[Paper Review] Extending species-area relationships (SAR) to diversity-area relationships (DAR)

Zhanshan Ma|arXiv (Cornell University)|Nov 16, 2017

Species Distribution and Climate Change34 references3 citations

TL;DR

This paper extends the classical species-area relationship (SAR) to a generalized diversity-area relationship (DAR) using Hill numbers to quantify alpha and beta diversity across multiple order parameters (q). By introducing three new profiles—DAR profile (z-q), PDO profile (g-q), and MAD profile (Dmax-q)—the framework integrates self-similarity and scaling laws into a unified model, validated on the American Gut Project dataset showing strong fit with power law and PLEC models, thereby enabling comprehensive biodiversity scaling analysis beyond species richness alone.

ABSTRACT

I extend the traditional SAR, which has achieved status of ecological law and plays a critical role in global biodiversity assessment, to the general (alpha- or beta-diversity in Hill numbers) diversity area relationship (DAR). The extension was motivated to remedy the limitation of traditional SAR that only address one aspect of biodiversity scaling, i.e., species richness scaling over space. The extension was made possible by the fact that all Hill numbers are in units of species (referred to as the effective number of species or as species equivalents), and I postulated that Hill numbers should follow the same or similar pattern of SAR. I selected three DAR models, the traditional power law (PL), PLEC (PL with exponential cutoff) and PLIEC (PL with inverse exponential cutoff). I defined three new concepts and derived their quantifications: (i)DAR profile: z-q series where z is the PL scaling parameter at different diversity order (q); (ii)PDO (pair-wise diversity overlap) profile: g-q series where g is the PDO corresponding to q; (iii) MAD (maximal accrual diversity) profile: Dmax-q series where Dmax is the MAD corresponding to q. Furthermore, the PDO-g is quantified based on the self-similarity property of the PL model, and Dmax can be estimated from the PLEC parameters. The three profiles constitute a novel DAR approach to biodiversity scaling. I verified the postulation with the American gut microbiome project (AGP) dataset of 1473 healthy North American individuals (the largest human dataset from a single project to date). The PL model was preferred due to its simplicity and established ecological properties such as self-similarity (necessary for establishing PDO profile), and PLEC has an advantage in establishing the MAD profile. All three profiles for the AGP dataset were successfully quantified and compared with existing SAR parameters in the literature whenever possible.

Motivation & Objective

• To address the limitation of traditional SAR, which only models species richness scaling, by extending it to encompass broader diversity measures such as alpha and beta diversity.
• To unify species-area relationships with diversity-area relationships (DAR) by expressing all Hill diversity measures in equivalent species units.
• To develop a scalable, mathematically consistent framework for biodiversity scaling that applies across different diversity orders (q).
• To validate the extended DAR framework using a large human microbiome dataset (American Gut Project, n=1,473 individuals).
• To establish new quantitative profiles—DAR, PDO, and MAD—that capture scaling patterns of diversity across spatial or sample-area gradients.

Proposed method

• Proposes a generalization of the power law (PL) model to diversity-area relationships (DAR) using Hill numbers as diversity measures, with all values expressed in species equivalents.
• Introduces the DAR profile as a z-q series, where z is the scaling exponent of the power law at each diversity order q.
• Defines the PDO (pairwise diversity overlap) profile as a g-q series, where g represents the overlap coefficient derived from self-similarity of the PL model.
• Introduces the MAD (maximal accrual diversity) profile as a Dmax-q series, where Dmax is estimated from PLEC model parameters to represent maximum diversity accrual.
• Uses the PLEC (power law with exponential cutoff) and PLIEC (power law with inverse exponential cutoff) models to model diversity scaling, with PLEC favoring MAD profile estimation.
• Employs the self-similarity property of the PL model to analytically derive the PDO-g relationship, enabling quantification of diversity overlap across q.

Experimental results

Research questions

• RQ1 Can the species-area relationship (SAR) be generalized to model not only species richness but also broader diversity measures such as alpha and beta diversity across spatial or sample-area gradients?
• RQ2 How do scaling exponents (z) of the power law model vary across different diversity orders (q) when using Hill numbers as diversity metrics?
• RQ3 What is the relationship between pairwise diversity overlap (PDO) and diversity order (q), and can it be quantified using self-similarity in the power law model?
• RQ4 Can the maximal accrual diversity (MAD) be estimated from PLEC model parameters, and how does it vary with diversity order (q)?
• RQ5 To what extent do the proposed DAR profiles (z-q, g-q, Dmax-q) provide a consistent and empirically verifiable framework for biodiversity scaling in real-world datasets?

Key findings

• The power law (PL) model was preferred for its simplicity and self-similarity, which enabled analytical derivation of the PDO-g profile.
• The PLEC model provided a better fit for estimating maximal accrual diversity (MAD), allowing robust Dmax-q profile estimation.
• The DAR profile (z-q) showed that scaling exponents z varied systematically with diversity order q, indicating non-uniform scaling across diversity types.
• The PDO-g profile revealed that pairwise diversity overlap decreased with increasing diversity order q, reflecting higher sensitivity to sampling at higher q values.
• The MAD profile (Dmax-q) demonstrated that maximum diversity accrual varied with q and could be reliably estimated from PLEC parameters, supporting the model's predictive power.
• All three profiles were successfully quantified and validated using the American Gut Project dataset, confirming the framework’s empirical applicability to large-scale human microbiome data.