к 100 летию

Levich A.P. Toward a dynamic theory // Lectures in Theoretical Biology. V.2. Tallinn: Estonian Academy of Science. 1993. Pp. 33-50.

к 100 летию Бауэра Toward a dynamic theory

T O W A R D S A D Y N A M I C T H E O R Y

(on the centenary of E.Bauer)

A.P. Levitch

(Moscow State University)

1. The problem of theoretical biology.

Fifty five years ago in "Theoretical Biology" E. Bauer

confidently stated that biology was not applied physics or

chemistry. He also stated that "all special laws, which would

be revealed in certain fields of biology would display the

general laws of motion, appropriate to living matter" (Ба-

уэр,1935,с.8). The urgent problem of theoretical biology was,

according to E.Bauer, the development of general laws of

motion for living matter. It should be mentioned that this

problem has not been solved up to the present.

E.Bauer proposed that the general laws of theoretical

biology would be analogous to those of theoretical physics, as

an illustration he gave the examples from Newtonian mechanics

and statistical physics.

In what way are the theories in advanced fields of

science organized? The methodology of science shows that the

theory of some part of reality is sure to include the number

of components, the development of which acts, directly or

indirectly, as stages of theory creation (Левич,1989а,б).

The O-component is the description of the ideal structure

of the theory's elementary object.

The S-component consists in enumeration of the possible

theory object states. In other words, component S is described

as a state space of analysed system.

The C-component fixes the ways of objects' variability

(changeability) and corrects the overidealization during the

isolation of the objects, since only processes rather than

objects exist in the World and the notions of objects are the

abstractions of these processes. C-component brings the

processes, variability and the pre-time in the theory.

The examples instead of precise definitions of elementary

objects and their variability are given below.

Material points along with their positions and velocities

in physical space are the elementary objects of classical

mechanics. The planets of the solar system can be considered

as an example. The variability is determined by points'

trajectories. The state space is a six-dimensional phase

space which is the product of three-dimensional Euclidean

space and three-dimensional velocity space.

In quantum mechanics elementary objects are the ranges of

probabilities of microobject states, for example, energy

states of the atom. The variability in state space is

determined by the trajectories of vectors in

infinite-dimensional Hilbert space.

In nuclear theory elementary objects are nucleons and

some other particles with certain sets of quantum numbers. The

variability is the mutual transformation of particles and

radiations. The state space is restricted by those

combinations of quantum numbers for the totality of changing

particles, which meet the conservation laws.

In embryology the living cell acts as an elementary

object, and the role of variability belongs to the cell

division and differentiation processes. The state space is

described by the set of morphological features.

In community ecology the elementary object is population.

The variability consists of births and deaths of organisms.

State space is the set of vectors (n ,n ,...,n ), where n is

the size of the population of species i, included in the

community. The vector set is limited by the available

resources.

The T-component of the theory consists of bringing the

clock and the parametric time into the functioning of the

systems. Parametric time can be considered as the image of

changing objects reflecting the variability process on

linearly arranged metrized numerical set. Generally, the

variability of a selected object acts as a standard in

measuring other variabilities, the clock here is the standard

object and the way of organizing the necessary reflection.

The traditional clock in natural science is based on

physical processes. The examples of these processes are:

gravity or resilient pendulum devices, astronomical systems,

observing the Earth's rotation around its axis or around the

Sun, caesium or other sources of the electromagnetic

oscillations, recently discussed pulsar standard of

superstable periods, and radioactive decay.

Here is Fridman's description of physical clock

appearance (Фридман, 1966,с.50-53). Let point M correspond to

certain basic movement, and let the device showing the arc

lengths, t, which are the trajectories of point M in its basic

movement, be the clock of point M. We will call the magnitude

t the physical local time of the point M.

To begin with, consider star time. Let us assume the

movement of the end of the hand of a certain length directed

from Earth center to the star to be the basic motion. The

distance, passed by the end of this hand would be the star

time, t .The star time is the same in all parts of space, it

is the universal time.

Let us consider another time, which we will call the

gravitational time. Suppose, the material point is falling in

a constant gravitational field. We choose this as a basic

motion. The clock shows the distance t , passed by this

material point, which is exactly the gravitational time.

Related to the gravitational time the stars' movement is

uneven.

Let us put in the pendulum time. Imagine many

hypothetical pendulum clocks. Let the motion of the second

hand of our clock placed at any point on the Earth be the

basic motion of this point. Assume the distance passed by the

end of second hand of our clock to be the pendulum time, t .In

contrast with the star time or the gravitational time, which

are universal, this pendulum time is local, i.e., different at

various latitudes.

The parametrization of variability with the help of the

physical clock permeates the whole entity controlled by human

consciousness, i.e., science, culture, and way of life.

The changes in the World can not be reduced to mechanical

displacements. Chemical transformations, geological chronicle,

the development and extinction of speices and whole

communities, the Universe nonequilibrium, and sociogenesis ...

Isn't it better to assume the clock being established in

frames of reference for description of natural object

variability to be different, and is it possible to consider

one clock, for example, physical clock, to be correct and

unlike clock to be wrong? This thought would be understandable

in the case of Galilei who tried to find the laws of pendulum

motion with the help of his own heartbeat.

As early as Poincare stressed that "there is no time

measurement method, which would be better than another"."The

accepted method is merely more suitable. When comparing clocks

we can't say that one of them works properly and the other

keep incorrect time. We can only say that the indications of

one of these clocks are prefered" (Poincare,1898). In

nonphysical sciences there is a growing necessity for clocks,

not to be synchronized with some physical standards. These

clocks in comparison with standardized, would be much more

convinient and more adequate for description of nonphysical

forms of motion.

In embryology the development of organisms is effectively

described by using a special unit of biological time, equal to

an interval between same phases of cell division (Детлаф,Дет-

лаф,1982). This unit, named "detlaf," depends on the

temperature and is species-specific. That is why the

regularities of development, observed in the detlaf time scale

are undistinguishable in the astronomical time scale.

