11institutetext: Centre for Mathematical Modelling, University of Leicester,
Leicester, LE1 7RH, UK<EMAIL_ADDRESS>22institutetext: Siberian
Federal University, Krasnoyarsk, 660041, Russia

# Law of the Minimum Paradoxes

Alexander N. Gorban 11 Lyudmila I. Pokidysheva 22 Elena V. Smirnova 22 Tatiana
A. Tyukina 11

###### Abstract

Keywords: Liebig s Law, Adaptation, Fitness, Stress

Keywords: Liebig s Law, Adaptation, Fitness, Stress

##  1 Introduction

###  1.1 The Law of the Minimum

The “law of the minimum” states that growth is controlled by the scarcest
resource (limiting factor) Liebig1 . This law is usually believed to be the
result of Justus von Liebig’s research (1840) but the agronomist and chemist
Carl Sprengel published in 1828 an article that contained in essence the Law
of the Minimum and this law can be called the Sprengel–Liebig Law of the
Minimum. van der Ploeg1999 .

This concept is illustrated on Fig. 1.

Figure 1:  The Law of the Minimum. Coordinates $c_{1}$, $c_{2}$ are normalized
values of factors. For a given state $s=(c_{1}(s),c_{2}(s))$, the bold solid
line $\min\\{c_{1},c_{2}\\}=\min\\{c_{1}(s),c_{2}(s)\\}$ separates the states
with better conditions (higher productivity) from the states with worse
conditions. On this line the conditions do not differ significantly from $s$
because of the same value of the limiting factor. The dot dash line shows the
border of survival. On the dashed line the factors are equally important
($c_{1}=c_{2}$).

This concept was originally applied to plant or crop growth. Many times it was
criticized, rejected, and then returned to and demonstrated quantitative
agreement with experiments Liebig1 ; Liebig2+ ; Liebig3+ .

The law of the minimum was extended then to more a general conception of
factors, not only for the elementary physical description of available
chemical substances and energy. Any environmental factor essential for life
that is below the critical minimum, or that exceeds the maximum tolerable
level could be considered as a limiting one.

There were several attempts to create a general theory of factors and
limitation in ecology, physiology and evolutionary biology. Tilman Tilman1980
proposed an equilibrium theory of resource competition based on classification
of interaction in pairs of resources. They may be: (1) essential, (2) hemi-
essential, (3) complementary, (4) perfectly substitutable, (5) antagonistic,
or (6) switching. This interaction depends on spatial heterogeneity of
resource distributions. For various resource types, the general criterion for
stable coexistence of species was developed.

Bloom, Chapin and Mooney BloomChapinMooney1985 ; ChapinSchulzeMooney1990
elaborated the economical metaphor of ecological concurrency. This analogy
allowed them to merge the optimality and the limiting approach and to
formulate four “theorems”. In particular, Theorem 3 states that a plant should
adjust allocation so that, for a given expenditure in acquiring each resource,
it achieves the same growth response: Growth is equally limited by all
resources. This is a result of adjustment: adaptation makes the limiting
factors equally important. They also studied the possibility for resources to
substitute for one another (Theorem 4) and introduced the concept of “exchange
rate”.

For human physiology the observation that adaptation makes the limiting
factors equally important was supported by many data of human adaptation to
the Far North conditions (or, which is the same, disadaptation causes
inequality of factors and leads to appearance of single limiting factor)
GorSmiCorAd1st . The theory of factors – resource interaction was developed
and supported by experimental data. The results are used for monitoring of
human populations in Far North Sedov .

In their perspectives paper, Sih and Gleeson SihGleeson1995  considered three
inter-related issues which form the core of evolutionary ecology: (1) key
environmental factors; (2) organismal traits that are responses to the key
factors; (3) the evolution of these key traits. They suggested to focus on
’limiting traits’ rather than optimal traits. Adaptation leads to optimality
and equality of traits as well as of factors but under variations some traits
should be more limiting than others. From the Sih and Gleeson point of view,
there is a growing awareness of the potential value of the limiting traits
approach as a guide for studies in both basic and applied ecology.

The critics of the Law of the Minimum is usually based on the “colimitation”
phenomenon: limitation of growth and surviving by a group of equally important
factors and traits. For example, analysis of species-specific growth and
mortality of juvenile trees at several contrasting sites suggests that light
and other resources can be simultaneously limiting, and challenges the
application of the Law of the Minimum to tree sapling growth Kobe1996 .

The concept of multiple limitation was proposed for unicellular organisms
based on the idea of the nutritional status of an organism expressed in terms
of state variables vandenBerg1998 . The property of being limiting was defined
in terms of the reserve surplus variables. This approach was illustrated by
numerical experiments.

In the world ocean there are High Nutrient–Low Chlorophyll regions where
chlorophyll concentrations are lower than expected concentrations given the
ambient phosphate and nitrate levels. In these regions, limitations of
phytoplankton growth by other nutrients like silicate or iron have been
hypothesized and supported by experiments. This colimitation was studied using
a nine-component ecosystem model embedded in the HAMOCC5 model of the oceanic
carbon cycle Aumontatal2003 .

The double–nutrient–limited growth appears also as a transition regime between
two regimes with single limiting factor. For bacteria and yeasts at a constant
dilution rate in the chemostat, three distinct growth regimes were recognized:
(1) a clearly carbon-limited regime with the nitrogen source in excess, (2) a
double–nutrient–limited growth regime where both the carbon and the nitrogen
source were below the detection limit, and (3) a clearly nitrogen-limited
growth regime with the carbon source in excess. The position of the
double–nutrient–limited zone is very narrow at high growth rates and becomes
broader during slow growth EgliZinn2003 ; ZinnWitholtEgli2004 .

Decomposition of soil organic matter is limited by both the available
substrate and the active decomposer community. The colimitation effects
strongly affect the feedbacks of soil carbon to global warming and its
consequences WutzlerReichstein2008 .

Dynamics of communities leads to colimitation on community level even if
organisms and populations remain limited by single factors. Communities are
likely to adjust their stoichiometry by competitive exclusion and coexistence
mechanisms. It guaranties simultaneous limitation by many resources and
optimal use of them at the community scale. This conclusion was supported by a
simple resource ratio model and an experimental test carried out in microcosms
with bacteria Dangeratal2008 .

In spite of the long previous discussion of colimitation, in 2008 Saito and
Goepfert stressed that this notion is “an important yet often misunderstood
concept” SaitoGoepfert2008 . They describe the potential nutrient colimitation
pairs in the marine environment and define three types of colimitation:

  1. I.

Independent nutrient colimitation concerns two elements that are generally
biochemically mutually exclusive, but are also both found in such low
concentrations as to be potentially limiting. Example: nitrogen–phosphorus
colimitation.

  2. II.

Biochemical substitution colimitation involves two elements that can
substitute for the same biochemical role within the organism. Example:
zinc–cobalt colimitation.

  3. III.

Biochemically dependent colimitation refers to the limitation of one element
that manifests itself in an inability to acquire another element. Example:
zinc–carbon colimitation.

The experimental colimitation examples of the first type do not refute the Law
of the Minimum completely but rather support the following statement: the
ecological systems of various levels, from an organism to a community, may
avoid the monolimitation regime either by the natural adjustment of their
consumption structure BloomChapinMooney1985 ; SemSem  or just by living in the
transition zone between the monolimitation regimes. From the general point of
view SihGleeson1995 , such a transition zone is expected to be quite narrow
(as a vicinity of a surface where factors are equal) but in some specific
situations it may be broad, for example, for slow growth regimes in the
chemostat EgliZinn2003 ; ZinnWitholtEgli2004 .

The type II and type III colimitations should be carefully separated from the
usual discussion of the Law of the Minimum limitation. For these types of
colimitation, two (or more) nutrients limit growth rates simultaneously,
either through the effect of biochemical substitution (type II) or by
depressing the ability for the uptake of another nutrient (type III)
SaitoGoepfert2008 . The type II and type III colimitations give us examples of
the “non-Liebig” organization of the system of factors.

The Law of the Minimum is one of the most important tools for mathematical
modeling of ecological systems. It gives a clue for making of the first model
for multi-component and multi-factor systems. This clue sounds rather simple:
first of all, we have to take into account the most important factors which
are, probably, limiting factors. Everything else should be excluded and return
back only in a case when a ’sufficient reason’ is proved (following the famous
“Principle of Sufficient Reason” by Leibnitz, one of the four recognized laws
of thought).

It is suggested to consider the Liebig production function as the “archetype”
for ecological modeling Nijlandatal2008 . The generalizations of the Law of
the Minimum were supported by the biochemical idea of limiting reaction steps
(see, for example, Brown  or recent review GorRadLim ). Three classical
production functions, the Liebig, Mitscherlich and Liebscher relations between
nutrient supply and crop production, are limiting cases of an integrated model
based on the Michaelis–Menten kinetic equation Nijlandatal2008 .

