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 controlled not by the total amount of resources available, but by the scarcest resource (limiting factor).

Applications

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."

Example

Liebig's barrel

Dobenecks^[1] 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.

Scientific applications

Liebig's Law has been extended to biological populations (and is commonly used in ecosystem models). 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.^[2]

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

\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)

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

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 multigenerational approach to resource consumption (See:sustainability/sustainable development).

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.
— Isaac Asimov, "Life's Bottleneck", Fact and Fancy

Where no substitutes exist, such as phosphorus, recycling will be necessary. This may require careful long-term planning and government 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.

Biotechnology

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.^[3] 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.

References

↑ 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.
↑ Sinclair, Thomas R. (1999). "Limits to Crop Yield". Plants and Population: is there time?. Colloquium. Washington DC: National Academy of Sciences.
↑ 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.

Modelling ecosystems – trophic components

General	Abiotic component Abiotic stress Behaviour Biogeochemical cycle Biomass Biotic component Biotic stress Carrying capacity Competition Ecosystem Ecosystem ecology Ecosystem model Keystone species List of feeding behaviours Metabolic theory of ecology Productivity Resource

Producers	Autotrophs Chemosynthesis Chemotrophs Foundation species Mixotrophs Myco-heterotrophy Mycotroph Organotrophs Photoheterotrophs Photosynthesis Photosynthetic efficiency Phototrophs Primary nutritional groups Primary production

Consumers	Apex predator Bacterivore Carnivores Chemoorganotroph Foraging Generalist and specialist species Intraguild predation Herbivores Heterotroph Heterotrophic nutrition Insectivore Mesopredator release hypothesis Omnivores Optimal foraging theory Predation Prey switching

Decomposers	Chemoorganoheterotrophy Decomposition Detritivores Detritus

Microorganisms	Archaea Bacteriophage Environmental microbiology Lithoautotroph Lithotrophy Microbial cooperation Microbial ecology Microbial food web Microbial intelligence Microbial loop Microbial mat Microbial metabolism Phage ecology

Food webs	Biomagnification Ecological efficiency Ecological pyramid Energy flow Food chain Trophic level

Example webs	Cold seeps Hydrothermal vents Intertidal Kelp forests Lakes North Pacific Subtropical Gyre Rivers San Francisco Estuary Soil Tide pool

Processes	Ascendency Bioaccumulation Cascade effect Climax community Competitive exclusion principle Consumer-resource systems Copiotrophs Dominance Ecological network Ecological succession Energy quality Energy Systems Language f-ratio Feed conversion ratio Feeding frenzy Mesotrophic soil Nutrient cycle Oligotroph Paradox of the plankton Trophic cascade Trophic mutualism Trophic state index

Defense/counter	Animal coloration Antipredator adaptations Camouflage Deimatic behaviour Herbivore adaptations to plant defense Mimicry Plant defense against herbivory Predator avoidance in schooling fish

Modelling ecosystems – other components

Population ecology	Abundance Allee effect Depensation Ecological yield Effective population size Intraspecific competition Logistic function Malthusian growth model Maximum sustainable yield Overpopulation in wild animals Overexploitation Population cycle Population dynamics Population modeling Population size Predator–prey (Lotka–Volterra) equations Recruitment Resilience Small population size Stability

Species	Biodiversity Density-dependent inhibition Ecological effects of biodiversity Ecological extinction Endemic species Flagship species Gradient analysis Indicator species Introduced species Invasive species Latitudinal gradients in species diversity Minimum viable population Neutral theory Occupancy-abundance relationship Population viability analysis Priority effect Rapoport's rule Relative abundance distribution Relative species abundance Species diversity Species homogeneity Species richness Species distribution Species-area curve Umbrella species

Species interaction	Antibiosis Biological interaction Commensalism Community ecology Ecological facilitation Interspecific competition Mutualism Storage effect

Spatial ecology	Biogeography Cross-boundary subsidy Ecocline Ecotone Ecotype Disturbance Edge effects Foster's rule Habitat fragmentation Ideal free distribution Intermediate Disturbance Hypothesis Island biogeography Landscape ecology Landscape epidemiology Landscape limnology Metapopulation Patch dynamics r/K selection theory Source–sink dynamics

Niche	Ecological niche Ecological trap Ecosystem engineer Environmental niche modelling Guild Habitat Marine habitats Limiting similarity Niche apportionment models Niche construction Niche differentiation

Other networks	Assembly rules Bateman's principle Bioluminescence Ecological collapse Ecological debt Ecological deficit Ecological energetics Ecological indicator Ecological threshold Ecosystem diversity Emergence Extinction debt Kleiber's law Liebig's law of the minimum Marginal value theorem Thorson's rule Xerosere

Other	Allometry Alternative stable state Balance of nature Biological data visualization Constructal theory Ecocline Ecological economics Ecological footprint Ecological forecasting Ecological humanities Ecological stoichiometry Ecopath Ecosystem based fisheries Endolith Evolutionary ecology Functional ecology Industrial ecology Macroecology Microecosystem Natural environment Regime shift Systems ecology Theoretical ecology

List of ecology topics

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