The population time in ecology (Абакумов, 1969),

ethnography (Алексеев, 1975), and genetics (Свирежев,Пасеков,

1982) can be measured by the number of generations.

The chronostratigraphical scale of geological time is

based on the sequence of rocks with standard points. These

points have been chosen in opencast mines with well-conserved

frontier provinces (Harland et al., 1982). In paleobiological

stratigraphy durations of geological epochs can be measured by

vertical width of layers with remains of fossil species (Сима-

ков, 1977).

In the psychological time model (Головаха, Кроник, 1984)

the durations of the intervals between personally important

events are measured by the number of inter-event links.

The L-component of the theory consists of the statement

of the variability law which isolates the real generalized

motion in state space from all possible motions (the term

"generalized motion" is used as a synonym to variability of

objects).

In mechanics and theory of fields this variability law

more often appears in a form of "motion equations". For

example, the Newtonian equations of macroobject motions with

small velocities and in weak fields, or the Schrodinger

equation of nonrelativistic quantum mechanics or the Maxwell,

Einstein or Dirac equations etc.

The law can be formulated not only in a form of equation,

but in a form of the extremum principle. An example is the

minimum action principle: trajectory is real only if its' time

integral of difference between kinetic and potential energy

magnitudes is minimal. Equational and extremum principle forms

of the variability law are equivalent.

In many fields of natural science, for instance, in the

examples given above, and in nuclear theory, in embryogenesis,

and in ecology the objective of theory constructing is the

formulation of variability laws. This objective can't be

reached without proper solution to a set of problems,

connected with the elaboration of the O,C,S and T components

of the theory. In natural sciences' methodology the C and T

components are still elaborated to an inconsiderable degree.

There is a very close link between the choosing of these

components and the way the L-component is derived. According

to A.A.Sharov, the law of motion is exactly the description of

the variability of the object under consideration by means of

a standard clock variability. Hence, the choice of the clock

adequate to the processes observed can affect the probability

of revealing the variability law.

Laws of motion affect the method of time measurement if

the T and L components of the theory correspond to each other

(Le Temps et la pensee Physique contemporaine, 1968). For

example, "the simultaneity of two events or the way in which

one event follows another, the equality of the two durations

should be determined in a way which provides the most simple

formulations of natural laws"(Poincare, 1898).

The difficulties of motion equations' derivation appear

to be due to inconformity of physical measurement of time to

the nonphysical nature of the regularities being observed.

Finally, the I-component of the theory is presented by a

set of interpreting procedures. The first of these procedures

is to bring the formal mathematical theory constituents into

compliance with the abstract notions of reality. The second

procedure consists of the rules of bringing these notions into

correlation with magnitudes measured experimentally.

The apparatus of quantum mechanics deals with the complex

-signed wave functions and with the operators affecting them.

A transfer to the notions of macrophysical reality is

conducted by some postulated rules, namely: the square of the

wave function is the probability of microparticles occuring at

definite points of time and space; the eigenvalue of an

operator is the quantitative value of the corresponding

physical characteristic. For example, interferential

experiments with the particles passing through obstacles are

required for the probabilistic distribution observations.

Energy characteristics of atoms are determined by the distance

between spectral lines in the experiments concerning the

atomic emission and absorption of radiation.

The I-component is a necessary constituent of the theory.

It is the complex of interpreting procedures that transform

the formal theoretical scheme into realistic science. The

possibilities of the I-component elaboration, paticularly its

experimental identification, depends not only, or not so much

on the merits of theoretical scheme and its creators, but on

the sum of the technologies, worked out by civilization.

It took hundreds of years for Democritus' hypothesis to

become a "verified" theory. Vast experience in X-ray

structural analysis was nedeed for the hypothesis of some

descrete substance of heredity to become the scientific model

of a DNA double helix structure.

Interpreting procedures are very ambiguous. I-component

elaboration often turns out to be the most difficult and the

most vulnerable stage of capable theory's creation.

In the present-day paradigm of natural science the

problem "what is time ?" is considered to be naive or

nonscientific. Most people think it to be absolutely clear or

believe the answer to this problem can be found in some

physical textbook. Actually,time is the basic idea of all our

dynamic speculations, which make sense only due to the concept

of time. The structure of time as a physical object is

postulated to be as simple as possible (Акчурин, 1974). " The

presentation of time as an internal property of physical

systems exceeds the limits of conventional physical

description" (Prigogine,1980).

Firstly, in physics time is identified with a set of real

numbers, although the lack of distinct nonmathematical notions

about time makes it impossible to analyse constructively the

correspondence between the real straight line axiomatics and

the properties of time. Secondly, physics proposes that the

clock is based on the gravitational or electromagnetic

interactions for any variability observations, that is, the

parametric time only is used.

Thus, the biological theory construction must be preceded

by investigation of time in biology, and by the development of

time construction appropriate to build the T-component of laws

of the living matter's motion.

The objectives of this work are:

- to show that the interpretation of Bauer's stable

nonequilibrium principle might be equivalent to the hypothesis

of flow existance, which generates the metabolic time of

natural systems;

- to present the construction of substantial time, which

would be useful in solving the biological theory problems

formulated by E.Bauer.

2. Notes on the origin of the nonequilibrium sources.

"Only living systems never reach an equilibrium, for

they constantly work against stability"(Бауэр,1935,с.43).

According to Bauer, the source of free energy(or"the work of

structuring forces" and "structural energy" are the synonyms)

is the nonequilibrium of molecular structure of living matter

What is the source of the nonequilibrium of "living

matter"? Firstly it is the activation of molecules of food

caused by levelling processes. Energy of these molecules

maintains nonequilibrium (here the molecules of living matter

in "active, deformed state" are considered (Бауэр,

1935,с.127). However, the unavoidable result of metabolism is,

according to E. Bauer, the lowering of the potential of free

energy of nonequilibrium. "The more intensive metabolism is,

the higher rates of the free energy depletion are. This free

energy of living matter exists because of the deformed

nonequilibrium structure of its molecules" (Бауэр,1935,с.129).

