A multifractal model for spatial variation in species richness

Models for species-area relationships up to now have focused on the mean richness as a function of area. We present MFp1p2, a self-similar multifractal. It explicitly models both trend and variation in richness as a function of area, and is a generalisation of the model of scaling of mean species richness due to Harte et al (1999). The construction is based on a cascade of bisections of a rectangle. The two parameters of the model are p1, the proportion of species that occur in the richer half, and p2, the proportion of species that occur in the poorer half. Equivalent parameterisations are a = (p1 + p2)/2 and b = p1/p2. These parameters are interpreted as follows: a gives the scaling of mean density, b gives the scaling of spatial variability. Several properties of MFp1p2 are derived, a generalisation is noted and some applications are suggested.

💡 Research Summary

The paper introduces MFp1p2, a self‑similar multifractal framework that simultaneously captures the mean species‑area relationship (SAR) and the spatial variability of species richness across scales. Traditional SAR models, typified by the power‑law S = c A^z, describe only the average number of species (S) as a function of area (A). Empirical observations, however, reveal substantial heterogeneity: two plots of identical size can differ dramatically in species count. MFp1p2 addresses this gap by modeling the entire distribution of richness as area is recursively halved.

The construction begins with a rectangular study region that is bisected at each iteration, generating a binary cascade of sub‑rectangles. At every bifurcation two parameters are defined: p₁, the proportion of the parent’s species that occupy the richer half, and p₂, the proportion that occupy the poorer half (p₁ ≥ p₂, p₁ + p₂ ≤ 1). After n bisections the region consists of 2ⁿ cells, each inheriting a subset of species determined by the cascade of p₁ and p₂ choices along its lineage.

Two alternative parameterisations are presented for interpretability: a = (p₁ + p₂)/2 and b = p₁/p₂. The parameter a governs the scaling of the mean density. Because each halving reduces the expected species count by a factor a, the mean richness after n steps is (2a)ⁿ S₀, where S₀ is the richness of the whole region. Translating to the classic SAR form yields the exponent z = −log₂ a, directly linking the multifractal model to Harte et al.’s (1999) mean‑field scaling.

The second parameter, b, quantifies spatial variability. When b = 1 the cascade is symmetric and the distribution of richness across cells collapses to a single value; as b increases, the disparity between the richer and poorer halves widens, producing a broader, heavy‑tailed distribution. Analytically, the variance after n steps is Var