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.
We study the dynamics of correlation and variance in systems under the load of environmental factors. A universal effect in ensembles of similar systems under the load of similar factors is described: in crisis, typically, even before obvious symptoms of crisis appear, correlation increases, and, at the same time, variance (and volatility) increases too. This effect is supported by many experiments and observations of groups of humans, mice, trees, grassy plants, and on financial time series.A general approach to the explanation of the effect through dynamics of individual adaptation of similar non-interactive individuals to a similar system of external factors is developed. Qualitatively, this approach follows Selye's idea about adaptation energy. (Alexander N. Gorban), seleval2008@yandex.ru (Elena V. Smirnova), tt51@le.ac.uk (Tatiana A. Tyukina).
a b s t r a c tWe study the dynamics of correlation and variance in systems under the load of environmental factors. A universal effect in ensembles of similar systems under the load of similar factors is described: in crisis, typically, even before obvious symptoms of crisis appear, correlation increases, and, at the same time, variance (and volatility) increases too. This effect is supported by many experiments and observations of groups of humans, mice, trees, grassy plants, and on financial time series.A general approach to the explanation of the effect through dynamics of individual adaptation of similar non-interactive individuals to a similar system of external factors is developed. Qualitatively, this approach follows Selye's idea about adaptation energy.In many areas of practice, from physiology to economics, psychology, and engineering we have to analyze the behavior of groups of many similar systems, which are adapting to the same or similar environment. Groups of humans in hard living conditions (Far North city, polar expedition, or a hospital, for example), trees under the influence of anthropogenic air pollution, rats under poisoning, banks in financial crisis, enterprises in recession, and many other situations of that type provide us with plenty of important problems, problems of diagnostics and prediction.For many such situations, it was found that the correlations between individual systems are better indicators than the value of attributes. More specifically, in thousands of experiments it was shown that in crisis, typically, even before obvious symptoms of crisis appear, the correlations increase, and, at the same time, the variance (volatility) increases too (Fig. 1).On the other hand, situations with inverse behavior were predicted theoretically and found experimentally [1]. For some systems, it was demonstrated that after the crisis achieves its bottom, it can develop into two directions: recovering (both the correlations and the variance decrease) or fatal catastrophe (the correlations decrease, but the variance continues to increase) (Fig. 1). This makes the problem more intriguing.If we look only on the state but not on the history then the only difference between comfort and disadaptation in this scheme is the value of variance: in the disadaptation state the variance is larger and the correlations in both cases are low. Qualitatively, the typical behavior of an ensemble of similar systems, which are adapting to the same or similar environment looks as follows:• In a well-adapted state, the deviations of the systems' state from the average value have relatively low correlations; • Under increasing of the load of environmental factors some of the systems leave the low correlated comfort cloud and form a low-dimensional highly correlated group (an order parameter appears). With further increasing of the load more * Corresponding address: Centre for Mathematical Modelling,
In 1938, Selye proposed the notion of adaptation energy and published 'Experimental evidence supporting the conception of adaptation energy.' Adaptation of an animal to different factors appears as the spending of one resource. Adaptation energy is a hypothetical extensive quantity spent for adaptation. This term causes much debate when one takes it literally, as a physical quantity, i.e. a sort of energy. The controversial points of view impede the systematic use of the notion of adaptation energy despite experimental evidence. Nevertheless, the response to many harmful factors often has general non-specific form and we suggest that the mechanisms of physiological adaptation admit a very general and nonspecific description. We aim to demonstrate that Selye׳s adaptation energy is the cornerstone of the top-down approach to modelling of non-specific adaptation processes. We analyze Selye׳s axioms of adaptation energy together with Goldstone׳s modifications and propose a series of models for interpretation of these axioms. Adaptation energy is considered as an internal coordinate on the 'dominant path' in the model of adaptation. The phenomena of 'oscillating death' and 'oscillating remission' are predicted on the base of the dynamical models of adaptation. Natural selection plays a key role in the evolution of mechanisms of physiological adaptation. We use the fitness optimization approach to study of the distribution of resources for neutralization of harmful factors, during adaptation to a multifactor environment, and analyze the optimal strategies for different systems of factors.
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