Fragile $\mathit{vs}$ robust Multiple Equilibria phases in generalized Lotka-Volterra model with non-reciprocal interactions

TL;DR

This work analyzes fragile versus robust multiple equilibria phases in a generalized Lotka–Volterra model with non-reciprocal random interactions. It deploys a replicated Kac–Rice framework under Replica Symmetry to compute the topological complexity of fixed points, distinguishing uninvadable and total equilibria and tracking their abundance, similarity, and stability as a function of diversity. A key result is the identification of two ME phases—fragile and robust—separated by a curve $\gamma_{\rm FR}(\sigma)$, with robust ME supporting exponentially many internally stable and uninvadable fixed points, while fragile ME lacks such points; a quenched–annealed matching point further clarifies when the cavity or annealed descriptions apply. These findings illuminate how high-dimensional ecological communities with non-reciprocal couplings can exhibit qualitatively different dynamical regimes, guiding intuition about the link between fixed-point structure and chaotic dynamics. The analysis also establishes a framework to extend complexity calculations to structured interactions and relates equilibrium statistics to dynamical outcomes in non-equilibrium ecological and neural systems.

Abstract

We investigate the Multiple Equilibria phase of generalized Lotka-Volterra dynamics with random, non-reciprocal interactions. We compute the topological complexity of equilibria, which quantifies how rapidly the number of equilibria of the dynamical equations grows with the total number of species. We perform the calculation for arbitrary degree of non-reciprocity in the interactions, distinguishing between configurations that are dynamically stable to invasions by species absent from the equilibrium, and those that are not. We characterize the properties of typical (i.e., most numerous) equilibria at a given diversity, including their average abundance, mutual similarity, and internal stability. This analysis reveals the existence of two distinct ME phases, which differ in how internally stable equilibria behave under invasions by absent species. We discuss the implications of this finding for the system's dynamical behavior.

Figures (8)

• Figure 1: Phase diagram in the variability-reciprocity space $(\sigma,\gamma)$, highlighting the existence of three distinct phases. The black dotted line marks the boundary between the Unique Fixed Point (UFP) phase (gray), where a unique uninvadable and internally stable equilibrium equilibrium exists, and the Multiple Equilibria (ME) phase. The latter is split into a Fragile (orange) phase, where all internally stable equilibria are unstable to invasions, and a Robust (blue) phase, where uninvadable internally stable equilibria exist in exponential number. The red crosses identify the critical line $\gamma_{FR}$, the red full line is the hyperbolic fit $\gamma_{FR} \approx 0.318/\sigma + 0.549$.
• Figure 2: Total complexity $\Sigma^{(t)}_{\sigma, \gamma}(\phi)$ (upper curves) and uninvadable complexity $\Sigma^{(u)}_{\sigma, \gamma}(\phi)$ (lower curves) for $\sigma=5$ and various $\gamma$. The gray vertical band indicates the range of $\phi_{\rm May}=[{\sigma^2(1+\gamma)^2}]^{-1}$ for $\sigma=5$ and $\gamma \in [0,1]$.
• Figure 3: Order parameters of the typical equilibria at given diversity $\phi$, for $\sigma=2, \mu=1$ and various $\gamma$. The parameters are the average abundance $m$ (top left), the average growth rate $p$ (top right), the self similarity $q_1$ (bottom left) and the mutual similarity $q_0$ (bottom right). These values refer to invadable equilibria ($\alpha=t$), as also indicated by $p>0$.
• Figure S1: All quantities in is plot are with respect to total complexity. Left. Relative difference between annealed and quenched total complexities for different $\gamma$ and $\phi$ at $\sigma=5$, showing convergence for small $\phi$. Right. Convergence of $\beta_1\beta_2 - \beta_3^2$ to zero as $\phi \to \phi^+_{\text{Match}}(\gamma,\sigma)$, indicating divergence of the Gaussian determinant.
• Figure S2: Behavior of $\lambda^{\text{app}}=r^{\text{app}}\sqrt{{\beta_2^{\text{app}}}/{\beta_1^{\text{app}}}}$ (left) and $x_1^{\text{app}}, x_2^{\text{app}}$ (right) as a function of $\hat{\phi}$, for $\sigma=4.3, \gamma=1/2$ and $\alpha=u$. Here $x_1^{\text{app}}, x_2^{\text{app}}, r^{\text{app}}, \beta_1^{\text{app}}, \beta_2^{\text{app}}$ solve the approximate equations obtained plugging \ref{['eq:mIntD0']}-\ref{['eq:PhiIntD0']} into \ref{['eq:VRtop']}-\ref{['eq:VRbottom']}, and coincide with the solutions of the exact equations only at $\hat{\phi}= \hat{\phi}_{\text{Match}}= 1.079$, where $\Delta^{\text{app}}=\beta_1^{\text{app}} \beta_2^{\text{app}}- (\beta_3^{\text{app}})^2=0$ (inset). At this point, $\lambda^{\text{app}}=1$ and $x_1^{\text{app}}= x_2^{\text{app}}$.