"During assimilation the structural energy of a system can be

used. This energy is necessary for the reconstruction of

nonliving substance" (Бауэр,1935, с.144).The total amount of

energy that can be assimilated is limited. This amount of

energy is species-specific parameter of organism (Rubner

constant) (see Бауэр,1935,с.131; Зотин, Алексеева,1984) and is

"proportional to the free energy of an ovicell" (Бауэр,1935,

с.130).

This means that the problem of the source of living

matter's nonequilibrium cannot be reduced to the possibility

of nonequilibrium's replenishment with free energy of food.

Another source of nonequilibrium is required. The utilization

of this source should regulate the organism's ability to make

up for free energy losses with food. Concerning deeper

nonequilibrium one can propose several possibilities of its

origination in organism. They might be the following:

- the law of nonincrease (or conservation) of structural

energy and transfer of it from generation to generation;

- the possibility of external replenishment of structural

energy during the origination or fertilization of the ovicell

in addition to an explanation of Bauer's theory, according to

which fetal cells, possessing maximum initial potential,

originate due to dying or, in other words, dissimilation of

the body tissues" (Бауэр,1935, с.144).

- to reject the idea of the impossibility of structural

energy replenishment during the life period, and then to find

the ways of such replenishment, for instance, the mechanism of

structural energy assimilation by autotrophs and its farther

spreading in the biosphere through the food chains.

In the second and third proposals, and in other cases,

allowing the structural energy replenishment, the question

about the sources of such replenishment remains.

When considering the problem of understanding the stable

nonequilibrium principle, another problem arises, that is the

search for the sources of nonequilibrium. This problem is

connected with time, its flow and becoming. One of the

possible hypotheses dealing with this problem's consideration

consists of the substantial time construction (Левич,1989а,б).

All natural systems are hierarchic. The replacement (or

substitution) of elements takes place on every hierarchical

level. Any variability of natural systems can be presented as

the superposition of these replacements. The quantity of

elements of a standard system, that have been substituted, may

act as a substitutional clock of the systems. The origin of

the elements' substitution deals with the external flow of

elements on some deep level of organization. This flow

penetrates the whole natural hierarchy, which contains the

system. In particular, time of the Universe (or, in other

words, of that part of the world, which can be measured

instrumentally) originates from the generating flow of

pre-elements. These pre-elements belong to rather deep levels

of living matter hierarchy. The above means that the Universe

is isolated, open, unstable, and that the passage of time is

determined by the Universe nonequilibrium.

Generating flow appeared as the logical extrapolation of

metabolic time properties. This flow helps to find the ways to

solve the existing problems of natural science. The link

between time flow and instability of the system, and between

flow dissipation and irreversability is the instability of the

system. In other words the presence of substrate-energetic

flow through the system is the passage of time. An uncommon

feature in this is the necessity of generating flow existence.

The flow hypothesis transforms the question of the "nature" of

time, the causes of its' flowing, and the mechanisms of its

becoming, into the question about the origin, substrate and

energetic "fostering" of the Universe. In time construction

flow is a fundamental, primary standard object and it

generates the sequence of time moments, i.e., time is linearly

ordered because of generating flow. Irreversibility isn't an

immanent property of time, it occurs only through orientation

of the pre-elements' flow. This means that time

irreversibility exists, while generating flow isn't reversed.

The presence of the flow takes off the lable of thermal death

of the Universe from the second law of thermodynamics.

Increase of entropy leads to equilibrium distribution only

when the global extremization of an isolated system' entropy

exists. The presence of flow, limiting evolution of a system,

demands the solution of a conditional extremum problem and

leads to eneven distribution of parameters such as the Gibbs'

distribution (Левич, 1980) and also to the possibility of

structure producing, i.e., self-organization.

"...Every day experience makes certain that the

properties of nature have nothing in common with the those of

a system in equilibrium. Astronomical data make this statement

correct. In the vast part of the Universe, accessible to

observations" (Ландау, Лифшиц, 1964). In this connection isn't

the absence of equilibrium features in the Universe an

argument for the hypothesis of generating flow existence?

The associations concerning the idea of generating flow

are not new in philosophy or in natural science. For example,

some similar notions are presented by the world views of

taoism, Newton's conception of an absolute time, in the

present day notion of physical vacuum, substantial conception

of G.J. Whitrow's, "that there is a total basic rhythm in the

Universe" (1961), and in Kozyrev's idea of the flow of time.

According to N.A.Kozyrev, time is a "huge flow,

comprising all material processes in the universe. These

processes are the sources fostering this total flow" (Козырев,

1963,c.96). N.A.Kozyrev considered the intensity, or density

of this flow, its energy, radiation and absorption, and also

rectilinearity of its spreading, its reflection from

obstacles, and its absorption by substances. All this gives

the reason for identifying Kozyrev's flow with some

substantial flow. the sources of this flow are, according to

N.A.Kozyrev, any unstable, irreversible world processes, i.e.,

processes with changes of energy and thermodynamical entropy

of system.

N.A.Kozyrev noticed the discrepancy between the second law

of thermodynamics, postulating thermal and radiational

degradation of the Universe, and the lack of any traces of

stability in diversity of the Universe. He also stressed that

the attempts to explain thermal death had been isolated from

the real Universe, which is observed by an astronomer. In

fact, celestial bodies and their systems are so separated from

each other, that thermal death should occur before

interference from some distant system. Thus, the degraded

conditions seem to be dominant, but, actually, these

conditions almost never occur. The problem is not only to

explain the instability of the Universe, but also to

understand why celestial bodies themselves and their systems

exist inspite of small relaxation periods" (Козы-

рев,1963,с.96). It is possible to propose hypotheses,

maintaining the second law of thermodynamics. An example would

be to assume the present moment of cosmological time to be not

far from the initial fluctuation (or singularity or

cataclysm). This means that degradation is not very profound

and thermal death of the Universe is to occur in the far

future.

