Liebig's law of the minimum

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 is a function of the minimum of three Michaelis-Menten terms representing limitation by factors , and . 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.