Revisiting Taylor's law, rediscovering Smith's law: A dual framework for scaling heterogeneity

Zhanshan (Sam) Ma, Lianwei Li, Shu Liu, Aaron Ellison

Taylor's (1961) power law (TPL)—variance scales with mean density—originated in agricultural entomology but now spans the natural and social sciences and even the humanities. Cited over 8,000 times and featured in Nature for decades, its reach far exceeds its ecological origins. Yet TPL has been dormant in Nature for the past two decades, though it remains active in PNAS. Smith's (1938) law (SPL)—crop yield variance scales with plot area—tells a different story. Cited approximately 500 times, almost exclusively in the agricultural literature, Smith's law has itself never appeared in Nature or PNAS. This disparity is not a minor oversight. It is a missing half. SPL captures environmental heterogeneity — the structured properties of the abiotic template. TPL captures ecological heterogeneity — the interacting distribution of organisms. Together, they form a dual framework for scaling heterogeneity across agents and their environment. We propose a hierarchical, testable model: Taylor's exponent depends on Smith's exponent and species traits shaped by evolution. The parallel to AI is striking. OpenAI's scaling law (agent) and DeepMind's refinement (template) mirror TPL and SPL. Both have played critical roles in guiding LLM training — yet neither cited Taylor or Smith. This convergence suggests a general principle: complex systems — from evolving ecosystems to engineered AI — exhibit dual scaling laws for agents and the templates (policies, environments, rules) that shape them. Smith's law was not wrong. It was forgotten. Rediscovering it offers a unified lens to predict heterogeneity across space, time, and disciplines — from ecology to economics to AI governance.