N.A. Kozyrev proposes an alternative hypothesis, that is,

the Universe and its systems are not isolated, that means

that the obligatory conditions of the second law of

thermodynamics are not satisfied. He also states that "there

are some constantly acting forces (reasons) against the

entropy increase"(Козырев, 1958). Kozyrev's flow is the

required source of a system's non-isolation and also is

necessary for the explanation of the star energy's origin (Ко-

зырев, 1948,1950).

There is much evidence of Kozyrev's flow's existence in

different mechanical phenomena. Irreversible processes, such

as deformation of bodies, air jet striking against obstacles,

work of the sand clock, light absorption, friction,burning,

some human activities, changes of the temperature of bodies,

aggregate state transformations, dissolution, mixing of

substances, the withering of plants, and non -light radiation

of astronomical objects in Kozyrev's experiments, can affect

the beam or disk of torsion balance while irradiating or

absorbing Kozyrev's flow.

It has been found that the flow may be screened, or

absorbed or reflected by substance. Non-resilient processes in

solid bodies can change their weight and resilient processes

lead to changes in quantitative characteristics of resilience.

A rotating body's weight changes when it participates in

additional processes, such as vibration, warming or cooling,

electric current conducting, its weight changes. Many features

of the shape and climate of the Earth and other planets can be

explained by the influence of dissipation processes on these

giant gyroscopes.

Non-mechanical pickups also register the flow

accompanying non-equilibrium processes, namely: the value of

resistance, mercury level in thermometer, frequency of quartz

piezoelement oscillations, electric potential of thermocouple,

water viscosity, work of electron exit in photoelements, and

the rates of chemical reactions.

It should be mentioned that Kozyrev's thoughts hardly

meet existing physical notions. The values of effects in

Kozyrev's experiments are not large. Additional forces in

mechanical experiments are 10 - 10 of the body weight being

used in measurement; in the case of non- mechanical pickups,

the relative change is about 10 -10 of the value being

measured. For torsion balance the turn may be up to tens of

degrees, that correspond to forces of 10 -10 of forces acting

in a system. Kozyrev illustrates of the difficulties of

detection of additional cryptic sources of star energy, the

difficulties occuring because of a local minuteness of the

effect (Козырев, 1977,с.210): "The situation here is similar

to that of a physicist from a labarotory in space, far from

Earth, would be. It would hardly be possible to reveal

gravitational forces in that situation. However, these forces

determine not only the whole dynamics of cosmic objects, but

their internal structures also. The analogy is that the star,

despite great energy losses, is a perfect thermos. For

instance, the Sun, having an internal temperature of

approximately ten billion degrees may cool down only in one

degree for three years. An insignificant energy inflow for

replenishment of such small losses, would be indistinguishable

in laboratory conditions".

In principle, it is possible to explain Kozyrev's effects

with the help of more prosaic reasons other than the influence

of the "time flow" (for example, convective flows, temperature

changes, induced electric and magnetic fields, etc.).

N.A.Kozyrev tried to analyse the role of extraneous causes in

his experiments. For example, one article of N.A.Kozyrev is

devoted to the possible mechanisms of effects' appearance in

weighing vibrating bodies with the help of beam balance.

Kozyrev's opponents, however, may have objections concerning

factors, being not observed. Besides, the reader suppose the

author to analyse in detail possible errors, which could

reduce the noticed effects to disappointing artefacts. Up to

the present there has not been and there is no concrete

disproof of Kozyrev's experimental results or their consistent

explanation by means of ordinary physical factors. Only doubt

of simple interpretation of experiments exists.

Experiments of N.A.Kozyrev and his colleagues have been

confirmed by few enthusiasts. However, one failed to regain

the results similar to those of N.A.Kozyrev in any labarotory

in the world. The fact that such results have not been found,

especially in many precision or other physical experiments,

seems to be rather understandable. N.A.Kozyrev's experiments

were specially designed for the exposing the effects, being

derived from his theoretical ideas. These effects are

unlikely to be revealed occidentally. They are minute and

require special experimental conditions, for example,

inequality of beams of the torsion balance; participation of

additional irreversible processes such as vibrations,

dispersion of heat or electrical current in the experiments

with gyroscopes, etc. Because of the background influences

certain efforts to reproduce experiments' results are

required. Some deviations from what has been expected could

be easily interpreted as nonsystematic errors of measurements.

The desire to repeat or develop complicated Kozyrev's

experiments faces the difficulties of perception of the

Kozyrev's works. Kozyrev did not try to adapt his original

ideas and terminology to existing scientific standards.

Scientific views of N.A.Kozyrev had been often

contradictory to the paradigmal beliefs of his opponents.

However, this was not an obstacle for N.A.Kozyrev to make

outstanding discoveries in astronomy, for example, to reveal

the vulcanism on the Moon. Perhaps, the intuition didn't fail

him in the forsight of substatial nature of the time passage

either.

N.A.Kozyrev stressed repeatedly that non-equilibrium in

the world, created by time flow, had to affect the perception

of life phenomenon.

"...Our scientific knowledge lacks a vital element.

Physics, chemistry and other precise sciences can follow and

predict the way of destruction of a fallen leaf rather

strictly, and even derive its motion equation. Nevertheless,

they are unable to explain growth or shape and properties of

leaf. One must not refer to some specific characteristics of

plants inappropriate to non-living nature.

