Taylor's Law, Smith's Law, and Diversity Power Laws: A Novel Triple Power Law Methodology for Scaling Diversity and Heterogeneity

Taylor's power law (TPL) and Smith's power law (SPL) are two foundational scaling laws describing how variance scales with mean density and plot area, respectively. While TPL captures ecological heterogeneity (variation among interacting organisms), SPL captures environmental heterogeneity (variation in the abiotic template). Diversity scaling, traditionally approached through species-area relationships, has been extended to diversity-area relationships (DAR) using Hill numbers. Here we integrate TPL, SPL, and DPL, including three newly proposed models [diversity-mean, diversity-variance, and diversity-heterogeneity relationships (DHR)], into a unified triple power law methodology for scaling diversity and heterogeneity in microbial ecosystems. Using human gut and vaginal microbiome datasets, we systematically vary two orthogonal factors: scale (unit vs. multi-unit) and accrual (without vs. with sample accrual). Our results show that TPL and SPL are complementary: classic TPL captures cross-sectional heterogeneity at the community scale, while accrual TPL, a new extension based on sample accrual, captures heterogeneity accumulation at the metacommunity scale and appears less scale-dependent. SPL provides a tool for relating environmental heterogeneity to diversity, supporting the reciprocity principle with ecological heterogeneity. Among the four diversity power laws, DHR, using the variance-to-mean ratio as a direct heterogeneity metric, is most aligned with the diversity-heterogeneity nexus. The triple power law methodology reveals that heterogeneity scaling predicts diversity scaling in the majority of models, with the strongest predictive relationships observed for accrual TPL at higher diversity orders. Nevertheless, the commonly assumed scale-invariance proved elusive, occurring in fewer than 20% of the power law models tested. This framework may extend beyond microbiome ecosystem to any complex system where heterogeneity and diversity arise from interacting components, from ecosystems to economies to artificial intelligence.