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Law of the Minimum Paradoxes

Authors
  • Gorban, Alexander N.1
  • Pokidysheva, Lyudmila I.2
  • Smirnova, Elena V.2
  • Tyukina, Tatiana A.1
  • 1 University of Leicester, Centre for Mathematical Modelling, Leicester, LE1 7RH, UK , Leicester (United Kingdom)
  • 2 Siberian Federal University, Krasnoyarsk, 660041, Russia , Krasnoyarsk (Russia)
Type
Published Article
Journal
Bulletin of Mathematical Biology
Publisher
Springer-Verlag
Publication Date
Nov 19, 2010
Volume
73
Issue
9
Pages
2013–2044
Identifiers
DOI: 10.1007/s11538-010-9597-1
Source
Springer Nature
Keywords
License
Yellow

Abstract

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.

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