Law of the Minimum Paradoxes

TL;DR

This paper explores how adaptation effects can lead to violations of Liebig's Law of the Minimum in ecosystems, revealing paradoxical outcomes where well-adapted systems do not follow the law.

Contribution

It introduces models based on adaptation energy to explain the paradoxical violations of Liebig's Law in natural and experimental ecosystems.

Findings

Adaptation tends to equalize the pressure of essential factors.

In synergistic systems, adaptation results in limitations by fewer factors.

Empirical data support the theoretical models and interdisciplinary applications.

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) 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.