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 (1828) 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 concept 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, to put it more plainly, "A chain is only as strong as its weakest link."

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.[1]

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.

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".[2] 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. Francis Moore Lappe published Diet for a Small Planet in 1971 which popularized protein combining using grains, legumes, and dairy products.

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."[3]

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.

Liebig's barrel

Liebig's barrel

Dobenecks[4] used the image of a barrel—often called "Liebig's barrel"—to explain Liebig's law. Just as the capacity of a barrel with staves of unequal length is limited by the shortest stave, so a plant's growth is limited by the nutrient in shortest supply.

If a system satisfies the law of the minimum then adaptation will equalize the load of different factors because the adaptation resource will be allocated for compensation of limitation.[5] Adaptation systems act as the cooper of Liebig's barrel and lengthens the shortest stave to improve barrel capacity. Indeed, in well-adapted systems the limiting factor should be compensated as far as possible. This observation follows the concept of resource competition and fitness maximization.[6]

Due to the law of the minimum paradoxes, if we observe the Law of the Minimum in artificial systems, then under natural conditions adaptation will equalize the load of different factors and we can expect a violation of the law of the minimum. Inversely, if artificial systems demonstrate significant violation of the law of the minimum, then we can expect that under natural conditions adaptation will compensate this violation. In a limited system life will adjust as an evolution of what came before.[5]


One example of technological innovation is in plant genetics whereby the biological characteristics of species can be changed by employing genetic modification to alter biological dependence on the most limiting resource. Biotechnological innovations are thus able to extend the limits for growth in species by an increment until a new limiting factor is established, which can then be challenged through technological innovation.

Theoretically there is no limit to the number of possible increments towards an unknown productivity limit.[7] This would be either the point where the increment to be advanced is so small it cannot be justified economically or where technology meets an invulnerable natural barrier. It may be worth adding that biotechnology itself is totally dependent on external sources of natural capital.

See also


  1. ^ Sinclair, Thomas R. (1999). "Limits to Crop Yield". Plants and Population: is there time?. Colloquium. Washington DC: National Academy of Sciences.
  2. ^ W.C. Rose (1931) Feeding Experiments, Journal of Biological Chemistry 94: 155–65
  3. ^ Asimov, Issac (1962). "Life's Bottleneck". Fact and Fancy. Doubleday.
  4. ^ Whitson, A.R.; Walster, H.L. (1912). Soils and soil fertility. St. Paul, MN: Webb. p. 73. OCLC 1593332. 100. Illustration of Limiting Factors. The accompanying illustration devised by Dr. Dobenecks is intended to illustrate this principle of limiting factors.
  5. ^ a b A.N. Gorban, L.I. Pokidysheva, E.V. Smirnova, T.A. Tyukina. Law of the Minimum Paradoxes, Bull Math Biol 73(9) (2011), 2013–2044
  6. ^ D. Tilman, Resource Competition and Community Structure, Princeton University Press, Princeton, NJ (1982).
  7. ^ Reilly, J.M.; Fuglie, K.O. (6 July 1998). "Future yield growth in field crops: what evidence exists?". Soil and Tillage Research. 47 (3–4): 275–290. doi:10.1016/S0167-1987(98)00116-0.
Algonquian peoples

The Algonquian are one of the most populous and widespread North American native language groups. Today, thousands of individuals identify with various Algonquian peoples. Historically, the peoples were prominent along the Atlantic Coast and into the interior along the Saint Lawrence River and around the Great Lakes. This grouping consists of the peoples who speak Algonquian languages.

Before Europeans came into contact, most Algonquian settlements lived by hunting and fishing, although quite a few supplemented their diet by cultivating corn, beans and squash (the "Three Sisters"). The Ojibwe cultivated wild rice.The Algonquians of New England (who spoke Eastern Algonquian) practiced a seasonal economy. The basic social unit was the village: a few hundred people related by a clan kinship structure. Villages were temporary and mobile. The people moved to locations of greatest natural food supply, often breaking into smaller units or gathering as the circumstances required. This custom resulted in a certain degree of cross-tribal mobility, especially in troubled times.

In warm weather, they constructed portable wigwams, a type of hut usually with buckskin doors. In the winter, they erected the more substantial longhouses, in which more than one clan could reside. They cached food supplies in more permanent, semi-subterranean structures.

In the spring, when the fish were spawning, they left the winter camps to build villages at coastal locations and waterfalls. In March, they caught smelt in nets and weirs, moving about in birch bark canoes. In April, they netted alewife, sturgeon and salmon. In May, they caught cod with hook and line in the ocean; and trout, smelt, striped bass and flounder in the estuaries and streams. Putting out to sea, the men hunted whales, porpoises, walruses and seals.dubious The women and children gathered scallops, mussels, clams and crabs, all the basis of menus in New England today.

From April through October, natives hunted migratory birds and their eggs: Canada geese, brant, mourning doves and others. In July and August they gathered strawberries, raspberries, blueberries and nuts. In September, they split into small groups and moved up the streams to the forest. There, the men hunted beaver, caribou, moose and white-tailed deer.