Applications of the Law of the Minimum to the ecological modeling are very
broad. The quantitative theories of the bottom–up control of the phytoplankton
dynamics is based on the influence of limiting nutrients on growth and
reproduction. The most used is the Droop model and its generalizations
Droop1973 ; LegovicCruzado1997 ; Ballantyneatal2008 .

The Law of the Minimum was combined with the evolutionary dynamics to analyze
the “Paradox of the plankton” Shoreshatal2008  formulated by Hutchinson
Hutchinson1961  in 1961: How it is possible for a number of species to coexist
in a relatively isotropic or unstructured environment all competing for the
same sorts of materials… According to the principle of competitive exclusion…
we should expect that one species alone would outcompete all of the others. It
was shown that evolution exacerbates the paradox and it is now very far from
the resolution.

The theory of evolution from monolimitation toward colimitation was developed
that takes into account the viruses attacks on the phytoplankton receptors
Menge2009 . In the classic theory Tilman1982 , evolution toward colimitation
decreases equilibrium resource concentrations and increases equilibrium
population density. In contrary, under influence of viruses, evolution toward
co-limitation may have no effect on equilibrium resource concentrations and
may decreases the equilibrium population density Menge2009 .

The Law of the Minimum was used for modeling of microcolonial fungi growth on
rock surfaces Chertovatal2004 . The analysis demonstrated, that a continued
lack of organic nutrition is a dominating environmental factor limiting growth
on stone monuments and other exposed rock surfaces in European temperate and
Mediterranean climate.

McGill McGill2005  developed a model of coevolution of mutualisms where one
resource is traded for another resource. The mechanism is based on the Law of
the Minimum in combination with Tilman’s approach to resource competition
Tilman1980 ; Tilman1982 . It was shown that resource limitations cause
mutualisms to have stable population dynamics.

The Law of the Minimum produces the piecewise linear growth functions which
are non-smooth and very far from being linear. This nonlinearity transforms
normal or uniform distributions of resource availabilities into skewed crop
yield distribution and no natural satisfactory motivation exists in favor of
any simple crop yield distribution Hennessy2009 . With independent, identical,
uniform resource availability distributions the yield skew is positive, and it
is negative for normal distributions.

The standard linear tools of statistics such as generalized linear models do
not work satisfactory for systems with limiting factors. Conventional
correlation analysis conflicts with the concept of limiting factors. This was
demonstrated in a study of the spatial distribution of Glacier lily in
relation to soil properties and gopher disturbance Thomsonatal1996 . For
systems with limiting factors, quantile regression performs much better. It
has strong theoretical justification in Law of the Minimum Austin2007 .

Some of the generalizations of the Law of the Minimum went quite far from
agriculture and ecology. The law of the minimum was applied to economics
EcolEcon  and to education, for example EcolEdu .

Recently, a strong mathematical background was created for the Law of the
Minimum. Now the limiting factors theory together with static and dynamic
limitation in chemical kinetics GorRadLim ; GorbanRadZin2010  are considered
as the realization of the Maslov dequantization KoMa97 ; LitvinovMaslov2005 ;
Litvinov2007  and idempotent analysis. Roughly speaking, the limiting factor
formalism means that we should handle any two quantities $c_{1},c_{2}$ either
as equal numbers or as numbers connected by the relation $\gg$: either
$c_{1}\gg c_{2}$ or $c_{1}\ll c_{2}$. Such a hard non-linearity can arise in
the smooth dynamic models because of the time-scale separation vandenBerg1998
.

Dequantization of the traditional mathematics leads to a mathematics over
tropical algebras like the max-plus algebra. Since the classical work of
Kleene Kleene  these algebras are intensively used in mathematics and computer
science, and the concept of dequantization and idempotent analysis opened new
applications in physics and other natural sciences (see the comprehensive
introduction in Litvinov2007 ). Liebig’s and anti-Liebig’s (see Definition 1
below) systems of factors may be considered as realizations of max-plus or
min-plus asymptotics correspondingly.

###  1.2 Fitness Convexity, Concavity and Various Interactions Between Factors

There exist an opposite type of organization of the system of factors, which,
from the first glance, seems to be symmetric to Liebig’s type of interaction
between them. In Liebig’s systems, the factor with the worst value determines
the growth and surviving. The completely opposite situation is: the factor
with the best value determines everything. We call such a system “anti-
Liebig’s” one. Of course, it seems improbable that all the possible factors
interact following the Law of the Minimum or the fully opposite anti-Liebig’s
rule. Interactions between factors in real systems are much more complicated
SaitoGoepfert2008 . Nevertheless, we can state a question about hierarchical
decomposition of the system of factors in elementary groups with simple
interactions inside, then these elementary groups can be clustered into
super–factors with simple interactions between them, and so on.

Let us introduce some notions and notations. We consider organisms that are
under the influence of several factors $F_{1},...F_{q}$. Each factor has its
intensity $f_{i}$ ($i=1,...q$). For convenience, we consider all these factors
as negative or harmful. This is just a convention about the choice of axes
directions: a wholesome factor is just a “minus harmful” factor.

At this stage, we do not specify the nature of these factors. Formally, they
are just inputs in the adaptation dynamics, the arguments of the fitness
functions.

The fitness function is the central notion of the evolutionary and ecological
dynamics. This is a function that maps the environmental factors and traits of
the organism into the reproduction coefficient, that is, its contribution, in
offspring to its population. Fisher proposed to construct fitness as a
combination of independent individual contribution of various traits
Fisher1930 . Haldane Haldane1932  criticized the approach based traits on
independent actions of traits. Modern definitions of fitness function are
based on adaptation dynamics. For the structured populations, the fitness
should be defined through the dominant Lyapunov exponents G1984 ;
MetzNisbetGeritz1992 . In the evolutionary game theory Maynard-Smith1982 ,
payoff represents Darwinian fitness and describes how the use of the strategy
improves an animal’s prospects for survival and reproduction. Recently, the
Fisher and Haldane approaches are combined WaxmanWelch2005 : Haldane’s concern
is incorporated into Fisher’s model by allowing the intensity of selection to
vary between traits.

It is a nontrivial task to measure the fitness functions and action of
selection in nature, but now it was done for many populations and phenotypical
traits KingsolverPfennig2007 . Special statistical methods for life-history
analysis for inference of fitness and population growth are developed and
tested Shawatal2008 .

In our further analysis we do not need exact values of fitness but rather it
existence and some qualitative features.

First of all, let us consider an oversimplified situation with identical
organisms. Given phenotypical treats, fitness $W$ is a function of factors
loads: $W=W(f_{1},\ldots,f_{q})$. This assumption does not take into account
physiological adaptation that works as a protection system and modifies the
factor loads. This modification is in the focus of our analysis in the follow-
up sections, and now we neglect adaptation. The convention about axes
direction means that all the partial derivatives of $W$ are non-positive
$\partial W/\partial f_{i}\leq 0$.

By definition, for Liebig’s system of factors $W$ is a function of the worst
(maximal) factor intensity: $W=W(\max\\{f_{1},\ldots,f_{q}\\})$ (Fig. 2a) and
for anti-Liebig’s system it is the function of the best (minimal) factor
intensity $W=W(\min\\{f_{1},\ldots,f_{q}\\})$ (Fig. 2c). Such representations
as well as the usual formulation of the Law of the Minimum require special
normalization of factor intensities to compare the loads of different factors.

For Liebig’s systems of factors the superlevel sets of $W$ given by
inequalities $W\geq w_{0}$ are convex for any level $w_{0}$ in a convex domain
(Fig. 2a). For anti-Liebig’s systems of factors the sublevel sets of $W$ given
by inequalities $W\leq w_{0}$ are convex for any level $w_{0}$ in a convex
domain (Fig. 2c).

(a) Liebig’s system

(b) Generalized Liebig’s system

(c) Anti-Liebig’s system

(d) Synergistic system

Figure 2: Various types of organization of the system of factors. For a given
state $s$ the bold solid line is given by the equation $W(f_{1},f_{2})=W(s)$.
This line separates the area with higher fitness (“better conditions”) from
the line with lower fitness (“worse conditions”). In Liebig’s (a) and
generalized Liebig’s systems (b) the area of better conditions is convex, in
“anti-Liebig’s” systems (c) and the general synergistic systems (d) the area
of worse conditions is convex. The dot dash line shows the border of survival.
On the dashed line the factors are equally important ($f_{1}=f_{2}$).

These convexity properties are essential for optimization problems which arise
in the modeling of adaptation and evolution. Let us take them as definitions
of the generalized Liebig and anti-Liebig systems of factors:

Definition 1.