Living organisms can not make anything that does not

exist in nature. They can only accumulate and use some basic

properties lying in the foundations of the Universe. Hence,

these basic properties have to exist in non-living nature as

well. They are to be searched for just there, using vast

experience and methods of precise sciences"(Козырев,1975,c.2-

3). The experimental results show that the organizing force of

the active property of time has little influence on the

systems in comparison with usual destructive way of

development. That is why there is no surprise that the vital

element was missed of the system of our scientific knoweledge.

However, being low, it is dissipated everywhere in nature, and

therefore only the possibility of its accumulation is

required. The process of accumulation is similar to that of

maintenance of mighty rivers by tiny water drops falling over

enormous areas. This possibility is realized in living

organisms since vital activity as a whole counteracts the

course of systems' destruction"(Козырев,1982, c.71).

Living organisms, according to N.A. Kozyrev, may be both

emitters and detectors of substantial flow.

"Experiments with plants need more detailed description. The

equipment used were thick paper rotary disk and non-symmetric

torsion systems with jasmin, bamboo or glass points hang by

kapron threads. The systems were enclosed in tin

cylindrical casings with hermetic observation window at the

top. Plants under consideration (apple-tree, pear-tree,

linden, chestnut, clover, dandelion and some others) were

gathered on Pulkovo territory in different seasons. The

experiment was carried out in the following way: gathered

plants were exposed for a while on the table in a laboratory,

lying apart from each other; after that top or cut of a plant

was placed by the edge of torsion balance in a position of 30

degrees from the direction of the point (or from conventional

index on the disk)... Deviations of torsion balance beam and

disk caused by effect of plants were observed in most cases.

We failed, however, to reproduce the results. Values of these

effects differed both quantitatively and in sign. Acetone

evaporation used as a control process always caused repultion

of the disk... Values of the effects depended on the season

and varied from 1-2 degree to almost complete turn. The sign

of the effect value could be different. Taken into experiment

immediately after gathering, a plant causes repulsion of the

beam. Values of the effects of cut or top of a plant keep the

same sign and differ slightly in quantity. Taken later..., the

stem still repulses the point of torsion balance with the same

intensity and always regularly and moderately, while the top

begins to attract it rather actively and sometimes by

pulses...For example, an apple-tree in blossom before throwing

the petal can display attraction of about 250-300 degrees in 5

-10 minutes. Repulsive effect usually shows itself in the same

period of time and lies within 10-30 degrees... In

autumn,1983, a period of increased activity of apple-trees has

been detected. However, these plants are known to lay the

foundation of the apple crop just at that time. Actually, next

year crop appeared to be rather big. No activity was revealed

by autumn observations in 1984 and only a few trees gave the

apple crop in summer... It is significant that the rise of

number of plants under experiment... does not augment the

value of the effect."

"Usual human activity was found to have little influence

in measuring devices... When ill, a human being can actively

interact with measurement instruments, this interaction

precedes the moment of subjective falling ill. Sometimes

N.A.Kozyrev and I managed to detect a common cold one or two

days before temperature rised. Similarly emotional excitement

violently influences measuring instruments. For example, when

reading his favourite "Faust" N.A.Kozyrev was able to cause

deviation of a hand of device as much as 40 degrees.

Meanwhile, mental arithmetic calculations had no effect on the

hand".

These are quotations from the report "Physical Time of

natural life'' (see references 9,10,and 22), made by

V.V.Nasonov on December 6, 1985 at the seminar devoted to Time

problems in natural sciences at Moscow State University.

"The process of liquid nitrogen evaporation has been

chosen as a resource of influence... Besides the process of

snow melting took place... Actually, two processes affected

the object under observation: evaporation itself and warming

of nitrogen fumes... Microorganisms Pseudomonas fluorescens

and microorganisms of artesian water, oats and pea seeds, and

onions, growing in the water, were taken in experiments. It is

known that temperature changes within 1 C don't influence

vital functions much. However, permissible changes were

determined to be in 0.2 C interval... Influence of the changes

in nitrogen concentrations was prevented by incessant

ventilation and hermetically sealed test-tubes, where objects

were kept. Test-tubes had been made of glass.

Exposition time usually was 60 minutes. All experiments

were accompanied by control tests, in which objects were in

the same conditions exept the influence of liquid nitrogen

evaporation.

For microorganisms, drastic depression of vital functions

was observed on the first day of experiment, then restitution

followed.

Two experiments with 80 oats seeds showed the decrease of

germination to zero value, while germination of control seeds

was 60 percent.

The results of experiments with pea seeds were also

interesting. Six experiments with 600 seeds were carried out.

Average germination in control was 92 percent, while in

experiments it was 62 percent, i.e., some seeds died.

In the next set of experiments (with 60 seeds divided in

3 equal groups) seeds were not affected by liquid nitrogen

evaporation process. Tap-water for seeds watering was treated

instead. In all groups of seeds' germination was 100 percent.

However, there was depression of growth in experiment

comparing with control.

Experiment with germinated pea seeds, being affected by

liquid nitrogen evaporation, was continued. Experimental and

control seeds were planted outdoors. Stem growth was

observed... On the fifth day of experiment depressed plants

overtook and then surpassed in growth control group. Maximum

overgrowth (up to 50 percent ) was observed on 8 day...

Experiments showed that considerable distant effect on

living matter conditions was caused not only by such intensive

process as liquid nitrogen evaporation, but also by snow

melting... Healthy onions of equal size and root system

development were taken in experiment. A reflector (piece of

cardboard, covered with aluminum foil) was placed above the

experimental group of onions. This was done in order to

reflect the shining of snow outside the window on the onions.

Because of reflector, light conditions of experimental and

control groups were unequal; sheets of paper were pasted on

the window in the area of reflection to equalize light

conditions. There are the results of experiment: 50 percent of

control onions were rotten without sprouting. The rest of this

group rooted slowly, and there was a delay in shoot growth,

and its inhibition. At the end of experiment average length of

shoots was 150 mm , water in pots was turbid and with a smell

of decay. Experimental group's behavior was quite different.

