Generalized Lotka-Volterra systems with quenched random interactions and saturating nonlinear response

TL;DR

This work addresses the unphysical unbounded growth in generalized Lotka-Volterra models with quenched randomness by introducing a Monod-type saturating nonlinear response $J(x)=\dfrac{x}{1+hx}$. Using Dynamical Mean Field Theory, it derives a closed-form fixed-point solution and the corresponding Species Abundance Distribution (SAD) in the Unique Fixed Point phase, and characterizes the loss of stability that leads to chaotic dynamics in the Multiple Attractors phase. It identifies two sub-regimes within the MA phase (MA I and MA II) controlled by the interaction-symmetry parameter $\gamma$, distinguished by volatility and trajectory similarity, and validates predictions with large-scale simulations. The work provides a more ecologically realistic framework for disordered ecosystems and highlights how nonlinearity and symmetry govern diversity and resilience, with implications for extending DMFT to more complex network structures.

Abstract

The generalized Lotka-Volterra (GLV) equations with quenched random interactions have been extensively used to investigate the stability and dynamics of complex ecosystems. However, the standard linear interaction model suffers from pathological unbounded growth, especially under strong cooperation or heterogeneity. This work addresses that limitation by introducing a Monod-type saturating nonlinear response into the GLV framework. Using Dynamical Mean Field Theory, we derive analytical expressions for the species abundance distribution in the Unique Fixed Point phase and show the suppression of unbounded dynamics. Numerical simulations reveal a rich dynamical structure in the Multiple Attractor phase, including a transition between high-dimensional chaotic and low-volatility regimes, governed by interaction symmetry. These findings offer a more ecologically realistic foundation for disordered ecosystem models and highlight the role of nonlinearity and symmetry in shaping the diversity and resilience of large ecological communities.

Figures (8)

• Figure 1: Qualitative behavior of the QGLV model with saturating nonlinear response, as a function of the strength of interactions $\sigma$ and the correlation parameter $\gamma$. The other parameters are set to $\mu = -3$, $h = 0.1$, $\lambda = 10^{-8}$. The insets show the trajectories of 8 random species among the $400$ used for the simulations for a total simulation time of $80$. The specific values of the parameters used are $\sigma = 5$ and $\gamma = -0.9, 0, 0.9$. The solid line marks the separation of between the Unique Fixed Point phase and the Multiple Attractors phase and is determined from the self-consistent condition Eq.(\ref{['eq:critical_condition']}). The dashed line marks the separation between the qualitatively different behaviors in the multiple attractors phase and is determined approximately with the order parameters shown in Fig. \ref{['fig:panel4']}.
• Figure 2: Species Abundance Distribution.(a) Comparison between the theoretical species abundance distribution and the histogram of the stationary samples obtained from $10$ simulations with $1000$ species. The values $x^*$ from the simulations are averaged over the last $5\%$ of the trajectories that last for a total simulation time of $100$. The used parameters are $\mu = -3$, $\sigma = 1$, $\gamma = 0$, $h=0.8$ and $\lambda = 10^{-8}$. The solid line represents the survival distribution $P_{\text{surv}}(x^*)$ while the point at $x^*=0$ is given by the sum of the probability of extinction $\phi$ and the fraction of non-extinct species with abundances contained within the first bin, both normalized by the bin length. (b) Comparison among the theoretical species abundance distributions for different values of $h$. Other parameters are fixed as $\mu = -3$, $\sigma = 1$, $\gamma = 0$ and $\lambda = 10^{-8}$.
• Figure 3: Different projections of the phase diagram. The lines separate the Unique Fixed Point phase (below) from the Multiple Attractor phase (above). (a) $\gamma$-$\sigma$ phase diagram with $\mu =-3$. Black dashed line is the result obtained in Ref. galla2018dynamically. (b) $\mu$-$\sigma$ phase diagram with $\gamma =0$. Black dashed line is the result obtained in Ref. galla2018dynamically. (c) $\mu$-$\sigma$ phase diagram with $h =0.1$.
• Figure 4: Comparison of the order parameter evaluated along a projection of the phase diagram, obtained fixing $\mu = -3$, $\sigma = 5$, $h = 0.1$ and varying $\gamma$. Each point is obtained as the average over $225$ realizations of the interaction matrix with $800$ species. The solid line marks the separation of between the Unique Fixed Point phase and the Multiple Attractors phase and is determined from the self-consistent condition Eq.(\ref{['eq:stability_condition']}). The dashed line marks the separation between the qualitatively different behaviors in the multiple attractors phase and is determined approximately.
• Figure 5: Species Abundance Distribution for different combinations of the control parameters of the model. Comparison between the theoretical species abundance distribution and the histogram of the stationary samples obtained from $10$ simulations with $1000$ species. The values $x^*$ from the simulations are averaged over the last $5\%$ of the trajectories that last for a total simulation time of $100$.