  * •

A system of factors is the generalized Liebig system in a convex domain $U$,
if for any level $w_{0}$ the superlevel set $\\{f\in U\ |\ W(f)\geq w_{0}\\}$
is convex (Fig. 2b).

  * •

A system of factors is the generalized anti-Liebig system in a convex domain
$U$, if for any level $w_{0}$ the sublevel set $\\{f\in U\ |\ W(f)\leq
w_{0}\\}$ is convex (Fig. 2d).

We call the generalized anti-Liebig systems of factors the synergistic systems
because this definition formalizes the idea of synergy: in the synergistic
systems harmful factors superlinear amplify each other.

(a) Liebig’s system

(b) Generalized Liebig’s system

(c) Anti-Liebig’s system

(d) Synergistic system

Figure 3: Conditional optimisation for various systems of factors. Because of
convexity conditions, fitness achieves its maximum on an interval $L$ for
Liebig’s system (a) on the diagonal (the factors are equally important), for
generalized Liebig’s systems (b) near the diagonal, for anti-Liebig’s system
(c) and for the general synergistic system (d) this maximum is one of the ends
of the interval $L$.

Conditional maximization of fitness destroys the symmetry between Liebig’s and
anti-Liebig’s systems as well as between generalized Liebig’s systems and
synergistic ones. Following the geometric approach of Tilman1980 ; Tilman1982
we illustrate this optimization on Fig. 3. The picture may be quite different
from the conditional maximization of a convex function near its minima point
(compare, for example, Figs. 3c,3d to Fig. from SihGleeson1995 ).

Individual adaptation changes the picture. In the next subsection we discuss
possible mechanism of these changes.

###  1.3 Adaptation Energy and Factor–Resource Models

The reaction of an organism to the load of a single factor may have plateaus
(intervals of tolerance considered in Shelford’s “law of tolerance”, Odum ,
Chapter 5). The dose–response curves may be nonmonotonic Colborn  or even
oscillating. Nevertheless, we start from a very simple abstract model that is
close to the usual factor analysis.

We consider organisms that are under the influence of several harmful factors
$F_{1},...F_{q}$ with intensities $f_{i}$ ($i=1,...q$). Each organism has its
adaptation systems, a “shield” that can decrease the influence of external
factors. In the simplest case, it means that each system has an available
adaptation resource, $R$, which can be distributed for the neutralization of
factors: instead of factor intensities $f_{i}$ the system is under pressure
from factor values $f_{i}-a_{i}r_{i}$ (where $a_{i}>0$ is the coefficient of
efficiency of factor $F_{i}$ neutralization by the adaptation system and
$r_{i}$ is the share of the adaptation resource assigned for the
neutralization of factor $F_{i}$, $\sum_{i}r_{i}\leq R$). The zero value
$f_{i}-a_{i}r_{i}=0$ is optimal (the fully compensated factor), and further
compensation is impossible and senseless.

For unambiguity of terminology, we use the term “factor” for all factors
including any deficit of available external resource or even some illnesses.
We keep the term “resource” for internal resources, mostly for the
hypothetical Selye’s “adaptation energy”.

We represent the organisms, which are adapting to stress, as the systems which
optimize distribution of available amount of a special adaptation resource for
neutralization of different aggressive factors (we consider the deficit of
anything needful as a negative factor too). These factor–resource models with
optimization are very convenient for the modeling of adaptation. We use a
class of models many factors – one resource.

Interaction of each system with a factor $F_{i}$ is described by two
quantities: the factor $F_{i}$ pressure $\psi_{i}=f_{i}-a_{i}r_{i}$ and the
resource $r_{i}$ assigned to the factor $F_{i}$ neutralization. The first
quantity characterizes, how big the uncompensated harm is from that factor,
the second quantity measures, how intensive is the adaptation answer to the
factor (or how far the system was modified to answer the factor $F_{i}$
pressure).

Already one factor–one resource models of adaptation produce the tolerance
law. We demonstrate below that it predicts the separation of groups of
organism into two subgroups: the less correlated well–adapted organisms and
highly correlated organisms with a deficit of the adaptation resource. The
variance is also higher in the highly correlated group of organisms with a
deficit of the adaptation resource.

This result has a clear geometric interpretation. Let us represent each
organism as a data point in an $n$-dimensional vector space. Assume that they
fall roughly within an ellipsoid. The well-adapted organisms are not highly
correlated and after normalization of scales to unit variance the
corresponding cloud of points looks roughly as a sphere. The organisms with a
deficit of the adaptation resource are highly correlated, hence in the same
coordinates their cloud looks like an ellipsoid with remarkable eccentricity.
Moreover, the largest diameter of this ellipsoid is larger than for the
well–adapted organisms and the variance increases together with the
correlations.

This increase of variance together with correlations may seem counterintuitive
because it has no formal backgrounds in definitions of the correlation
coefficients and variance. This is an empirical founding that under stress
correlations and variance increase together, supported by many observations
both for physiological and financial systems. The factor-resource models give
a plausible explanation of this phenomena.

The crucial question is: what is the resource of adaptation? This question
arose for the first time when Selye published the concept of adaptation energy
and experimental evidence supporting this idea SelyeAEN ; SelyeAE1 . Selye
found that the organisms (rats) which demonstrate no differences in normal
environment may differ significantly in adaptation to an increasing load of
environmental factors. Moreover, when he repeated the experiments, he found
that adaptation ability decreases after stress. All the observations could be
explained by existence of an universal adaptation resource that is being spent
during all adaptation processes.

Selye’s ideas allow the following interpretation: the aggressive influence of
the environment on the organism may be represented as action of independent
factors. The system of adaptation consists of subsystems, which protect the
organism from different factors. These subsystems consume the same resource,
the adaptation energy. The distribution of this resource between the
subsystems depends on environmental conditions.

Later the concept of adaptation energy was significantly improved GP_AE1952 ,
plenty of indirect evidence supporting this concept were found, but this
elusive adaptation energy is still a theoretical concept, and in the modern
“Encyclopedia of Stress” we read: “As for adaptation energy, Selye was never
able to measure it…” AEencicl . Nevertheless, the notion of adaptation energy
is very useful in the analysis of adaptation and is now in wide use (see, for
example, BreznitzAEappl ; SchkadeOccAdAE2003 ). It should be specially
stressed that the adaptation energy is neither physical energy nor a
substance. This idealization describes the experimental results: in many
experiments it was demonstrated, that organisms under load of various factors
behave as if they spend a recourse, which is the same for different factors.
This recourse may be exhausted and then the organism dies.

The idea of exchange can help in the understanding of adaptation energy: there
are many resources, but any resource can be exchanged for another one. To
study such an exchange an analogy with the currency exchange is useful.
Following this analogy, we have to specify, what is the exchange rate, how
fast this exchange could be done (what is the exchange time), what is the
margin, how the margin depends on the exchange time. There may appear various
limitations of the amount of the exchangeable resource, and so on. The
economic metaphor for ecological concurrency and adaptation was elaborated in
1985 BloomChapinMooney1985 ; ChapinSchulzeMooney1990  but much earlier, in
1952, it was developed for physiological adaptation GP_AE1952 .

Market economics seems closer to the idea of resource universalization than
biology is, but for biology this exchange idea also seems useful. Of course
there exist some limits on the possible exchanges of different resources. It
is possible to include the exchange processes into models, but many questions
appear immediately about unknown coefficients. Nevertheless, we can follow
Selye’s arguments and postulate the adaptation energy as a universal
adaptation resource.

The adaptation energy is neither physical energy nor a substance. This is a
theoretical construction, which may be considered as a pool of various
exchangeable resources. When an organism achieves the limits of resource
exchangeability, the universal non-specific stress and adaptation syndrome
transforms (disintegrates) into specific diseases. Near this limit we have to
expect the critical retardation of exchange processes.

Adaptation optimizes the state of the system for given available amounts of
the adaptation resource. This idea seems very natural, but it may be a
difficult task to find the objective function that is hidden behind the
adaptation process. Nevertheless, even an assumption about the existence of an
objective function and about its general properties helps in analysis of
adaptation process.

Assume that adaptation should maximize a fitness function $W$ which depends on
the compensated values of factors, $\psi_{i}=f_{i}-a_{i}r_{i}$ for the given
amount of available resource:

$\left\\{\begin{array}[]{l}W(f_{1}-a_{1}r_{1},f_{2}-a_{2}r_{2},...f_{q}-a_{q}r_{q})\
\to\ \max\ ;\\\ r_{i}\geq 0$, $f_{i}-a_{i}r_{i}\geq 0$,
$\sum_{i=1}^{q}r_{i}\leq R\ .\end{array}\right.$ (1)

The only question is: how can we be sure that adaptation follows any
optimality principle? Existence of optimality is proven for microevolution
processes and ecological succession. The mathematical backgrounds for the
notion of “natural selection” in these situations are well–established after
work by Haldane (1932) Haldane1932  and Gause (1934) Gause . Now this
direction with various concepts of fitness (or “generalized fitness”)
optimization is elaborated in many details (see, for example, review papers
Bom02 ; Oechssler02 ; GorbanSelTth ).