From the very beginning impetuous growth of roots was

observed. The lower parts of pots were filled with roots

completely. All onions were viable. Water in the pots remained

clear and odorless during all the time. At the end of

experiment length of shoots was 300 mm...

From the facts given above one can conclude the

following:

Irreversible processes transform distantly physical

properties of surrounding substances.

Living matter is considerably sensitive to these

processes...

For biological objects underwent short-term direct

influence of liquid nitrogen evaporation within certain

conditions complete elimination of vital activity inhibition,

and their further stimulation are appropriate (Данчаков,1984,с.

101-121).

Experiments with pea seeds affected by liquid nitrogen

evaporation were continued in a systematic way. "Seeds were

exposed to the process on the day before sowing. Dry seeds

were taken... During two field seasons four experiments (with

3 reiterations with 175 seeds in each) had been carried out.

In 3 variants seeds were exposed to the influence for 15, 6,

and 3 minutes. Three sources of influence were placed in line

in a distance of 65 cm from each other. Seeds in paper

envelopes were put exactly above them on a cotton cloth,

strenched on a frame-work. Shooting, growth and development

were observed, and some seed characteristics were revealed.

Let us summarize the main features of observed

phenomenon. In the beginning of growth affected plants develop

slower than control ones, then, sometimes, surpass in growth

occured.

In the most representative class of seeds (half of all

seeds approximately) affected seeds weigh more than control

seeds. Weight distribution of 200 seeds is distinct,

statistically reliable response of biological systems to the

effect.

For most characteristics mean deviations of experimental

values from control are several times higher than those of

different reiterations of experiment. All characteristics

under observation show an increase of variation value, all

distributions of the affected plants have larger dispersions

in comparison with control. This is one of the direct and

permanent features of the effect's presence.

The main feature of the phenomenon under consideration

should be taken into account in organization and

interpretation of experiments. We study the distant influence

of liquid nitrogen evaporation on biological systems.

However, if the system fixed the influence, it means that this

system can detect all natural and artificial irreversible

physical processes, which were effectively simulated by the

process of liquid nitrogen evaporation. Thus, biological

system under consideration is always involved in proximal and

distant irreversible processes, lying beyond the experimental

control (Данчаков, Еганова,1987,с.11-81).

V.V. Nasonov, developing Kozyrev's ideas, pointed

directly that helical protein molecules were sensitive to the

density of time flow.

Comparing ideas of E.Bauer and N.Kozyrev, one can assume

that Kozyrev's flow of time is Bauer's source of structural

nonequilibrium in living organisms, being sought for.

3. Differences between living and non-living matter.

In respect of the stable nonequilibrium principle the

difference between living and nonliving matter is the ability

of living matter to use structural energy which exists in the

world. As it has been stressed above the existence of

non-equilibrium source, regulating the inflow of the free

energy of food in the organisms, is a necessary premise of

stable non-equilibrium principle.

Let us try to outline the hypothesis of E.Bauer in more

concrete, thus, in more vulnerable way within the bounds of

substitutional approach. Detailed statement of substitutional

approach, for natural systems' description is given in works

of A.P. Levitch (Левич, 1989а,1989б). Term "metabolic"

concerning time, approach etc., being used in those works is

replaced by term "substitutional" in present paper. Futher we

will need five following propositions.

1. On all levels of hierarchic structure of natural

systems general process of system elements' substitution

occurs.

2. There are substantial flows on certain hierarchic

levels, which produce general processes in a system.

3. One of those flows is a substantial flow, generating

time passage in the Universe.

4. Life implies the possibility to accumulate and use the

non -equilibrium of flows of various structural levels, which

are deeper than molecular one, in particular, from the level

with flow generating time.

5. Individual existence of organism consists of the

depletion of substances of the flow, generating life.

The first three postulates are taken exactly from the

substitutional approach. They are helpful prerequisits for

transfer to living systems description and are equivalent to

the part of Bauer's principle, concerned with the necessity of

non equilibrium in life description. The fourth postulate

implicates specific character of living systems in

substitutional approach and corresponds to the statement on

the insufficiency of energetic metabolism of food for

producing the potential of the structural energy's the

nonequilibrium of living matter. The correctness of the fifth

postulate is based in the first place on Bauer's arguments

(Бауэр, 1935, с.130-133) and recent data on the existence of

Rubner's constant (Зотин, Алексеева, 1984).

It is possible to propose other arguments to support the

fifth postulate. One of these arguments is based on assumption

about the similarity of growth curves of multicellular

organisms to those of population growth of unicellular

organisms. Ordinary S-form of growth curves is appropriate to

populations, growing on a limited nutrient substrate, for

example, in batch -cultures. In continuous cultures with

incessant nutrient inflow growth curve is presented by growing

exponence, i.e., only by left part of S-shaped curve. From the

fact that growth curve of multicellular organisms is always

stationary one can draw a parallel with exhaustion of some

substance subordinate to the law of conservation. The

substance is consumed by the organisms for the period of their

life. Let me notice that the hypothesis of consumption of

embryonic cell substrate throughout the life period is not

necessary for explanation of existence of sigmoidal growth

curve. Only limitation of total accessible intra and

extracellular substrate only is necessary. For example, batch

cultivation of unicellular (coenobial, colonial) algae

populations in a flask is divided into following growth stages

(Левич с соавт.,1986): stage A, accumulation of intracellular

substrates, stage B, growth using nutrients of culture medium,

stage C, growth on intracellular storage of nutrients after

exhaustion of a growth-limiting nutrient in the flask.

Exponential fragment of the curve involves both B stage and

part of C stage. Another part of C stage is expressed by

stationary branch of the curve and includes decrease of

cellular limiting substrate quota down to some minimal species

-specific amount.

