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Spatiotemporal noise stabilizes unbounded diversity in strongly-competitive communities

Amer Al-Hiyasat, Daniel W. Swartz, Jeff Gore, Mehran Kardar

Abstract

Classical ecological models predict that large, diverse communities should be unstable, presenting a central challenge to explaining the stable biodiversity seen in nature. We revisit this long-standing problem by extending the generalized Lotka-Volterra model to include both spatial structure and environmental fluctuations across space and time. We find that neither space nor environmental noise alone can resolve the tension between diversity and stability, but that their combined effects permit arbitrarily many species to stably coexist despite strongly disordered competitive interactions. We analytically characterize the noise-induced transition to coexistence, showing that spatiotemporal noise drives an anomalous scaling of abundance fluctuations, known empirically as Taylor's law. At the community level, this manifests as an effective sublinear self-inhibition that renders the community stable and asymptotically neutral in the high-diversity limit. Spatiotemporal noise thus provides a novel resolution to the diversity-stability paradox and a generic mechanism by which complex communities can persist.

Spatiotemporal noise stabilizes unbounded diversity in strongly-competitive communities

Abstract

Classical ecological models predict that large, diverse communities should be unstable, presenting a central challenge to explaining the stable biodiversity seen in nature. We revisit this long-standing problem by extending the generalized Lotka-Volterra model to include both spatial structure and environmental fluctuations across space and time. We find that neither space nor environmental noise alone can resolve the tension between diversity and stability, but that their combined effects permit arbitrarily many species to stably coexist despite strongly disordered competitive interactions. We analytically characterize the noise-induced transition to coexistence, showing that spatiotemporal noise drives an anomalous scaling of abundance fluctuations, known empirically as Taylor's law. At the community level, this manifests as an effective sublinear self-inhibition that renders the community stable and asymptotically neutral in the high-diversity limit. Spatiotemporal noise thus provides a novel resolution to the diversity-stability paradox and a generic mechanism by which complex communities can persist.
Paper Structure (5 sections, 15 equations, 4 figures)

This paper contains 5 sections, 15 equations, 4 figures.

Table of Contents

  1. gLV metacommunities with spatiotemporal noise
  2. Noise-induced transition to coexistence
  3. Effective dynamics of mean abundances
  4. Fixed points of the $\theta$-gLV model
  5. Discussion

Figures (4)

  • Figure 1: Spatiotemporal noise stabilizes diversity. (a) Generalized Lotka-Volterra metacommunity with fluctuating growth rates. (b) Sample abundance traces on a single patch for different dispersal rates, $D$, and noise magnitudes, $T$. The same interaction matrix is used in all cases and a particular focal species is emphasized in black. Initial pool size is $S = 64$. (b) Surviving richness $S^\star$ as a function of initial pool size $S$ for the different combinations of $T$ and $D$ in (b). (d) A metacommunity with $S=16$ is prepared (grey) and new species are introduced at fixed intervals (colored curves), with and without noise. In all panels, $r=1$, $P=2^{14}$, $A_{ij} \sim \mathcal{N}(0.7, 0.2^2)$ and $N_c = 10^{-15}$.
  • Figure 2: Noise-induced transition to full coexistence. (a) Survival fraction $\phi = S^\star/S$ as a function of noise strength $T$, showing a transition from $\phi=0$ to $\phi=1$ which sharpens as $P$ is increased. Inset shows collapse of the curves near $T_c$ upon rescaling with $\sqrt{P}$. $D=r=1, S = 512$. (b) Survival fraction $\phi$ as a function of dispersal rate and noise magnitude, showing different phases of community diversity. $S=256, P=2^{14}$. In both panels, $A_{ij} \sim \mathcal{N}(2, 1), N_c = 10^{-15}$.
  • Figure 3: Species abundance distribution in the coexistence phase. (a) Truncated power law abundance distribution over patches for a particular species, showing agreement with Eq. \ref{['eq:powerlaw']}. $S=256$. (b) Verification of the closure $\langle N_{i} N_j \rangle = \langle N_i \rangle \langle N_j \rangle$ (left) and the anomalous scaling $\langle N_i^2 \rangle \propto \langle N_i \rangle^{1+\beta D}$ (right), demonstrating the emergence of Taylor's law. Data combines simulations with $S = 128$ and $S=256$ for the three indicated values of $\beta D$. In all panels, $P = 2^{22}$, $A_{ij} \sim \mathcal{N}(4, 4)$, and $D = 2r$.
  • Figure 4: Effective $\theta$-logistic gLV dynamics of mean abundances. (a) Patch-averaged abundance trajectories from metacommunity simulations (left) compared to the prediction of Eq. \ref{['eq:effectiveDynamics']} (right). For clarity, six species are emphasized within a background of $256$ in the coexistence case $T>2D$. (b) Survival fraction as a function of $(T, D)$ obtained from metacommunity simulations and plotted on a linear axis. The boundaries at $T=D$ and $T=2D$ are indicated, representing the breakdown of effective gLV dynamics followed by the transition to coexistence ($\theta<1/2$). (c) Distribution (over species) of patch-averaged abundances, obtained from simulations of the effective $\theta$-gLV dynamics. The distribution narrows as $S$ is increased, demonstrating the emergent neutrality of the model. (d) Stability spectrum of the $\theta$-gLV model: each point is an eigenvalue of the fixed-point Jacobian, obtained numerically from one simulation at each indicated pool size, $S$. Black circles show the prediction of Eq. \ref{['eq:stability']}. Parameter values are given in the SI.