Skip to main content
U.S. flag

An official website of the United States government

Publication Date
16 September 2021

Finding Liebig’s Law of the Minimum

The addtive model and synthesizing unit model are better than the widely used Liebig’s Law of the Minimum model for representing multiple-substrate co-limited growth of plants and microbes.
Print / PDF
Powerpoint Slide

Representing multiple-substrate co-limited growth is critical for modeling nutrient-constrained biogeochemistry-climate feedback in earth system models (ESMs). These limitations are generally applied using Liebig’s law of the minimum (LLM) model. Starting from the state-of-the-art Law of Mass Action model, we here derived three approximation models: (1) synthesizing unit, (2) additive, and (3) LLM. We then evaluated these three models with microbial and plant growth data and found the LLM model is the least accurate. We also explained why the LLM model is not appropriate for modeling biological growth under variable elemental stoichiometry conditions. Given the additive model’s low complexity and good accuracy, we recommend it to represent and analyze multiple-substrate co-limited growth in ESM land models.


Our theoretical analysis and observational benchmarks indicate that (1) the widely used LLM model is not appropriate for analyzing and modeling multiple-substrate co-limited growth; (2) the synthesizing unit model and the additive model are more accurate with the same number of model parameters as used in the LLM model; and (3) biogeochemical models and empirical analyses should revisit their conclusions based on Liebig’s law of the minimum model.


Ecosystem biogeochemical modeling and analysis are often faced with the question of how nutrients limit plant and microbial growth. The LLM model has been used for decades to address this question. However, the theoretical underpinning of the LLM model has never been presented. In this paper, we derived the LLM, synthesizing unit, and additive models from the fundamental theory of the Law of Mass Action. Through theoretical analysis and empirical benchmarks, we show that the LLM model is not appropriate to represent multiple-substrate co-limited growth with variable elemental stoichiometry. However, the synthesizing unit and additive models are more accurate while using the same number of model parameters as the LLM model. We advocate a revision to the existing biogeochemical representations of nutrient-constrained plant and microbial growth models in ESMs and recommend a comprehensive uncertainty analysis of ecosystem biogeochemistry with these new insights on multiple-substrate co-limited biological growth.

Point of Contact
William J. Riley
Lawrence Berkeley National Laboratory (LBNL)
Funding Program Area(s)