The foundation of optimization is not so clear for such processes as
modifications of a phenotype, and for adaptation in various time scales. The
idea of genocopy–phenocopy interchangeability was formulated long ago by
biologists to explain many experimental effects: the phenotype modifications
simulate the optimal genotype (West-Eberhardgenocopy-phenocopy , p. 117). The
idea of convergence of genetic and environmental effects was supported by an
analysis of genome regulation ZuckerkandlConvergGenEnv  (the principle of
concentration–affinity equivalence). The phenotype modifications produce the
same change, as evolution of the genotype does, but faster and in a smaller
range of conditions (the proper evolution can go further, but slower). It is
natural to assume that adaptation in different time scales also follows the
same direction, as evolution and phenotype modifications, but faster and for
smaller changes. This hypothesis could be supported by many biological data
and plausible reasoning. (See, for example, the case studies of relation
between evolution of physiological adaptation Hoffman1978 ; Greene1999 , a
book about various mechanisms of plants responses to environmental stresses
Lerner1999 , a precise quantitative study of the relationship between
evolutionary and physiological variation in hemoglobin Miloatal2007  and a
modern review with case studies FuscoMinelli2010 .)

It may be a difficult task to find an explicit form of the fitness function
$W$, but for our qualitative analysis we need only a qualitative assumption
about general properties of $W$. First, we assume monotonicity with respect to
each coordinate:

$\frac{\partial W(\psi_{1},\ldots\psi_{q})}{\partial\psi_{i}}\leq 0\,.$ (2)

A system of factors is Liebig’s system, if

$W=W\left(\max_{1\leq i\leq q}\\{f_{i}-a_{i}r_{i}\\}\right)\ .$ (3)

This means that fitness depends on the worst factor pressure.

A system of factors is generalized Liebig’s system, if for any two different
vectors of factor pressures $\mathbf{\psi}=(\psi_{1},...\psi_{q})$ and
$\mathbf{\phi}=(\phi_{1},...\phi_{q})$ ($\mathbf{\psi}\neq\mathbf{\phi}$) the
value of fitness at the average point $(\mathbf{\psi}+\mathbf{\phi})/2$ is
greater, than at the worst of points $\mathbf{\psi}$, $\mathbf{\phi}$:

$W\left(\frac{\mathbf{\psi}+\mathbf{\phi}}{2}\right)>\min\\{W(\mathbf{\psi}),W(\mathbf{\phi})\\}\
.$ (4)

Any Liebig’s system is, at the same time, generalized Liebig’s system because
for such a system the fitness $W$ is a decreasing function of the maximal
factor pressure, the minimum of $W$ corresponds to the maximal value of the
limiting factor and

$\max\left\\{\frac{\psi_{1}+\phi_{1}}{2},\ldots,\frac{\psi_{q}+\phi_{q}}{2}\right\\}\leq\max\\{\max\\{\psi_{1},\ldots,\psi_{q}\\},\max\\{\phi_{1},\ldots,\phi_{q}\\}\\}\
.$

The opposite principle of factor organization is synergy: the superlinear
mutual amplification of factors. The system of factors is a synergistic one,
if for any two different vectors of factor pressures
$\mathbf{\psi}=(\psi_{1},...\psi_{q})$ and
$\mathbf{\phi}=(\phi_{1},...\phi_{q})$ ($\mathbf{\psi}\neq\mathbf{\phi}$) the
value of fitness at the average point $(\mathbf{\psi}+\mathbf{\phi})/2$ is
less, than at the best of points $\mathbf{\psi}$, $\mathbf{\phi}$:

$W\left(\frac{\mathbf{\psi}+\mathbf{\phi}}{2}\right)<\max\\{W(\mathbf{\psi}),W(\mathbf{\phi})\\}\
.$ (5)

A system of factors is anti-Liebig’s system, if

$W=W\left(\min_{1\leq i\leq q}\\{f_{i}-a_{i}r_{i}\\}\right)\ .$ (6)

This means that fitness depends on the best factor pressure. Any anti-Liebig
system is, at the same time a synergistic one because for such a system the
fitness $W$ is a decreasing function of the minimal factor pressure, the
maximum of $W$ corresponds to the minimal value of the factor with minimal
pressure and

$\min\left\\{\frac{\psi_{1}+\phi_{1}}{2},\ldots,\frac{\psi_{q}+\phi_{q}}{2}\right\\}\geq\min\\{\min\\{\psi_{1},\ldots,\psi_{q}\\},\min\\{\phi_{1},\ldots,\phi_{q}\\}\\}$

We prove that adaptation of an organism to Liebig’s system of factors, or to
any synergistic system, leads to two paradoxes of adaptation:

  * •

Law of the Minimum paradox: If for a randomly selected pair, ( State of
environment – State of organism ), the Law of the Minimum is valid (everything
is limited by the factor with the worst value) then, after adaptation, many
factors (the maximally possible amount of them) are equally important.

  * •

Law of the Minimum inverse paradox: If for a randomly selected pair, ( State
of environment – State of organism ), many factors are equally important and
superlinearly amplify each other then, after adaptation, a smaller amount of
factors is important (everything is limited by the factors with the worst non-
compensated values, the system approaches the Law of the Minimum).

In this paper, we discuss the individual adaptation. Other types of
adaptations, such as changes of the ecosystem structure, ecological succession
or microevolution lead to the same paradoxes if the factor–resource models are
applicable to these processes.

##  2 One-Factor Models, the Law of Tolerance, and the Order–Disorder
Transition

The question about interaction of various factors is very important, but,
first of all, let us study the one-factor models. Each organism is
characterized by measurable attributes $x_{1},\ldots x_{m}$ and the value of
adaptation resource, $R$.

###  2.1 Tension–Driven Models

In these models, observable properties of interest $x_{k}$ $(k=1,...m)$ can be
modeled as functions of factor pressure $\psi$ plus some noise $\epsilon_{k}$.

Let us consider one-factor systems and linear functions (the simplest case).
For the tension–driven model the attributes $x_{k}$ are linear functions of
tension $\psi$ plus noise:

$x_{k}=\mu_{k}+l_{k}\psi+\epsilon_{k}\ ,$ (7)

where $\mu_{k}$ is the expectation of $x_{k}$ for fully compensated factor,
$l_{k}$ is a coefficient, $\psi=f-ar_{f}\geq 0$, and $r_{f}\leq R$ is amount
of available resource assigned for the factor neutralization. The values of
$\mu_{k}$ could be considered as “normal” (in the sense opposite to
“pathology”), and noise $\epsilon_{k}$ reflects variability of norm.

If systems compensate as much of factor value, as it is possible, then
$r_{f}=\min\\{R,f/a\\}$, and we can write:

$\psi=\left\\{\begin{array}[]{ll}&f-aR\ ,\ \ {\rm if}\ \ f>aR\ ;\\\ &0,\ \ \
{\rm else.}\end{array}\right.$ (8)

Individual systems may be different by the value of factor intensity (the
local intensity variability), by amount of available resource $R$ and, of
course, by the random values of $\epsilon_{k}$. If all systems have enough
resource for the factor neutralization ($aR>f$) then all the difference
between them is in the noise variables $\epsilon_{k}$. Nobody will observe any
change under increase of the factor intensity, until violation of inequality
$F<r$.

Let us define the dose–response curve as

$M_{k}(f)=\mathbf{E}(x_{k}|f).$

Due to (7)

$M_{k}(f)=\mu_{k}+l_{k}\mathbf{P}(aR<f)(f-a\mathbf{E}(R|aR<f))\ ,$ (9)

where $\mathbf{P}(aR<f)$ is the probability of organism to have insufficient
amount of resource for neutralization of the factor load and
$\mathbf{E}(R|aR<f)$ is the conditional expectation of the amount of resource
if it is insufficient.

The slope $\mathrm{d}M_{k}(f)/\mathrm{d}f$ of the dose–response curve (9) for
big values of $f$ tends to $l_{k}$, and for small $f$ it could be much
smaller. This plateau at the beginning of the dose-response curve corresponds
to the law of tolerance (Victor E. Shelford, 1913, Odum , Chapter 5).

