Spatiotemporal noise stabilizes unbounded diversity in strongly-competitive communities

The gist The spatiotemporal noise stabilizes unbounded diversity in strongly-competitive communities through an emergent sublinear self-inhibition that renders interaction disorder irrelevant in the high-diversity limit

How it works

The authors extend the generalized Lotka-Volterra model to include both spatial structure and environmental fluctuations across space and time, finding that neither space nor environmental noise alone can resolve the tension between diversity and stability. They demonstrate that spatiotemporal noise drives power-law abundance fluctuations, leading to an anomalous scaling of moments known empirically as Taylor’s law. At the metacommunity level, this manifests as an emergent sublinear self-inhibition that stabilizes diversity and renders the interaction disorder irrelevant in the high-diversity limit.

Key Mechanisms

The stabilization mechanism involves two ubiquitous features of natural ecosystems: (1) spatial structure and (2) spatiotemporal environmental noise. The model is augmented by implementing spatial structure using a metacommunity comprising patches with local interactions within each patch and dispersal between them. Environmental noise is included through fluctuations in the species growth rates, which amounts to a multiplicative noise term proportional to the species abundance.

Noise-Induced Transition to Coexistence

Numerical simulations show that when the dispersal rate is nonzero and the noise is sufficiently strong, all species persist indefinitely, with abundances fluctuating between a migration floor set by D⟨Ni⟩/r, and an O(1) ceiling set by the carrying capacity. The patch-averaged abundances, ⟨Ni⟩, are O(1/S), which stabilizes a coexistence state in which an O(1) total biomass is shared among S species.

Anomalous Scaling and Taylor’s Law

Spatiotemporal noise generates heavytailed abundance distributions, giving rise to a manifestation of Taylor’s law. This leads to an emergent nonanalytic, sublinear self-inhibition at the community level that stabilizes unbounded diversity. The second moment of the stationary distribution exhibits anomalous scaling, specifically ⟨N2i⟩ ∝ ⟨Ni⟩(1+D/T) when T > Tc. This constitutes an instance of Taylor’s law, providing a mechanistic explanation for this empirical law.

Effective Dynamics and Coexistence Phase

The dynamics of the mean (patch-averaged) abundances are governed by the effective θ-logistic gLV equation, where the growth law is often referred to as θ-logistic or Richard’s growth. The authors find that for constant interactions Aij = µ, the coexisting fixed point is asymptotically stable whenever θ < 1, and this implies that the critical noise strength for coexistence is Tc = D.

Limits and Extensions

The analytical theory of the noise-stabilized coexistence phase relies on the adiabatic limit D ≫ r, where factorization approximation becomes valid. In this regime, they find that Taylor’s law scaling ⟨N2i⟩ ∝ ⟨Ni⟩(1+θ) is obeyed robustly, with an exponent that can be obtained by fitting. The results suggest that the coexistence phase persists in d = 2 but not d = 1.

Weak Interactions and Paradox Resolution

In weakly-interacting systems, where interactions scale as µ, σ2 ∝ 1/S, a finite survival fraction is obtained even without noise. The paper concludes that in metacommunities with environmental noise, stronger competition can paradoxically protect against competitive exclusion by suppressing total biomass.

Summary of Findings

The work shows that spatiotemporal noise provides a novel resolution to the diversity-stability paradox and a generic mechanism by which complex communities can persist. The combination of spatial structure and spatiotemporal environmental noise permits arbitrarily many species to stably coexist in a sufficiently large system, despite strongly disordered competitive interactions. The emergent θ-gLV model accounts for the coexistence transition and bears similarities to the sublinear growth model analyzed in Ref. [58]. The results establish Taylor’s law as a probe of dynamical stability.

Further Research Directions

Future work is needed to determine the detailed dependence of S∗ on P and to investigate whether the coexistence phase persists in finite dimensions, particularly in d = 1. They plan to pursue this question in a future work. The authors also note that the maximal sustainable richness grows without bound as P is increased, though the scaling of S∗max(P) with P appears nonuniversal. The framework invites several natural extensions relevant to real ecosystems, such as investigating the role of interspecific correlations in environmental noise or heterogeneity in dispersal rates and noise strengths across species.

Numerical Details

Simulations were performed using the Milstein method, though the standard Euler-Maruyama method achieves comparable performance for the observables reported in this work. The simulations are amenable to massively-parallel GPU acceleration, allowing for simulations where the number of variables is on the order of a billion. Most simulation parameter values are listed in the Figure captions; below, we list remaining ones that were left out of the captions for brevity. The measurements represent quasi-stationary values, defined by sampling at a time tmax such that S∗(tmax/2) differs from S∗(tmax) by less than 10%. All simulations in this work use Nc ≪ 1/S.

References

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1. Spatial-temporal noise stabilization of unbounded diversity: The AI system can model and predict community persistence in strongly-competitive settings by incorporating both spatial structure (metacommunity patches) and environmental fluctuations (spatiotemporal noise). This allows the system to maintain arbitrarily many species to stably coexist, resolving the diversity–stability paradox.

2. Taylor's Law prediction for abundance fluctuations: The system can analytically characterize how spatiotemporal noise drives power-law abundance fluctuations, leading to an anomalous scaling of moments known empirically as Taylor’s law, specifically predicting that variance scales as a power of the mean, potentially serving as a probe of dynamical stability.

3. Emergent sublinear self-inhibition: The AI system can identify the macroecological consequence where spatiotemporal noise drives heavytailed abundance distributions, giving rise to a manifestation of Taylor’s law, which leads to an emergent nonanalytic, sublinear self-inhibition at the community level that stabilizes unbounded diversity.

4. Transition from exclusion to coexistence: The system can predict the critical noise threshold, Tc(D), where the transition occurs between a low diversity competitive exclusion phase and a noise-stabilized coexistence phase, depending on whether dispersal rate D is much less than or much greater than generation time r.

5. Robustness to interaction disorder: The AI system can determine that in the large-S limit of the coexistence phase, interaction disorder is asymptotically irrelevant, as mean abundances converge to a state where all patch-averaged abundances ⟨Ni⟩ will, for large S, become sharply peaked around their species average.

6. Species selection based on growth rate: By incorporating growth rate disorder, the AI system can predict that dynamics select for faster-growing species, and this selection is more stringent for larger S, suggesting that the number of survivors S∗ appears to saturate as a function of the initial pool size S under bounded growth rate disorder.

7. Fitness-density covariance modeling: The AI system can model how fitness-density covariance develops during invasion, indicating that a rare invader can overcome negative mean fitness by accumulating in patches where its competitive fitness is instantaneously higher, facilitating successful invasion and coexistence.

Classical ecological models predict that 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 together they permit arbitrarily many species to stably coexist in a sufficiently large system, despite strongly disordered competitive interactions. We analytically characterize the noise-induced transition to coexistence, showing that spatiotemporal noise drives power-law abundance fluctuations, leading to an anomalous scaling of moments known empirically as Taylor's law. At the metacommunity level, this manifests as an emergent sublinear self-inhibition that stabilizes diversity and renders the interaction disorder irrelevant 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.