosis is almost independent on the temperature $T$. The parameters of the model are $\hat{\mu}=0.1$ and $N=256$.
• Figure 3: Left: Contour plots for the joint distribution $P(n,m)$, see Eq. \ref{['eq:disorder-average-joint']}, for $\hat{\sigma}=0.02$, in the Gaussian regime (top), and $\hat{\sigma}=0.20$, in the non-Gaussian regime (bottom). Right: connected correlation function $P(n,m)-\eta(n)\eta(m)$ for $\hat{\sigma}=0.02$ (top), and $\hat{\sigma}=0.20$ (bottom). Correlations are different from zero, at variance to what happens in the fully connected case. For all the panels $\hat{\mu}=0.1$, $T=1$ and $N=256$.
• Figure 4: Normalized sections of $P(n,n)$ and $\eta(n)\eta(n)$ for $\hat{\sigma}=0.02$, Gaussian regime, and $\hat{\sigma}=0.20$, non-Gaussian regime. Non trivial correlation effects are evident, differently from the fully connected case for which $P(n,n)=\eta(n)\eta(n)$. The parameters of the model are $\hat{\mu}=0.1$, $T=1$ and $N=256$.
• Figure 5: Connected correlation functions for exclusively competitive $P^{(+)}(n,m)-\eta(n)\eta(m)$ (top) and exclusively mutualistic $P^{(-)}(n,m)-\eta(n)\eta(m)$ (bottom) interactions. We are inside the strong-nonGaussian region $\hat{\sigma}=0.20$.