A geometric basis for surface habitat complexity and biodiversity


Structurally complex habitats tend to contain more species and higher total abundances than simple habitats. This ecological paradigm is grounded in first principles: species richness scales with area, and surface area and niche density increase with three-dimensional complexity. Here we present a geometric basis for surface habitats that unifies ecosystems and spatial scales. The theory is framed by fundamental geometric constraints between three structure descriptors—surface height, rugosity and fractal dimension—and explains 98% of surface variation in a structurally complex test system: coral reefs. Then, we show how coral biodiversity metrics (species richness, total abundance and probability of interspecific encounter) vary over the theoretical structure descriptor plane, demonstrating the value of the theory for predicting the consequences of natural and human modifications of surface structure.

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Fig. 1: Increasing fractal dimension (that is, space filling) while keeping surface rugosity constant results in a decline in a surface’s mean height range.
Fig. 2: Comparison of the geometric theory with field data.
Fig. 3: The geometric diversity of coral reef habitats.
Fig. 4: Geometric-biodiversity coupling of coral reef habitats.

Data availability

Source data for statistical analyses and figures are available at https://github.com/jmadinlab/surface_geometry.

Code availability

Code for data preparation, statistical analyses and figures is available at https://github.com/jmadinlab/surface_geometry.


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We thank A. Hoggett and L. Vail of the Lizard Island Research Station for their support. This work was supported by an Australian Research Council Future Fellowship (to J.S.M.), the John Templeton Foundation (to M.A.D. and J.S.M.), a Royal Society research grant and a Leverhulme Fellowship (to M.A.D.), an International Macquarie University Research Excellence Scholarship (to D.T.-P.), two Ian Potter Doctoral Fellowships at Lizard Island (to D.T.-P. and V.B.) and an Australian Endeavour Scholarship (to T.J.C.).

Author information




J.S.M., D.T.-P., M.A.D. and O.P. conceptualized the study. J.S.M. and O.P. developed the theory and J.S.M. ran the analyses. J.S.M., D.T.-P. and O.P. developed the software pipeline for the data and produced the visualizations. J.S.M., D.T.-P., M.A.D. and O.P. led the investigation. J.S.M. and M.A.D. led and funded the broader project, with additional field robotics resources provided by O.P. and S.W. D.T.-P., M.A.D., O.P., M.B., S.A.B., N.B., V.B., T.J.C., G.F., A.F., M.O.H., S.W., K.J.A.Z. and J.S.M. collected the data. J.S.M. wrote the first draft of the paper and all authors reviewed at least one draft.

Corresponding author

Correspondence to Joshua S. Madin.

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Extended data

Extended Data Fig. 1

Differences in fractal dimension when calculated empirically versus from the theory that assumes self-similarity across scales.

Extended Data Fig. 2 Geometric theory for surface habitats.

a, Schematic showing mean height variation as a function of scale. ΔH is the height range of the habitat surface for the extent L. ΔH0 is the mean height range of the surface at the smallest scale: the resolution L0. The two slopes S represent fractal dimensions D of 2 and 3 according to Eq. 2. For example, high D results when mean height ranges at the scale of the grain are large (that is, in the vicinity of ΔH), suggesting a highly convoluted surface. Conversely, low D occurs when mean height variations at the scale of the grain are very small, suggesting an approximately flat surface. b, Area A0 at the scale of the grain L0 is calculated as the minimum surface area given the mean height range at this scale.

Extended Data Fig. 3 Underlying empirical data and models fits (±95% confidence intervals) for all combinations of the three surface descriptors (x-axes) and species richness, abundance and diversity (PIE) (y-axes).

The overall pattern was captured by both statistical approaches: generalised additive models and linear models with second-order polynomials for each surface descriptors. r2 values were consistently better for GAMs and reported here and in Extended Data Fig. 8.

Extended Data Fig. 4

The 21 shallow reef flat locations (black points) and the large Trimodal plot (red box) at Lizard Island, the Great Barrier Reef, Australia.

Extended Data Fig. 5 Digital elevation models of the 21 reef sites capturing different habitats encircling Lizard Island (see Extended Data Fig. 4 for map).

The models show the 2 m by 2 m reef patches (white squares) and depth gradience from 1.5m (lightest green) to 5m depth (darkest blue).

Extended Data Fig. 6 The relationship between R when calculated from the geometric theory for habitat surfaces and when calculated using surface area calculated using a spatial statistics function.

Dashed line is the unity line. Black line is the fit of a linear model (intercept: 0.023 ± 0.007 95% confidence intervals; slope: 1.014 ± 0.046 95% confidence intervals which encapsulates 1).

Extended Data Fig. 7 Model parameter and smooth term estimates for the best-fit species richness (a), abundance (b) and diversity (PIE) GAMs (c).

N =255.

Extended Data Fig. 8 Adjusted r2 values for GAM models of species richness (a), total abundance (b) and diversity (PIE) (c) with each of the three geometric variables and the best combined model after model selection.

N = 255.

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Torres-Pulliza, D., Dornelas, M.A., Pizarro, O. et al. A geometric basis for surface habitat complexity and biodiversity. Nat Ecol Evol (2020). https://doi.org/10.1038/s41559-020-1281-8

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