If the factor value increases, and for some of the systems the factor
intensity $f$ exceeds the available compensation $aR$ then for these systems
$\psi>0$ and the term $l_{k}\psi$ in Eq. (7) becomes important. If the noise
of the norm $\epsilon_{k}$ is independent of $\psi$ then the correlation
between different $x_{k}$ increases monotonically with $f$.

With increase of the factor intensity $f$ the dominant eigenvector of the
correlation matrix between $x_{k}$ becomes more uniform in the coordinates,
which tend asymptotically to $\pm\frac{1}{\sqrt{m}}$.

For a given value of the factor intensity $f$ there are two groups of
organisms: the well–adapted group with $R\geq f$ and $\psi=0$, and the group
of organisms with deficit of adaptation energy and $\psi>0$. If the
fluctuations of norm $\epsilon_{k}$ are independent for different $k$ (or just
have small correlation coefficients), then in the group with deficit of
adaptation energy the correlations between attributes is much bigger than in
the well–adapted group. If we use metaphor from physics, we can call this two
groups two phases: the highly correlated phase with deficit of adaptation
energy and the less correlated phase of well-adapted organisms.

In this simple model (7) we just formalize Selye’s observations and
theoretical argumentation. One can call it Selye’s model. There are two other
clear possibilities for one factor–one resource models.

###  2.2 Response–Driven Models

What is more important for values of the observable quantities $x_{k}$: the
current pressure of the factors, or the adaptation to this factor which
modified some of parameters? Perhaps, both, but let us introduce now the
second simplest model.

In the response–driven model of adaptation, the quantities $x_{k}$ are modeled
as functions of adaptive response $ar_{f}$ plus some noise $\epsilon_{k}$:

$x_{k}=\mu_{k}+q_{k}ar_{f}+\epsilon_{k}\ .$ (10)

When $f$ increases then, after threshold $f=aR$, the term $l_{k}ar_{f}$
transforms into $l_{k}aR$ and does not change further. The observable
quantities $x_{k}$ are not sensitive to changes in the factor intensity $f$
when $f$ is sufficiently large. This is the significant difference from the
behavior of the tension-driven model (7), which is not sensitive to change of
$f$ when $f$ is sufficiently small.

###  2.3 Tension–and–Response Driven 2D One–Factor Models.

This model is just a linear combination of Eqs. (7) and (10)

$x_{k}=\mu_{k}+l_{k}\psi+q_{k}ar_{f}+\epsilon_{k}\ .$ (11)

For small $f$ (comfort zone) $\psi=0$, the term $l_{k}\psi$ vanishes,
$ar_{f}=f$ and the model has the form $x_{k}=\mu_{k}+q_{k}f+\epsilon_{k}$. For
intermediate level of $f$, if systems with both signs of inequality
$f\gtreqless aR$ are present, the model imitates 2D (two-factor) behavior.
After the threshold $f\geq aR$ is passed for all systems, the model
demonstrates 1D behavior again:
$x_{k}=\mu_{k}+l_{k}f+(q_{k}-l_{k})aR+\epsilon_{k}\ .$ For small $f$ the
motion under change of $f$ goes along direction $q_{k}$, for large $f$ it goes
along direction $l_{k}$.

Already the first model of adaptation (7) gives us the law of tolerance and
practically important effect of order–disorder transition under stress. Now we
have no arguments for decision which of these models is better, but the second
model (10) has no tolerance plateau for small factor values, and the third
model has almost two times more fitting parameters. Perhaps, the first choice
should be the first model (7), with generalization to (11), if the described
two-dimensional behaviour is observed.

##  3 Law of the Minimum Paradox

Liebig used the image of a barrel – now called Liebig’s barrel – to explain
his law. Just as the capacity of a barrel with staves of unequal length is
limited by the shortest stave, so a plant’s growth is limited by the nutrient
in shortest supply.

Adaptation system acts as a cooper and repairs the shortest stave to improve
the barrel capacity. Indeed, in well-adapted systems the limiting factor
should be compensated as far as this is possible. It seems obvious because of
very natural idea of optimality, but arguments of this type in biology should
be considered with care.

Assume that adaptation should maximize a objective function $W$ (1), which
satisfies the Law of the Minimum (3 and the monotonicity requirement (2) under
conditions $r_{i}\geq 0$, $f_{i}-a_{i}r_{i}\geq 0$, $\sum_{i=1}^{q}r_{i}\leq
R$. (Let us remind that $f_{i}\geq 0$ for all $i$.)

Description of the maximizers of $W$ gives the following theorem (the proof is
a straightforward consequence of the Law of the Minimum and monotonicity of
$W$).

Theorem 1. For any objective function $W$ that satisfies conditions (3) the
optimizers $r_{i}$ are defined by the following algorithm.

  1. 1.

Re-enumerate factors in the order of their intensities: $f_{1}\geq
f_{2}\geq...f_{q}$.

  2. 2.

Calculate differences $\Delta_{j}=f_{j}-f_{j+1}$ (take formally
$\Delta_{0}=\Delta_{q+1}=0$).

  3. 3.

Find such $k$ ($0\leq k\leq q$) that

$\sum_{j=1}^{k}\left(\sum_{p=1}^{j}\frac{1}{a_{p}}\right)\Delta_{j}\leq
R\leq\sum_{j=1}^{k+1}\left(\sum_{p=1}^{j}\frac{1}{a_{p}}\right)\Delta_{j}\ .$

For $R<\Delta_{1}$ we put $k=0$, for $R>\sum_{j=1}^{k+1}j\Delta_{j}$ we take
$k=q$.

  4. 4.

If $k<q$ then the optimal amount of resource $r_{j_{l}}$ is

$r_{l}=\left\\{\begin{array}[]{ll}&\frac{\Delta_{l}}{a_{l}}+\frac{1}{a_{l}\sum_{p=1}^{k}\frac{1}{a_{p}}}\left(R-\sum_{j=1}^{k}\left(\sum_{p=1}^{j}\frac{1}{a_{p}}\right)\Delta_{j}\right)\
,\ \ {\rm if}\ \ l\leq k+1\ ;\\\ &0\ ,\ \ \ \ \ {\rm if}\ \ l>k+1\
.\end{array}\right.$ (12)

If $k=q$ then $r_{i}=f_{i}/a_{i}$ for all $i$. $\square$

After all we have to restore the initial enumeration $f_{i_{1}}\geq
f_{i_{2}}\geq\ldots f_{i_{q}}$. This optimization is illustrated in Fig. 4.

Figure 4:  Optimal distribution of resource for neutralization of factors
under the Law of the Minimum. (a) histogram of factors intensity (the
compensated parts of factors are highlighted, $k=3$), (b) distribution of
tensions $\psi_{i}$ after adaptation becomes more uniform, (c) the sum of
distributed resources. For simplicity of the picture, we take here all
$a_{i}=1$.

Hence, if the system satisfies the law of the minimum then the adaptation
process makes more uniform the tension produced by different factors
$\psi_{i}=f_{i}-ar_{i}$ (Fig. 4). Thus adaptation decreases the effect from
the limiting factor and hides manifestations of the Law of the Minimum.

Under the assumption of optimality (1) the law of the minimum paradox becomes
a theorem: if the Law of the Minimum is true then microevolution, ecological
succession, phenotype modifications and adaptation decrease the role of the
limiting factors and bring the tension produced by different factors together.

The cooper starts to repair Liebig’s barrel from the shortest stave and after
reparation the staves are more uniform, than they were before. This cooper may
be microevolution, ecological succession, phenotype modifications, or
adaptation. For the ecological succession this effect (the Law of the Minimum
leads to its violation by succession) was described in Ref. SemSem . For
adaptation (and in general settings too) it was demonstrated in Ref.
GorSmiCorAd1st .

##  4 Law of the Minimum Inverse Paradox

The simplest formal example of “anti–Liebig’s” organization of interaction
between factors gives us the following dependence of fitness from two factors:
$W=-f_{1}f_{2}$: each of factors is neutral in the absence of another factor,
but together they are harmful. This is an example of synergy: the whole is
greater than the sum of its parts. (For our selection of axes direction,
“greater” means “more harm”.)

In according to Definition 1, the system of factors $F_{1},...F_{q}$ is
synergistic, in a convex domain $U$ of the admissible vectors of factor
pressure if for any level $w_{0}$ the sublevel set $\\{\psi\in U\ |\
W(\psi)\leq w_{0}\\}$ is convex. Another definition gives us the synergy
inequality (5). These definitions are equivalent. This proposition follows
from the definition of convexity and standard facts about convex sets (see,
for example, Rockafellar )

Proposition 1. The synergy inequality (5) holds if and only if all the
sublevel sets $\\{\mathbf{f}\ |\ W(\mathbf{f})\leq\alpha\\}$ are strictly
convex.$\square$

(The fitness itself may be a non-convex function.)

