Bulletin of Mathematical Biology

, Volume 73, Issue 9, pp 2013–2044 | Cite as

Law of the Minimum Paradoxes

  • Alexander N. GorbanEmail author
  • Lyudmila I. Pokidysheva
  • Elena V. Smirnova
  • Tatiana A. Tyukina
Original Article


The “Law of the Minimum” states that growth is controlled by the scarcest resource (limiting factor). This concept was originally applied to plant or crop growth (Justus von Liebig, 1840, Salisbury, Plant physiology, 4th edn., Wadsworth, Belmont, 1992) and quantitatively supported by many experiments. Some generalizations based on more complicated “dose-response” curves were proposed. Violations of this law in natural and experimental ecosystems were also reported. We study models of adaptation in ensembles of similar organisms under load of environmental factors and prove that violation of Liebig’s law follows from adaptation effects. If the fitness of an organism in a fixed environment satisfies the Law of the Minimum then adaptation equalizes the pressure of essential factors and, therefore, acts against the Liebig’s law. This is the the Law of the Minimum paradox: if for a randomly chosen pair “organism–environment” the Law of the Minimum typically holds, then in a well-adapted system, we have to expect violations of this law.

For the opposite interaction of factors (a synergistic system of factors which amplify each other), adaptation leads from factor equivalence to limitations by a smaller number of factors.

For analysis of adaptation, we develop a system of models based on Selye’s idea of the universal adaptation resource (adaptation energy). These models predict that under the load of an environmental factor a population separates into two groups (phases): a less correlated, well adapted group and a highly correlated group with a larger variance of attributes, which experiences problems with adaptation. Some empirical data are presented and evidences of interdisciplinary applications to econometrics are discussed.


Liebig’s Law Adaptation Fitness Stress 


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Copyright information

© Society for Mathematical Biology 2010

Authors and Affiliations

  • Alexander N. Gorban
    • 1
    Email author
  • Lyudmila I. Pokidysheva
    • 2
  • Elena V. Smirnova
    • 2
  • Tatiana A. Tyukina
    • 1
  1. 1.Centre for Mathematical ModellingUniversity of LeicesterLeicesterUK
  2. 2.Siberian Federal UniversityKrasnoyarskRussia

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