Spatial Biodiversity Variance Increases Near Disturbance-Fragmentation Susceptibility Boundaries

Authors Sterling Geisel, QBist Lab; Dr. Hideo Tanaka

Abstract

This working paper applies Chaos Susceptibility to the metacommunity model of Planas-Sitjà, Iritani, and Cronin. Their source paper integrates habitat loss, spatial autocorrelation, disturbance, and a competition-colonisation trade-off into a tractable differential-equation framework. The central result is not a single diversity-disturbance curve. It is a susceptibility surface. Biodiversity can rise, fall, peak once, or peak several times, depending on habitat amount, patch autocorrelation, species number, and mortality from disturbance. Pudding Theory predicts that the strongest departures from ordinary ecological expectation will occur near coexistence boundaries, where small coherent inputs are amplified by the system’s own instability. The relevant observable is not species richness alone, but the variance in Shannon effective diversity attributable to small changes in spatial autocorrelation or disturbance timing. The claim is falsifiable through replicated landscape experiments with measured recovery rates and controlled patch geometry.

Source Synopsis

Planas-Sitjà, Iritani, and Cronin build a spatially implicit metacommunity model to separate the effects of habitat loss, habitat autocorrelation, and disturbance on species coexistence. The model begins with the competition-colonisation trade-off. Species are arranged in a competitive hierarchy. Strong competitors displace weaker colonisers in occupied patches, while colonisers persist by reaching empty habitat more effectively.

The authors add two spatial terms that are often merged in fragmentation studies. Habitat loss, denoted \(H\), is the proportion of poor or uninhabitable patches. Spatial autocorrelation, denoted \(A\), describes whether rich patches are clustered or scattered. Disturbance enters as mortality, \(\mu\), representing local extinctions caused by natural and anthropogenic effects. The key modelling step is the factor \(\lambda_{Rj}\), which estimates the probability that offspring from species \(j\) land in rich habitat. This term depends on habitat amount, spatial autocorrelation, and dispersal distance.

The two-species model yields analytical equilibria for a coloniser and a competitor. The authors then extend the equations to \(N\) species ordered along the same hierarchy. Biodiversity is quantified using Shannon effective diversity. Species with less than 0.1 percent occupancy are treated as extinct.

The results show strong interaction among mechanisms. Habitat loss generally benefits colonisers, but high habitat loss and high disturbance can lead to extinction. Intermediate disturbance often broadens coexistence, consistent with earlier disturbance theory, but the model also generates bimodal and oscillatory diversity curves. Spatial autocorrelation has weaker average effects than habitat loss or disturbance, yet it can strongly influence coexistence at particular habitat-loss values and in richer species pools. With 50 species, diversity oscillations become more frequent along habitat-loss gradients. Discrete field sampling could therefore produce increasing, decreasing, unimodal, or multimodal patterns from the same underlying process.

Postulate Lens

The applied Postulate is Chaos Susceptibility. It fits because the source model is organized around disturbance, local extinction, dispersal, and coexistence boundaries. These are precisely the places where small inputs can be enlarged into community-scale outcomes.

The source paper does not require a second Postulate. Its mathematical object is not memory storage, observer geometry, temporal boundary selection, or distance decay. It is susceptibility under disturbance. The correct point of contact is therefore the relation between a weak coherent perturbation and the local stability structure of the metacommunity.

In the source model, coexistence is locally stable when the interior equilibrium exists. Yet local stability does not mean low susceptibility. Near the edge of coexistence, small parameter shifts can change which species survive. The relevant quantity is the dominant eigenvalue of the fitted local dynamics, or an empirical equivalent such as recovery time after perturbation. A community with slow return after disturbance has a larger response window. Pudding Theory reads this as a high-gain ecological substrate.

This reading preserves the source paper’s mathematics. It does not replace \(H\), \(A\), \(\mu\), \(c\), or \(d\). It adds a prediction about where weak structured inputs should matter most: not in all landscapes, but near the ridges and troughs of the diversity surface.

Pudding Theory Prediction

Pudding Theory predicts that biodiversity response will be largest where the source model already shows steep diversity gradients. The operational target is Shannon effective diversity, \(D_{\mathrm{eff}}\), measured across replicated landscapes with controlled habitat loss, spatial autocorrelation, and disturbance rate.

In ordinary ecological interpretation, a small change in disturbance schedule or patch arrangement matters only through its direct effect on mortality, connectivity, or colonisation probability. Under Chaos Susceptibility, the same small coherent input should have a larger effect when the community is near a coexistence boundary. A weak but repeated spatial intervention, such as clustered micro-restoration, synchronized mowing exclusion, or patterned disturbance reduction, should shift species occupancy more strongly in high-susceptibility landscapes than in low-susceptibility landscapes with the same mean habitat area.