This proposition immediately implies that the synergy inequality is invariant
with respect to increasing monotonic transformations of $W$. This invariance
with respect to nonlinear change of scale is very important, because usually
we don’t know the values of function $W$.

Proposition 2. If the synergy inequality (5) holds for a function $W$, then it
holds for a function $W_{\theta}=\theta(W)$, where $\theta(x)$ is an arbitrary
strictly monotonic function of one variable.$\square$

Already this property allows us to study the problem about optimal
distribution of the adaptation resource without further knowledge about the
fitness function.

Assume that adaptation should maximize an objective function
$W(f_{1}-r_{1},...f_{q}-r_{q})$ (1) which satisfies the synergy inequality (5)
under conditions $r_{i}\geq 0$, $f_{i}-a_{i}r_{i}\geq 0$,
$\sum_{i=1}^{q}r_{i}\leq R$. (Let us remind that $f_{i}\geq 0$ for all $i$.)
Following our previous convention about axes directions all factors are
harmful and $W$ is monotonically decreasing function

$\frac{\partial W(f_{1},...f_{q})}{\partial f_{i}}<0\ .$

We need also a technical assumption that $W$ is defined on a convex set in
$\mathbb{R}^{q}_{+}$ and if it is defined for a nonnegative point
$\mathbf{f}$, then it is also defined at any nonnegative point
$\mathbf{g}\leq\mathbf{f}$ (this inequality means that $g_{i}\leq f_{i}$ for
all $i=1,...q$).

The set of possible maximizers is finite. For every group of factors
$F_{i_{1}},...F_{i_{j+1}}$, ($1\leq j+1<q$) with the property

$\sum_{k=1}^{j}\frac{f_{i_{k}}}{a_{i_{k}}}<R\leq\sum_{k=1}^{j+1}\frac{f_{i_{k}}}{a_{i_{k}}}$
(13)

we find a distribution of resource
$\mathbf{r}_{\\{{i_{1}},...{i_{j+1}}\\}}=(r_{i_{1}},...r_{i_{j+1}})$:

$r_{i_{k}}=\frac{f_{i_{k}}}{a_{i_{k}}}\ \ (k=1,...j)\ ,\ \
r_{i_{j+1}}=R-\sum_{k=1}^{j}\frac{f_{i_{k}}}{a_{i_{k}}}\ ,\ \ r_{i}=0\ \ {\rm
for}\ \ i\notin\\{{i_{1}},...{i_{j+1}}\\}\ .$ (14)

This distribution (13) means that the pressure of $j$ factors are completely
compensated and one factor is partially compensated. For $j=0$, Eq. (13) gives
$0<R\leq f_{i_{1}}$ and there exists only one nonzero component in the
distribution (14), $r_{i_{1}}=R$. For $j=q$ all $r_{i}=f_{i}/a_{i}$,
$\sum_{i}r_{i}<R$ and all factors are fully compensated.

We get the following theorem as an application of standard results about
extreme points of convex sets Rockafellar .

Theorem 2. Any maximizer for $W(f_{1}-a_{1}r_{1},...f_{q}-a_{q}r_{q})$ under
given conditions has the form $\mathbf{r}_{\\{{i_{1}},...{i_{j+1}}\\}}$
(14).$\square$

To find the optimal distribution we have to analyze which distribution of the
form (13) gives the highest fitness.

If the initial distribution of factors intensities,
$\mathbf{f}=(f_{1},...f_{q})$, is almost uniform and all factors are
significant then, after adaptation, the distribution of effective tensions,
$\mathbf{\psi}=(\psi_{1},...\psi_{q})$ ($\psi_{i}=f_{i}-a_{i}r_{i}$), is less
uniform. Following Theorem 2, some of factors may be completely neutralized
and one additional factor may be neutralized partially. This situation is
opposite to adaptation to Liebig’s system of factors, where amount of
significant factors increases and the distribution of tensions becomes more
uniform because of adaptation. For Liebig’s system, adaptation transforms low
dimensional picture (one limiting factor) into high dimensional one, and we
expect the well-adapted systems have less correlations than in stress. For
synergistic systems, adaptation transforms high dimensional picture into low
dimensional one (less factors), and our expectations are inverse: we expect
the well-adapted systems have more correlations than in stress (this situation
is illustrated in Fig. 5; compare to Fig. 4). We call this property of
adaptation to synergistic system of factors the law of the minimum inverse
paradox.

Figure 5:  Typical optimal distribution of resource for neutralization of
synergistic factors. (a) Factors intensity (the compensated parts of factors
are highlighted, $j=2$), (b) distribution of tensions $\psi_{i}$ after
adaptation becomes less uniform (compare to Fig. 4), (c) the sum of
distributed resources. For simplicity of the picture, we take here all
$a_{i}=1$.

The fitness by itself is a theoretical construction based on the average
reproduction coefficient (instant fitness). It is impossible to measure this
quantity in time intervals that are much shorter than the life length and even
for the life–long analysis it is a non-trivial problem Shawatal2008 .

In order to understand which system of factors we deal with, Liebig’s or
synergistic one, we have to compare theoretical consequences of their
properties and compare them to empirical data. First of all, we can measure
results of adaptation, and use for analysis properties of optimal adaptation
in ensembles of systems for analysis (Fig. 4, Fig. 5).

##  5 Empirical data

In many areas of practice, from physiology to economics, psychology, and
engineering we have to analyze 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 influence of anthropogenic air pollution, rats under
poisoning, banks in financial crisis, enterprizes 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, variance. After the crisis achieves its bottom, it can develop
into two directions: recovering (both correlations and variance decrease) or
fatal catastrophe (correlations decrease, but variance continue to increase).

In this Sec. we review several sets of empirical results which demonstrate
this effect. Now, after 21 years of studying of this effect GorSmiCorAd1st ;
Sedov , we maintain that it is universal for groups of similar systems that
are sustaining a stress and have an adaptation ability. On the other hand,
situations with inverse behavior were predicted theoretically and found
experimentally Mansurov . This makes the problem more intriguing.

Below, to collect information about strong correlations between many
attributes in one indicator, we evaluate the non-diagonal part of the
correlation matrix and delete terms with values below a threshold 0.5 from the
sum:

$G=\sum_{j>k,\ |r_{jk}|>\alpha}|r_{jk}|.$ (15)

This quantity $G$ is a weight of the correlation graph. The vertices of this
graph correspond to variables, and these vertices are connected by edges, if
the absolute value of the correspondent sample correlation coefficient exceeds
$alpha$: $|r_{jk}|>alpha$. Usually, we take $\alpha=0.5$ (a half of the
maximum) if there is no reason to select another value.

###  5.1 Adaptation of Adults for Change of Climatic Zone

Activity of enzymes in human leukocytes was studied Bul1Limf ; Bul2Limf . We
analyzed the short-term adaptation (20 days) of groups of healthy 20-30 year
old men who change their climate zone:

  * •

From Far North to the South resort (Sochi, Black Sea) in summer;

  * •

From the temperate belt of Russia to the South resort (Sochi, Black Sea) in
summer.

Results are represented in Fig. 6. This analysis supports the basic hypothesis
and, on the other hand, could be used for prediction of the most dangerous
periods in adaptation, which need special care.

Figure 6: Weight of the correlation graphs of activity of enzymes in
leucocytes during urgent adaptation at a resort. For people from Far North,
the adaptation crisis occurs near the 15th day.

We selected the group of 54 people who moved to Far North, that had any
illness during the period of short-term adaptation. After 6 months at Far
North, this test group demonstrates much higher correlations between activity
of enzymes than the control group (98 people without illness during the
adaptation period). We analyzed the activity of enzymes (alkaline phosphatase,
acid phosphatase, succinate dehydrogenase, glyceraldehyde-3-phosphate
dehydrogenase, glycerol- 3-phosphate dehydrogenase, and glucose-6-phosphate
dehydrogenase) in leucocytes: $G=5.81$ in the test group versus $G=1.36$ in
the control group. To compare the dimensionless variance for these groups, we
normalize the activity of enzymes to unite sample means (it is senseless to
use the trace of the covariance matrix without normalization because normal
activities of enzymes differ in order of magnitude). For the test group, the
sum of the enzyme variances is 1.204, and for the control group it is 0.388.

###  5.2 Destroying of Correlations “on the Other Side of Crisis”: Acute
Hemolytic Anemia in Mice

It is very important to understand where the system is going: (i) to the
bottom of the crisis with possibility to recover after that bottom, (ii) to
the normal state, from the bottom, or (iii) to the “no return” point, after
which it cannot recover.

This problem was studied in many situation with analysis of fatal outcomes in
oncological MansurOnco  and cardiological Strygina  clinics, and also in
special experiments with acute hemolytic anemia caused by phenylhydrazine in
mice mice . The main result here is: when approaching the no-return point,
correlations destroy ($G$ decreases), and variance typically does continue to
increase.