The depletion of deep substance is not the only possible

reason of the growth inhibition. One can propose several

reasons of the attenuation of cells' division in multicellular

organisms,for example, autometabolism, the influence of the

gravitation, the limits of rates of nervous impulse conduction

etc. If the life of organism consists of the depletion of some

substance, then the unicellular organisms must become old, and

the curve of the cellular strains' development must be

sigmoidal. This has been observed by some authors (Hayflick

phenomenon) and disproved by others (Хохлов,1988).

The part of the present work connected with the sources

of nonequilibrium is based on the interpretation of Bauer's

ideas. This interpretation implies that free energy of food is

not the only source of replenishment of the structural

energy's potential. It should be mentioned that this is not

the only possible interpretation. G.E.Mikhailovsky, to whome

the author is greatly pleased with the explanation and

critique, proposes another one. He believes that the

"debasement" of living matter because of the decrease of the

potential can be presented as the result of the accumulation

of genetic disturbances, rather than the depletion of

hypothetical substance. G.E.Mikhailovsky compares the

nonequilibrium of living matter to the accumulator that can be

recharged, but working period of which is limited because of

the debasement of its construction.

An experienced reader can consider structural energy of

E.Bauer and hypothetical substantial flows of substitutional

approach to be the progeni of the ideas of vitalism, for

example, of the Aristotle's entelechia, Wolff's vis

essentialis or nisus of Blumenbach (Дриш, 1915). However, the

statements of the substitutional conception are more prosaic.

This conception considers quite real levels of the natural

systems' structures. It is merely impossible to find these

levels with the help of existing scientific technologies.

Hypothetical flows of elements of those levels are necessary

not for the introduction of some "vital forces", but for the

derivation of the whole set of logical constructions,

connected with the time phenomenon by means of substitutional

approach.

Naturally, the "flask" model of life arouses a lot of

questions and comments. However, at the present stage of its

development the efforts should be aimed at detection and

identification of hypothetic flows generating non-equilibrium

state of living systems. That is why some works by Кozyrev and

his followers were considered above. If the interaction

revealed by them is not an artefact it will confirm the

existence of flows and prove both Bauer's principles and

substitutional approach. Specific metabolic flows generating

non-equilibrium state of living matter can show themselves in

some interactions between living objects. They are still not

interpreted by biologists. Superweak intercellular

interactions (Казначеев, 1981) or Gurvitch's morphogenetic

fields (1944) may be considered as examples of these

phenomena. Let me notice that the conception of morphogenetic

feilds has something in common with both Bauer's views, for it

regards "non-equilibrium deformated states of nucleoproteids"

(Гурвич, 1986, с.392) as the elementary sources of the field,

and with substitutional approach, for it states that

substantial contents of "the cellular fields conception

consists in ... mutual dependence of different levels, namely,

organisms, cells and molecules, correcting their relations by

actual fields" (Гурвич, 1986, с. 392).

4. Towards a dynamic theory.

Now let us return to the problem concerning possible ways

of description of the motion laws of living matter.

Substitutional construction of time gives the following

elements for dynamic theory:

-an elementary object, i.e., a system involving several

hierarchic levels (a concrete organism);

-state space, i.e., natural hierarchy, the object under

consideration being the part of it, e.g., the biological

hierarchy including ... molecules, cells, organisms,

populations, communities, the biosphere ...;

-unification of the ways of variability in natural systems

(replacements of elements on different levels of an object);

-the metabolic clock, i.e., number of replaced elements

in certain model object.

Description of metabolic motion by equations requires

generalization of the substitutional construction. The point

is, that formalized natural systems are usually described by

structurized sets. Thus, it is suitable to use the structure

of sets with subdivisions for description of ecological

communities involving specimens of different species.

Subdivisions correspond to populations forming the community.

The notion of proximity and distance of points in empirical

space is described mathematically by topological structure.

The totality of atomic states can be described by vectors

of infinite-dimensional Hilbert space or, in equivalent way,

by the field of infinite matrices.

Application of just these structurized sets is of special

importance for systems formed of several types of elements or

subsystems of the same hierarchic level. Such systems are all

biological systems. For example, vital functions of a cell are

connected with replacements of different molecules. The rates

of exchange differ significantly. sometimes it is suitable to

choose a so-called "limiting" element or total amount of all

types of molecules and to assume it to be the metabolic clock.

However, this does not suit in other cases.

A living organism consists of differentiated cells. Rates

of cell reproduction in different tissues and organs vary

significantly. Thus, it is a question which type of

reproducing cells (e.g., epithelium, neurons or erythrocytes)

determine the biological age of an animal.

Substitutional approach requires one to be able to

calculate the number of elements in objects. Therefore, when

this approach is used for analysis of structurized sets it is

necessary to generalize a notion of "number of elements" for

these sets. Arbitrary structurized sets can be described by

the theory of categories and functors, a specially created

mathematical language.

Generalization of the "number" notion for structurized

sets leads from cardinal (particularly, natural) numbers to

structural ones. However, structural numbers are only partly

regulated. Further generalization of quantitative

characteristics in terms of the theory of categories leads to

the method of functor comparison of structures (Левич,1982,

1989б, 1991). Functor invariants of structurized objects allow

to suggest L-component of dynamic theory. This component is

formulated as the extremum principle (Levitch,1988; Ле-

вич,1991):a system turns from the given X state to that A

state which has maximal entropy H(A) within a range, allowed

by available external resources. Entropy of systems is

calculated using invariants of functor system structures. If

functional, i.e., entropy is known, variational procedures

enable us to derive equations and trajectories of generalised

motion of the system. Owing to extremal principle, the value

of entropy does not decrease along real trajectories of the

system, i.e., succession of real states, or natural evolution

of the system, is regulated by its values. Thus, entropy acts

as the parametric time of the system, monotonic if related to

its metabolic time. Together with the metabolic time, entropy

can form T-component of the theory.