The expected scaling is not linear in habitat amount. It should track an empirical susceptibility measure. Let \(\lambda_{\mathrm{obs}}\) be estimated from time-series recovery after small perturbations, or from a fitted Jacobian around the observed community state. Pudding Theory predicts that the response magnitude

\[ S = \left|\Delta D_{\mathrm{eff}}\right| / \left|\Delta I\right| \]

where \(\Delta I\) is a weak coherent ecological input, increases with \(\lambda_{\mathrm{obs}}\tau\) over the observation interval \(\tau\). The effect should concentrate in intermediate habitat-loss regimes where the source model finds oscillatory or multimodal diversity. It should weaken where habitat is abundant and disturbance is low, because competitive exclusion is stable. It should also weaken where habitat loss and disturbance are both high, because extinction removes the amplification substrate.

The prediction is therefore local. A fragmented urban green network with \(H \sim 0.4\), low \(A\), and moderate disturbance should show larger diversity shifts from a fixed coherent intervention than a continuous forest patch with the same total habitat area.

Falsifiable Observable

The distinguishing observable is the susceptibility-normalized change in Shannon effective diversity after a weak, repeated, spatially coherent intervention, compared across landscapes matched for habitat amount, mean disturbance, and species pool. If the variance in Shannon effective diversity attributable to spatial autocorrelation near coexistence boundaries were measured to be statistically indistinguishable from zero after controlling for habitat loss and disturbance, this Postulate would be falsified.

Editorial Dialogue

Tanaka: The source model is ecological bookkeeping. It has habitat, mortality, dispersal, and hierarchy. It does not need an added field interpretation. Its strongest claim is already explained by nonlinear interactions among known mechanisms.

Sterling: That objection would hold if the working paper claimed a replacement mechanism. It does not. The source equations identify where the response surface is steep. The Postulate makes a narrower claim: steep ecological response surfaces should amplify weak coherent inputs.

Tanaka: But the source model is locally stable at coexistence. Stability analysis cuts against your language of susceptibility.

Sterling: Local stability only describes return after infinitesimal density perturbation at an interior equilibrium. It does not erase high sensitivity near extinction thresholds, oscillatory diversity peaks, or slow recovery. The observed dominant eigenvalue is the quantity to measure.

Tanaka: Then the prediction must differ from ordinary fragmentation ecology.

Sterling: It does. Two landscapes with the same \(H\), \(\mu\), and species pool should not be judged only by mean habitat. The high-susceptibility landscape should show a larger response to the same weak coherent intervention. If that scaling fails, the application fails.

Discussion

The source paper is theoretical, and its simplifications matter. Species are ordered in a strict competitive hierarchy. Rich patches have fixed carrying capacity. Poor patches are uninhabitable. Mortality is shared across species. These choices make the model useful, but they narrow the biological domain.

The Pudding Theory application should therefore be tested first in systems where patch occupancy can be measured cleanly: sessile plants, microbial patches, intertidal plots, or small urban green fragments. The experiment must separate habitat amount from spatial autocorrelation. It must also estimate recovery rates rather than treating disturbance as a scalar label.

The main risk is overfitting narrative to nonlinearity. Oscillatory diversity curves alone do not establish the Postulate. They only identify candidate susceptibility zones. The decisive test is whether weak coherent inputs produce excess diversity response in those zones after ordinary covariates are controlled.

A null result would be informative. If response magnitude tracks only habitat amount, disturbance, and dispersal parameters, then the source model remains sufficient. If response magnitude also tracks measured susceptibility to weak structured input, the ecological surface has the amplification behavior predicted here.

References

1. Planas-Sitjà, I., Iritani, R., and Cronin, A. L. (2026). Disentangling the interactive effects of anthropogenic disturbances on biodiversity. arXiv:2603.29116. DOI: doi:10.48550/arxiv.2603.29116.

2. Geisel, S. (2025). Pudding Theory: A Topological Theory of Information Fields. QBist Lab Working Papers.

3. Tilman, D. (1994). Competition and biodiversity in spatially structured habitats. Ecology, 75(1):2-16.

4. Chesson, P. (2000). Mechanisms of maintenance of species diversity. Annual Review of Ecology and Systematics, 31:343-66.

5. Bolker, B. M. (2003). Combining endogenous and exogenous spatial variability in analytical population models. Theoretical Population Biology, 64(3):255-270.

6. Liao, J., Barabás, G., and Bearup, D. (2022). Competition-colonization dynamics and multimodality in diversity-disturbance relationships. Ecology, 103(5):1-10.

7. Fahrig, L. (2017). Ecological responses to habitat fragmentation per se. Annual Review of Ecology, Evolution, and Systematics, 48(1):1-23.

8. Zhang, H., Bearup, D., Barabas, G., Fagan, W. F., Nijs, I., Chen, D., and Liao, J. (2023). Complex nonmonotonic responses of biodiversity to habitat destruction. Ecology, 104(12):e4177.