In December, when the snows began, the people created larger winter camps in sheltered locations, where they built or reconstructed longhouses. February and March were lean times. The tribes in southern New England and other northern latitudes had to rely on cached food. Northerners developed a practice of going hungry for several days at a time. Historians hypothesize that this practice kept the population down, according to Liebig's law of the minimum. Northerners were food gatherers only.The southern Algonquians of New England relied predominantly on slash and burn agriculture. They cleared fields by burning for one or two years of cultivation, after which the village moved to another location. This is the reason the English found the region relatively cleared and ready for planting. By using various kinds of native corn (maize), beans and squash, southern New England natives were able to improve their diet to such a degree that their population increased and they reached a density of 287 people per 100 square miles as opposed to 41 in the north.Even with mobile crop rotation, southern villages were necessarily less mobile than northern ones. The natives continued their seasonal occupation but tended to move into fixed villages near their lands. They adjusted to the change partially by developing a gender-oriented division of labor. The women cultivated crops, and the men fished and hunted.

Scholars estimate that, by the year 1600, the indigenous population of New England had reached 70,000–100,000.


Bacterivores are free-living, generally heterotrophic organisms, exclusively microscopic, which obtain energy and nutrients primarily or entirely from the consumption of bacteria. Many species of amoeba are bacterivores, as well as other types of protozoans. Commonly, all species of bacteria will be prey, but spores of some species, such as Clostridium perfringens, will never be prey, because of their cellular attributes.

Carl Sprengel

Karl or Philipp Carl Sprengel (March 29, 1787 – April 19, 1859) was a German botanist from Schillerslage (now part of Burgdorf, Hanover).

Sprengel worked under Albrecht Thaer (1752–1828) in Celle. He then worked from 1804 to 1808 with Heinrich Einhof (1778–1808) in Möglin on agricultural studies. He travelled the world between 1810 and 1820, exploring agricultural ideas in Asia, Americas and Mesopotamia. Between 1821-1828 he studied natural sciences in Göttingen, where he eventually became professor.

In the early 1830s he moved to Regenwalde (Resko), where he accepted position of the Chairman of the Pomorskie Towarzystwo Ekonomiczne (Pomeranian Economic Society), which he held for the rest of his life. Having his financial needs satisfied, finally he could fulfil his dream and establish Regenwalde Akademie der Landwirtschaft (Academy of Agriculture in Resko), where he taught, studied and lived until his death in 1859.

Influenced by (one of the students at Regenwalde Akademie der Landwirtschaft) Felicjan Sypniewski theories, Sprengel was the first to formulate the "theory of minimum" in agricultural chemistry, meaning that plant growth is limited by the essential nutrient at the lowest concentration. This rule, often incorrectly attributed to Justus von Liebig as Liebig's law of the minimum, was instead only popularised later as a scientific concept by Liebig.

Consumer (food chain)

Consumers are organisms that eat organisms from a different population. These organisms are formally referred to as heterotrophs, which include animals, some bacteria and fungi. Such organisms may consume by various means, they are called primary consumers.


A copiotroph is an organism found in environments rich in nutrients, particularly carbon. They are the opposite to oligotrophs, which survive in much lower carbon concentrations.

Copiotrophic organisms tend to grow in high organic substrate conditions. For example, copiotrophic organisms grow in Sewage lagoons. They grow in organic substrate conditions up to 100x higher than oligotrophs.


Decomposers are organisms that break down dead or decaying organisms, and in doing so, they carry out the natural process of decomposition. Like herbivores and predators, decomposers are heterotrophic, meaning that they use organic substrates to get their energy, carbon and nutrients for growth and development. While the terms decomposer and detritivore are often interchangeably used, detritivores must ingest and digest dead matter via internal processes while decomposers can directly absorb nutrients through chemical and biological processes hence breaking down matter without ingesting it. Thus, invertebrates such as earthworms, woodlice, and sea cucumbers are technically detritivores, not decomposers, since they must ingest nutrients and are unable to absorb them externally.

Dominance (ecology)

Ecological dominance is the degree to which a taxon is more numerous than its competitors in an ecological community, or makes up more of the biomass. Most ecological communities are defined by their dominant species.

In many examples of wet woodland in western Europe, the dominant tree is alder (Alnus glutinosa).

In temperate bogs, the dominant vegetation is usually species of Sphagnum moss.

Tidal swamps in the tropics are usually dominated by species of mangrove (Rhizophoraceae)

Some sea floor communities are dominated by brittle stars.

Exposed rocky shorelines are dominated by sessile organisms such as barnacles and limpets.

Feeding frenzy

In ecology, a feeding frenzy occurs when predators are overwhelmed by the amount of prey available. For example, a large school of fish can cause nearby sharks, such as the lemon shark, to enter into a feeding frenzy. This can cause the sharks to go wild, biting anything that moves, including each other or anything else within biting range. Another functional explanation for feeding frenzy is competition amongst predators. This term is most often used when referring to sharks or piranhas. It has also been used as a term within journalism.

