# 李比希最低量定律

## 應用

### 科學應用

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.[2]

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 ${\displaystyle P}$.

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

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

## 李比希桶

Dobenecks[5]使用了一個木桶來描述李比希最低量定律，這個木桶也被稱為李比希桶。一個由長度不等木板做成的的木桶，其容量受到最短的木板的限制，因此植物的生長也受到最缺少的養分的限制。

## 生物技術

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.[9] 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.

## 參考資料

1. ^ Sinclair, Thomas R.; Park, Wayne I. Inadequacy of the Liebig Limiting‐Factor Paradigm for Explaining Varying Crop Yields. Agronomy Journal. 1993-05, 85 (3): 742–746. ISSN 0002-1962. doi:10.2134/agronj1993.00021962008500030040x （英语）.
2. ^ Sinclair, Thomas R. Limits to Crop Yield. Plants and Population: is there time?. Colloquium. Washington DC: National Academy of Sciences. 1999. ISBN 978-0-309-06427-9. doi:10.17226/9619. （原始内容存档于2011-07-03）.
3. ^ W.C. Rose (1931) Feeding Experiments页面存档备份，存于互联网档案馆）, 生物化學雜誌 94: 155–65
4. ^ Asimov, Issac. Life's Bottleneck. Fact and Fancy. Doubleday. 1962 [2021-06-01]. （原始内容存档于2021-06-17）.
5. ^ Whitson, A.R.; Walster, H.L. Soils and soil fertility. St. Paul, MN: Webb. 1912: 73. OCLC 1593332. 100. Illustration of Limiting Factors. The accompanying illustration devised by Dr. Dobenecks is intended to illustrate this principle of limiting factors.
6. Gorban, Alexander N.; Pokidysheva, Lyudmila I.; Smirnova, Elena V.; Tyukina, Tatiana A. Law of the Minimum Paradoxes. Bulletin of Mathematical Biology. 2011-09, 73 (9): 2013–2044. ISSN 0092-8240. doi:10.1007/s11538-010-9597-1 （英语）.
7. ^ D. Tilman, Resource Competition and Community Structure, Princeton University Press, Princeton, NJ (1982).
8. ^ MBA智库百科. 木桶原理. [2021-06-01]. （原始内容存档于2013-11-05）.
9. ^ Reilly, John M; Fuglie, Keith O. Future yield growth in field crops: what evidence exists?. Soil and Tillage Research. 1998-07, 47 (3-4): 275–290 [2022-02-16]. doi:10.1016/S0167-1987(98)00116-0. （原始内容存档于2022-03-03） （英语）.