Liebig's law of the minimum, often simply called Liebig's law or the law of the minimum, is a principle developed in agricultural science by Carl Sprengel (1840) and later popularized by Justus von Liebig. It states that growth is dictated not by total resources available, but by the scarcest resource (limiting factor). The law has also been applied to biological populations and ecosystem models for factors such as sunlight or mineral nutrients.

This was originally applied to plant or crop growth, where it was found that increasing the amount of plentiful nutrients did not increase plant growth. Only by increasing the amount of the limiting nutrient (the one most scarce in relation to "need") was the growth of a plant or crop improved. This principle can be summed up in the aphorism, "The availability of the most abundant nutrient in the soil is only as good as the availability of the least abundant nutrient in the soil." Or the rough analog, "A chain is only as strong as its weakest link." Though diagnosis of limiting factors to crop yields is a common study, the approach has been criticized.

Liebig's law has been extended to biological populations (and is commonly used in ecosystem modelling). For example, the growth of an organism such as a plant may be dependent on a number of different factors, such as sunlight or mineral nutrients (e.g., nitrate or phosphate). The availability of these may vary, such that at any given time one is more limiting than the others. Liebig's law states that growth only occurs at the rate permitted by the most limiting factor.

For instance, in the equation below, the growth of population ${\displaystyle O}$ is a function of the minimum of three Michaelis-Menten terms representing limitation by factors ${\displaystyle I}$, ${\displaystyle N}$ and ${\displaystyle P}$.

Where O is the biomass concentration or population density. μI,μN,μP are the specific growth rates in response to the concentrations of three different limiting nutrients, represented by I,N,P respectively. kI,kN,kP are the half-saturation constants for the three nutrients I,N,P respectively. These constants represent the concentration of the nutrient at which the growth rate is half of its maximum. I,N,P are the concentrations of the three nutrients /factors. m is the mortality rate or decay constant.

The use of the equation is limited to a situation where there are steady state ceteris paribus conditions, and factor interactions are tightly controlled.

In human nutrition, the law of the minimum was used by William Cumming Rose to determine the essential amino acids. In 1931 he published his study "Feeding experiments with mixtures of highly refined amino acids". Knowledge of the essential amino acids has enabled vegetarians to enhance their protein nutrition by protein combining from various vegetable sources. One practitioner was Nevin S. Scrimshaw fighting protein deficiency in India and Guatemala. Frances Moore Lappé published Diet for a Small Planet in 1971 which popularized protein combining using grains, legumes, and dairy products.

The law of the minimum was tested at University of Southern California in 1947. "The formation of protein molecules is a coordinated tissue function and can be accomplished only when all amino acids which take part in the formation are present at the same time." It was further concluded, that "'incomplete' amino acid mixtures are not stored in the body, but are irreversibly further metabolized." Robert Bruce Merrifield was a laboratory assistant for the experiments. When he wrote his autobiography he recounted in 1993 the finding: