Centre for Mathematical Modelling, University of Leicester, Leicester, LE1 7RH, UK.
Bull Math Biol. 2011 Sep;73(9):2013-44. doi: 10.1007/s11538-010-9597-1. Epub 2010 Nov 19.
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.
“最小定律”指出，生长受最稀缺资源（限制因素）控制。这一概念最初应用于植物或作物生长（尤斯图斯·冯·李比希，1840 年，索尔兹伯里，植物生理学，第 4 版，沃兹沃思，贝尔蒙特，1992 年），并得到了许多实验的定量支持。在此基础上提出了一些基于更复杂“剂量反应”曲线的概括。也有报道称，在自然和实验生态系统中存在对这一定律的违反。我们研究了在环境因素负荷下类似生物体集合中的适应模型，并证明对李比希定律的违反源于适应效应。如果生物体在固定环境中的适合度满足最小定律，那么适应就会使必需因素的压力均等化，从而违反李比希定律。这就是最小定律悖论：如果对于一个随机选择的“生物体-环境”对，最小定律通常成立，那么在一个适应良好的系统中，我们预计会违反这一定律。对于因素的相反相互作用（相互增强的协同因素系统），适应会导致因素等同化，并受到少数因素的限制。为了分析适应，我们根据塞利的普遍适应资源（适应能量）的思想开发了一个模型系统。这些模型预测，在环境因素的负荷下，一个种群会分为两个群体（阶段）：一个相关性较小、适应良好的群体，和一个具有较大属性方差、适应存在问题的相关性较大的群体。一些实证数据被呈现出来，并讨论了跨学科应用于计量经济学的证据。