There exist no formal criterion to recognize the situation “on the other side
of crisis”. Nevertheless, it is necessary to select situations for testing of
our hypothesis. Here can help the “general practitioner point of view”
GP_AE1952  based on practical experience. From such a point of view, the
situation described below is on the other side of crisis: the acute hemolytic
anemia caused by phenylhydrazine in mice with lethal outcome.

Figure 7: Adaptation and disadaptation dynamics for mice after phenylhydrazine
injection.

This effect was demonstrated in special experiments mice . Acute hemolytic
anemia caused by phenylhydrazine was studied in CBAxlac mice. After
phenylhydrazine injections (60 mg/kg, twice a day, with interval 12 hours)
during first 5-6 days the amount of red cells decreased (Fig. 7), but at the
7th and 8th days this amount increased because of spleen activity. After 8
days most of the mice died. Dynamics of correlation between hematocrit,
reticulocytes, erythrocytes, and leukocytes in blood is presented in Fig. 7.
Weight of the correlation graph increase precedeed the active adaptation
response, but $G$ decreased to zero before death (Fig. 7), while amount of red
cells increased also at the last day.

###  5.3 Grassy Plants Under Trampling Load

Table 1: Weight $G$ of the correlation graph for different grassy plants under various trampling load  Grassy Plant |  Group 1 |  Group 2 |  Group 3
---|---|---|---
Lamiastrum | 1.4 | 5.2 | 6.2
Paris (quadrifolia) | 4.1 | 7.6 | 14.8
Convallaria | 5.4 | 7.9 | 10.1
Anemone | 8.1 | 12.5 | 15.8
Pulmonaria | 8.8 | 11.9 | 15.1
Asarum | 10.3 | 15.4 | 19.5

The effect exists for plants too. The grassy plants in oak tree-plants are
studied RazzhevaikinTrava1996 . For analysis the fragments of forests are
selected, where the densities of trees and bushes were the same. The
difference between those fragments was in damaging of the soil surface by
trampling. Tree groups of fragments are studied:

  * •

Group 1 – no fully destroyed soil surface;

  * •

Group 2 – 25% of soil surface are destroyed by trampling;

  * •

Group 3 – 70% of soil surface are destroyed by trampling.

The studied physiological attributes were: the height of sprouts, the length
of roots, the diameter of roots, the amount of roots, the area of leafs, the
area of roots. Results are presented in Table 1.

###  5.4 Scots Pines Near a Coal Power Station

The impact of emissions from a heat power station on Scots pine was studied
KofmantREES . For diagnostic purposes the secondary metabolites of phenolic
nature were used. They are much more stable than the primary products and hold
the information about past impact of environment on the plant organism for
longer time.

The test group consisted of Scots pines (Pinus sylvestric L) in a 40 year old
stand of the II class in the emission tongue 10 km from the power station. The
station had been operating on brown coal for 45 years. The control group of
Scots pines was from a stand of the same age and forest type, growing outside
the industrial emission area. The needles for analysis were one year old from
the shoots in the middle part of the crown. The samples were taken in spring
in bud swelling period. Individual composition of the alcohol extract of
needles was studied by high efficiency liquid chromatography. 26 individual
phenolic compounds were identified for all samples and used in analysis.

No reliable difference was found in the test group and control group average
compositions. For example, the results for Proantocyanidin content (mg/g dry
weight) were as follows:

  * •

Total 37.4$\pm$3.2 (test) versus 36.8$\pm$2.0 (control);

Nevertheless, the variance of compositions of individual compounds in the test
group was significantly higher, and the difference in correlations was huge:
$G=17.29$ for the test group versus $G=3.79$ in the control group.

##  6 Comparison to Econometrics

The simplest Selye’s model (7) seems very similar to the classical one-factor
econometrics models Campbell  which assume that the returns of stocks
($\rho_{i}$) are controlled by one factor, the “market” return $M(t)$. In this
model, for any stock

$\rho_{i}(t)=a_{i}+b_{i}M(t)+\epsilon_{i}(t)$ (16)

where $\rho_{i}(t)$ is the return of the $i$th stock at time $t$, $a_{i}$ and
$b_{i}$ are real parameters, and $\epsilon_{i}(t)$ is a zero mean noise. In
our models, the factor pressure characterizes the time window and is slower
variable than the return.

The main difference between models (7) and (16) could be found in the
nonlinear coupling (8) between the environmental property (the factor value
$f$) and the property of individuals (the resource amount $R$). Exactly this
coupling causes separation of a population into two groups: the well-adapted
less correlated group and the highly correlated group with larger variances of
individual properties and amount of resource which is not sufficient for
compensation of the factor load. Let us check whether such a separation is
valid for financial data.

###  6.1 Data Description

For the analysis of correlations in financial systems we used the daily
closing values for companies that are registered in the FTSE 100 index
(Financial Times Stock Exchange Index). The FTSE 100 is a market-
capitalization weighted index representing the performance of the 100 largest
UK-domiciled blue chip companies which pass screening for size and liquidity.
The index represents approximately 88.03% of the UK s market capitalization.
FTSE 100 constituents are all traded on the London Stock Exchange s SETS
trading system. We selected 30 companies that had the highest value of the
capital (on the 1st of January 2007) and stand for different types of business
as well. The list of the companies and business types is displayed in Table 2.

Table 2: Thirty largest companies for analysis from the FTSE 100 index  Number | Business type | Company | Abbreviation
---|---|---|---
1 | Mining | Anglo American plc | AAL
2 |  | BHP Billiton | BHP
3 | Energy (oil/gas) | BG Group | BG
4 |  | BP | BP
5 |  | Royal Dutch Shell | RDSB
6 | Energy (distribution) | Centrica | CNA
7 |  | National Grid | NG
8 | Finance (bank) | Barclays plc | BARC
9 |  | HBOS | HBOS
10 |  | HSBC HLDG | HSBC
11 |  | Lloyds | LLOY
12 | Finance (insurance) | Admiral | ADM
13 |  | Aviva | AV
14 |  | LandSecurities | LAND
15 |  | Prudential | PRU
16 |  | Standard Chartered | STAN
17 | Food production | Unilever | ULVR
18 | Consumer | Diageo | DGE
19 | goods/food/drinks | SABMiller | SAB
20 |  | TESCO | TSCO
21 | Tobacco | British American Tobacco | BATS
22 |  | Imperial Tobacco | IMT
23 | Pharmaceuticals | AstraZeneca | AZN
24 | (inc. research) | GlaxoSmithKline | GSK
25 | Telecommunications | BT Group | BTA
26 |  | Vodafone | VOD
27 | Travel/leasure | Compass Group | CPG
28 | Media (broadcasting) | British Sky Broadcasting | BSY
29 | Aerospace/ | BAE System | BA
30 | defence | Rolls-Royce | RR

a)b)
c)
d)

Figure 8: Correlation graphs for six positions of sliding time window on
interval 10/04/2006 - 21/07/2006. a) Dynamics of FTSE100 (dashed line) and of
$G$ (solid line) over the interval, vertical lines correspond to the points
that were used for the correlation graphs. b) Thirty companies for analysis
and their distributions over various sectors of economics. c) The correlation
graphs for the first three points, FTSE100 decreases, the correlation graph
becomes more connective. d) The correlation graphs for the last three points,
FTSE100 increases, the correlation graph becomes less connective.

Data for these companies are available form the Yahoo!Finance web-site. For
data cleaning we use also information for the selected period available at the
London Stock Exchange web-site. Let $x_{i}(t)$ denote the closing stock price
for the $i$th company at the moment $t$, where $i=\overline{1,30}$, $t$ is the
discrete time (the number of the trading day). We analyze the correlations of
logarithmic returns: $x^{l}_{i}(t)=\ln\frac{x_{i}(t)}{x_{i}(t-1)}$, in sliding
time windows of length $p=20$, this corresponds approximately to 4 weeks of 5
trading days. The correlation coefficients $r_{ij}(t)$ for time moment $t$ are
calculated in the time window $[t-p,t-1]$, which strongly precedes $t$. Here
we calculate correlations between individuals (stocks), and for biological
data we calculated correlations between attributes. This corresponds to
transposed data matrix.

###  6.2 Who Belongs to the Highly Correlated Group in Crisis

For analysis we selected the time interval 10/04/2006 - 21/07/2006 that
represents the FTSE index decrease and restoration in spring and summer 2006
(more data are analyzed in our e-print GSTarXiv ). In Fig. 8 the correlation
graphs are presented for three time moments during the crisis development and
three moments of the restoration. The vertices of this graph correspond to
stocks. These vertices are connected by solid lines is the correspondent
correlation coefficient $|r_{jk}|\geq\sqrt{0.5}$
$(\sqrt{0.5}=\cos(\pi/4)\approx 0.707)$, and by dashed lines if
$\sqrt{0.5}>|r_{jk}|>0.5$.