Limiting factor

A limiting factor is a variable of a system that, if subject to a small change, causes a non-negligible change in an output or other measure of the system. A factor not limiting over a certain domain of starting conditions may yet be limiting over another domain of starting conditions, including that of the factor.


A lithoautotroph or chemolithoautotroph is a microbe which derives energy from reduced compounds of mineral origin. Lithoautotrophs are a type of lithotrophs with autotrophic metabolic pathways. Lithoautotrophs are exclusively microbes; macrofauna do not possess the capability to use mineral sources of energy. Most lithoautotrophs belong to the domain Bacteria, while some belong to the domain Archaea. For lithoautotrophic bacteria, only inorganic molecules can be used as energy sources. The term "Lithotroph" is from Greek lithos (λίθος) meaning "rock" and trōphos (τροφοσ) meaning "consumer"; literally, it may be read "eaters of rock". Many lithoautotrophs are extremophiles, but this is not universally so.

Lithoautotrophs are extremely specific in using their energy source. Thus, despite the diversity in using inorganic molecules in order to obtain energy that lithoautotrophs exhibit as a group, one particular lithoautotroph would use only one type of inorganic molecule to get its energy.

Mesotrophic soil

Mesotrophic soils are soils with a moderate inherent fertility. An indicator of soil fertility is its base status, which is expressed as a ratio relating the major nutrient cations (calcium, magnesium, potassium and sodium) found there to the soil's clay percentage. This is commonly expressed in hundredths of a mole of cations per kilogram of clay, i.e. cmol (+) kg−1 clay.


A mycotroph is a plant that gets all or part of its carbon, water, or nutrient supply through symbiotic association with fungi. The term can refer to plants that engage in either of two distinct symbioses with fungi:

Many mycotrophs have a mutualistic association with fungi in any of several forms of mycorrhiza. The majority of plant species are mycotrophic in this sense. Examples include Burmanniaceae.

Some mycotrophs are parasitic upon fungi in an association known as myco-heterotrophy.


An organotroph is an organism that obtains hydrogen or electrons from organic substrates. This term is used in microbiology to classify and describe organisms based on how they obtain electrons for their respiration processes. Some organotrophs such as animals and many bacteria, are also heterotrophs. Organotrophs can be either anaerobic or aerobic.

Antonym: Lithotroph, Adjective: Organotrophic.


A planktivore is an aquatic organism that feeds on planktonic food, including zooplankton and phytoplankton.

Recruitment (biology)

In biology, especially marine biology, recruitment occurs when a juvenile organism joins a population, whether by birth or immigration, usually at a stage whereby the organisms are settled and able to be detected by an observer.There are two types of recruitment: closed and open.In the study of fisheries, recruitment is "the number of fish surviving to enter the fishery or to some life history stage such as settlement or maturity".

Relative abundance distribution

In the field of ecology, the relative abundance distribution (RAD) or species abundance distribution describes the relationship between the number of species observed in a field study as a function of their observed abundance. The graphs obtained in this manner are typically fitted to a Zipf–Mandelbrot law, the exponent of which serves as an index of biodiversity in the ecosystem under study.

Resource (biology)

In Biology and Ecology, a resource is a substance or object in the environment required by an organism for normal growth, maintenance, and reproduction. Resources can be consumed by one organism and, as a result, become unavailable to another organism. For plants key resources are light, nutrients, water, and place to grow. For animals key resources are food, water, and territory.

Shelford's law of tolerance

Shelford's law of tolerance is a principle developed by American zoologist Victor Ernest Shelford in 1911. It states that an organism's success is based on a complex set of conditions and that each organism has a certain minimum, maximum, and optimum environmental factor or combination of factors that determine success.

C. The further elaboration on the theory of tolerance is credited to Ronald Good.

Points out the second limitation of Liebig's law of the minimum - that factors act in concert rather than in isolation. A low level of one factor can sometimes be partially compensated for by appropriate levels of other factors.

A corollary to this is that two factors may work synergistically (e.g. 1 + 1 = 5), to make a habitat favorable or unfavorable.

Geographic distribution of sugar maple.

It cannot tolerate average monthly high temperatures above 24–27 °C or winter temperatures below −18 °C. The western limit is determined by dryness, and this coincides with the western limits of forest vegetation in general.

Because temperature and rainfall interact to determine the availability of water, sugar maple tolerates lower annual precipitation at the edge of its northern range (by about 50 cm).

Good restated the theory of tolerance as: Each and every species is able to exist and reproduce successfully only within a definite range of environmental conditions.

The law of tolerance, or theory of tolerance, is best illustrated by a bell shaped curve.

The range of the optimum.

Tolerance ranges are not necessarily fixed. They can change as:

Seasons change.

Environmental conditions change.

Life stage of the organism changes.

Example – blue crabs. The eggs and larvae require higher salinity than adults.

The range of the optimum may differ for different processes within the same organism.

Photosynthesis and growth in the pea plant

Food webs
Example webs
Ecology: Modelling ecosystems: Other components

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