Liebig's law of the minimum

Applications

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.

Scientific applications

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 ${\displaystyle P}$.

${\displaystyle \frac{dO}{dt}=O\left(min\left(\frac{{\mu }_{I}I}{k_I+I},\frac{{\mu }_{N}N}{k_N+N},\frac{{\mu }_{P}P}{k_P+P}\right)-m\right)}$

Where ${\displaystyle O}$ is the biomass concentration or population density; ${\displaystyle \{{\mu }_I,{\mu }_N,{\mu }_P\}}$ are the specific growth rates in response to the concentrations of three different limiting nutrients, represented by ${\displaystyle I}$, ${\displaystyle N}$, and ${\displaystyle P}$ respectively; ${\displaystyle \{k_I,k_N,k_P\}}$ are the half-saturation constants for the three nutrients ${\displaystyle I}$, ${\displaystyle N}$, and ${\displaystyle P}$ respectively—these constants represent the concentration of the nutrient at which the growth rate is half of its maximum; ${\displaystyle \{I,N,P\}}$ are the concentrations of the three nutrient factors; and ${\displaystyle m}$ 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.

Protein nutrition

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:

We showed that no net growth occurred when one essential amino acid was omitted from the diet, nor did it occur if that amino acid was fed several hours after the main feeding with the deficient diet.

Marine primary productivity

Phytoplankton in marine environments are limited by various micro- and macronutrients. These nutrients make it possible for phytoplankton to grow and reproduce. Some of the most important macronutrients for phytoplankton are nitrogen and phosphorus. These are considered macronutrients, as phytoplankton need a relatively large amount to function and grow. Some of the most common micronutrients for phytoplankton are zinc, cobalt, and iron. These are considered micronutrients because phytoplankton require smaller quantities of them compared to the macronutrients. However, these micronutrients are just as important as macronutrients, and both can be limiting.

The nutrient that is limiting for each species of phytoplankton can depend on many factors, including time of year and coastal proximity. For instance, it has been shown that in coastal upwelling zones, where macronutrients like phosphorus and nitrogen are abundant due to ocean circulation patterns and terrestrial runoff, iron can be limiting. Additionally, in open ocean environments, where there are very few nutrients, macronutrients like phosphorus and nitrogen can be limiting. In general, the limiting nutrient depends on each species’ life history strategy.

Other applications

More recently Liebig's law is starting to find an application in natural resource management where it surmises that growth in markets dependent upon natural resource inputs is restricted by the most limited input. As the natural capital upon which growth depends is limited in supply due to the finite nature of the planet, Liebig's law encourages scientists and natural resource managers to calculate the scarcity of essential resources in order to allow for a multi-generational approach to resource consumption.

Neoclassical economic theory has sought to refute the issue of resource scarcity by application of the law of substitutability and technological innovation. The substitutability "law" states that as one resource is exhausted—and prices rise due to a lack of surplus—new markets based on alternative resources appear at certain prices in order to satisfy demand. Technological innovation implies that humans are able to use technology to fill the gaps in situations where resources are imperfectly substitutable.

A market-based theory depends on proper pricing. Where resources such as clean air and water are not accounted for, there will be a "market failure". These failures may be addressed with Pigovian taxes and subsidies, such as a carbon tax. While the theory of the law of substitutability is a useful rule of thumb, some resources may be so fundamental that there exist no substitutes. For example, Isaac Asimov noted, "We may be able to substitute nuclear power for coal power, and plastics for wood ... but for phosphorus there is neither substitute nor replacement."

Where no substitutes exist, such as phosphorus, recycling will be necessary. This may require careful long-term planning and governmental intervention, in part to create Pigovian taxes to allow efficient market allocation of resources, in part to address other market failures such as excessive time discounting.