The correlation graphs from Fig. 8 show that in the development of this crisis
(10/04/2006 - 21/07/2006) the correlated group was formed mostly by two
clusters: a financial cluster (banks and insurance companies) and an energy
(oil/gas) – mining – aerospace/defence and travel cluster. At the bottom of
crisis the correlated phase included almost all stocks. The recovery followed
a significantly different trajectory: the correlated phase in the recovery
seems absolutely different from that phase in the crisis development: there
appeared the strong correlation between financial sector and industry. This is
a sign that after the crisis bottom the simplest Selye’s model is not valid
for a financial market. Perhaps, interaction between enterprizes and
redistribution of resource between them should be taken into account. We need
additional equations for dynamics of the available amounts of resource $R_{i}$
for $i$th stock. Nevertheless, appearance of the highly correlated phase in
the development of the crisis in the financial world followed the predictions
of Selye’s model, at least, qualitatively.

Asymmetry between the drawups and the drawdowns of the financial market was
noticed also in the analysis of the financial empirical correlation matrix of
the 30 companies which compose the Deutsche Aktienindex (DAX)
DrozdComCollNoise2000 .

The market mode was studied by principal component analysis Stanley2002 .
During periods of high market volatility values of the largest eigenvalue of
the correlation matrix are large. This fact was commented as a strong
collective behavior in regimes of high volatility. For this largest
eigenvalue, the distribution of coordinates of the correspondent eigenvector
has very remarkable properties:

  * •

It is much more uniform than the prediction of the random matrix theory
(authors of Ref. Stanley2002  described this vector as “approximately
uniform”, suggesting that all stocks participate in this “market mode”);

  * •

Almost all components of that eigenvector have the same sign.

  * •

A large degree of cross correlations between stocks can be attributed to the
influence of the largest eigenvalue and its corresponding eigenvector

Two interpretations of this eigenvector were proposed Stanley2002 : it
corresponds either to the common strong factor that affects all stocks, or it
represents the “collective response” of the entire market to stimuli. Our
observation supports this conclusion at the bottom of the crisis. At the
beginning of the crisis the correlated group includes stocks which are
sensitive to the factor load, and other stocks are tolerant and form the less
correlated group with the smaller variance. Following Selye’s model we can
conclude that the effect is the result of nonlinear coupling of the
environmental factor load and the individual adaptation response.

##  7 Functional Decomposition and Integration of Subsystems

In the simple factor–resource Selye models the adaptation response has no
structure: the organism just distributes the adaptation resource to
neutralization of various harmful factors. It is possible to make this model
more realistic by decomposition. The resource is assigned not directly
“against factors” but is used for activation and intensification of some
subsystems.

We need to define the hierarchical structure of the organism to link the
behavior in across multiple scales. In integrative and computational
physiology it is necessary to go both bottom–up and top–up approaches. The
bottom–up approach goes from proteins to cells, tissues, organs and organ
systems, and finally to a whole organism Cramlinatal2004 .

The top-down approach starts from a bird’s eye view of the behavior of the
system – from the top or the whole and aims to discover and characterize
biological mechanisms closer to the bottom – that is, the parts and their
interactions Cramlinatal2004 .

There is a long history of discussion of functional structure of the organism,
and many approaches are developed: from the Anokhin theory of functional
systems Sudakov2004  to the inspired by the General Systems approach theory of
“Formal Biological Systems” Chauvet1999 .

The notion of functional systems represents a special type of integration of
physiological functions. Individual organs and tissue elements, are
selectively combined into self-regulating systems organizations to achieve the
necessary adaptive results important for the whole organism. The self-
organization process is ruled by the adaptation needs.

For decomposition of the models of physiological systems, the concept of
principal dynamic modes was developed Marmarelis1997 ; Marmarelis2004 .

In this section, we demonstrate how to decompose the factor–resource models of
the adaptation of the organism to subsystems.

In general, the analysis of interaction of factors is decomposed to
interaction of factors and subsystems. Compensation of the harm from each
factor $F_{i}$ requires activity of various systems. For every system $S_{j}$
a variable, activation level $I_{j}$ is defined. Level 0 corresponds to a
fully disabled subsystem (and for most of essentially important subsystems it
implies death). For each factor $F_{i}$ and every subsystem $S_{j}$ a
“standard level” of activity $\varsigma_{ij}$ is defined. Roughly speaking,
this level of activation of the subsystem $S_{j}$ is necessary for
neutralization of the unit value of the pressure of the factor $F_{i}$. If
$\varsigma_{ij}=0$ then the subsystem $S_{j}$ is not involved in the
neutralization of the factor $F_{i}$.

The compensated value of the factor pressure $F_{i}$ is

$\psi_{i}=f_{i}-\min_{j,\ \varsigma_{ij}\neq
0}\left\\{\frac{I_{j}}{\varsigma_{ij}}\right\\}\ .$ (17)

In this model resources are assigned not to neutralization of factors but for
activation of subsystems. The activation intensity of the subsystem $S_{j}$
depends on the adaptation resource $r_{j}$, assigned to this subsystem:

$I_{j}=\alpha_{j}r_{j}\ .$ (18)

For any given organization of the system of factors, optimization of fitness
together with definitions (17) and (18) lead to a clearly stated optimization
problem. For example, for Liebig’s system of factors we have to find
distributions of $r_{j}$ that either are maximizers in a problem:

$\min_{i}\left\\{f_{i}-\min_{j,\ \varsigma_{ij}\neq
0}\left\\{\frac{\alpha_{j}r_{j}}{\varsigma_{ij}}\right\\}\right\\}\ \ {\rm
for}\ \ r_{j}\geq 0,\ \sum_{j}r_{j}\leq R\ $ (19)

if this minimum is nonnegative, or give a solution to the system of
inequalities

$f_{i}-\min_{j,\ \varsigma_{ij}\neq
0}\left\\{\frac{\alpha_{j}r_{j}}{\varsigma_{ij}}\right\\}\leq 0;\ r_{j}\geq
0,\ \sum_{j}r_{j}\leq R$ (20)

if the minimum in (19) is negative.

For the study of integration in experiment we use principal component analysis
and find, parameters of which systems give significant inputs in the first
principal components. Under the stress, the configuration of the subsystems,
which are significantly involved in the first principal components, changes
Svetlichnaia1997 .

We analyzed interaction of cardiovascular and respiratory subsystems under
exercise tolerance tests at various levels of load. Typically, we observe the
following dynamics of the first factor composition. With increase of the load,
coordinates both the correlations of the subsystems attributes with the first
factor increase up to some maximal load which depend on the age and the health
in the group of patients. After this maximum of integration, if the load
continues to increase then the level of integration decreases Svetlichnaia1997
.

Generalization of Selye’s models by decomposition creates a rich and flexible
system of models for adaptation of hierarchically organized systems. Principal
component analysis Jolliffe2002  with its various nonlinear generalizations
Gorbanatal2008 ; GorbanZinovyev2009  gives a system of tools for extracting
the information about integration of subsystems from the empirical data.

##  8 Conclusion

Due to the law of the minimum paradoxes, if we observe the Law of the Minimum
in artificial systems, then under natural conditions adaptation will equalize
the load of different factors and we can expect a violation of the law of the
minimum. Inversely, if an artificial systems demonstrate significant violation
of the law of the minimum, then we can expect that under natural conditions
adaptation will compensate this violation.

This effect follows from the factor–resource models of adaptation and the idea
of optimality applied to these models. We don’t need an explicit form of
generalized fitness (which may be difficult to find), but use only the general
properties that follow from the Law of the Minimum (or, oppositely, from the
assumption of synergy).

Another consequence of the factor–resource models is the prediction of the
appearance of strongly correlated groups of individuals under an increase of
the load of environmental factors. Higher correlations in those groups do not
mean that individuals become more similar, because the variance in those
groups is also higher. This effect is observed for financial market too and
seems to be very general in ensembles of systems which are adapting to
environmental factors load.

Decomposition of the factor–resource models for the hierarchy of subsystems
allows us to discuss integration of the subsystems in adaptation. For the
explorative analysis of this integration in empirical data the principal
component analysis is the first choice: for the high level of integration
different subsystems join in the main factors.

The most important shortcoming of the factor–resource models is the lack of
dynamics. In the present form it describes adaptation as a single action, the
distribution of the adaptation resource. We avoid any kinetic modeling.
Nevertheless, adaptation is a process in time. We have to create a system of
dynamical models.

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