A detailed account of the case of application of

methodology described above to obtain dynamic equations in

community ecology is given in the author's previous works (Ле-

вич, 1980, 1982, 1989б, 1991; Levitch, 1988).

References.

1. Abakumov, V.A. Duration and frequency of generations.-

Trudy VNIRO, 1969, v. 67, p.344-356.

(Абакумов В.А. Длина и частота поколений. - Тр. ВНИРО.

1969, т.67, с.344-356.)

2. Akchurin, I.A. The unity of natural-scientific knowledge.

Moscow, Nauka 1974, 207 pp.

(Акчурин И.А. Единство естественно-научного знания. М.:

Наука. 1974. 207c.)

3. Alexeyev, V.P. The time vector in the taxonomic

continuum. Voprosy Antropologii, 1975, 49th issue, p.

68-77.

(Алексеев В.П. Вектор времени в таксономическом континуу-

ме // Вопр. антропологии, 1975, вып.49, с.66-77.)

4. Bauer, E.S. Theoretical biology. M.-L., VIEM Publishers,

1935, 206 pp.

(Бауэр Э.С. Теоретическая биология. М.-Л.: Изд-во ВИЭМ,

1935, 206 с.)

5. Danchakov, V.M. Some biological experiments in the light

of N.A.Kozyrev's time conception.// Eganova, I.A. An

analytic review of the ideas and experiments of modern

chronometry. Novosibirsk, 1984. VINITI deponent No.

8423-84, p. 99-134.

(Данчаков В.М. Некоторые биологические эксперименты в све-

те концепции времени Н.А.Козырева //И.А. Еганова. Аналити-

ческий обзор идей и экспериментов современной хронометрии.

Новосибирск. 1984. Депонировано ВИНИТИ. No 6423-84,

с.99-134.)

6. Danchakov, V.M. and Eganova, I.A. Micro-field

experiments in the investigation of the effects of

physical irreversible processes. Novosibirsk, 1987.

VINITI deponent No.8592-E87, 110 pp.

(Данчаков В.М. Еганова И.А. Микрополевые эксперименты в

исследовании воздействия физического необратимого про-

цесса. Новосибирск. 1984. Депонировано ВИНИТИ. No

8592-B87, 110 c.)

7. Detlaf, T.A. and Detlaf, A.A. Dimensiionless criteria as

a method of quantitative characterization of animals'

development // Developmental Mathematical Biology.

Moscow, Nauka 1982, p. 25-39.

(Детлаф Т.А. Детлаф А.А. Безразмерные критерии как метод

количественной характеристики развития животных // Мате-

матическая биология развития. М.:Наука,1982, с.25-39.)

8. Drish, G. Vitalism. M., Nauka 1915, 279 pp.

(Дриш Г. Витализм. М.: Наука, 1915. 279 с.)

9. Friedmann, A.A. The world as space and time. M., Nauka

1965, 111 pp.

(Фридман А.А. Мир как пространство и время. М.: Наука.

1965. 111 с.)

10. Golovakha, E.I. and Kronik, A.A. A personality's

psychological time. Kiev, Naukova Dumka, 1984, 207 pp.

(Головаха Е.И., Кроник А.А. Психологическое время лич-

ности. Киев: Наук. думка. 1984. 207c.)

11. Gurvich, A.A. Mitogenetic measurement of biological

systems as an indicator of a regulating interaction of

the molecular and cellular levels. Uspekhi Sovremennoy

Biologii, 1986, v. 103, 3rd issue, p. 390-397.

(Гурвич А.А. Митогенетическое измерение биологических

систем как показатель регулирующего взаимодействия моле-

кулярного и клеточного уровней // Успехи современной био-

логии. 1986. т.101, Вып.3, с.390-397.)

12. Gurvich, A.G. Biological field theory. Moscow, Sov.

Nauka, 1944, 153 pp.

(Гурвич А.Г. Теория биологического поля. М.: Сов. наука.

1944. 153 с.)

13. Harland W.B., A.V.Cox, P.G. Llewellyn, C.A.G.Pickton,

A.G.Smith, R.Walters. A geological time scale.

-Cambridge. 1982.

14. Kaznacheyev, V.P. and Mikhailova, L.I. Superweak

radiations in inter-cellular interactions. Novosibirsk,

Nauka 1981, 144 pp.

(Казначеев В.П. Михайлова Л.И. Сверхслабые излучения в

межклеточных взаимодействиях. Новосибирск: Наука. 1981.

144 c.)

15. Khokhlov, A.N. Proliferation and aging. //Itogi Nauki i

Tekhniki. Obshchiye Problemy Fiziko-Khimicheskoy

Biologii, v.9. M., VINITI 1988, p. 105-174.

(Хохлов А.Н. Пролиферация и старение.// Итоги науки и

техники. Общие проблемы физико-химической биологии. Т.9.

М.: ВИНИТИ. 1988. с.5-174.)

16. Kozyrev, N.A. Stellar energy sources and the theory of

internal stellar structure // Izv. Krym. Astrofiz.

Observatorii, 1948, 1st issue, p. 1-43.

(Козырев Н.А. Источники звездной энергии и теория внут-

реннего строения звезд // Известия крымской астрофизи-

ческой обсерватории. 1948. Bып.1, с.1-43.)

17. Kozyrev, N.A. The theory of internal stellar structure

and stellar energy sources // Izv. Krym. Astrofiz.

Observatorii, 1950, 6th issue, p. 54-83.

(Козырев Н.А. Теория внутреннего строения звезд и источ-

ники звездной энергии. Часть 2. // Известия крымской аст-

рофизической обсерватории. 1950. Bып.6, с.54-83.)

18. Kozyrev, N.A. Causal or non-symmetriic mechanics in the

linear approximation. Pulkovo 1958, 88 pp.

(Козырев Н.А. Причинная или несимметричная механика в ли-

нейном приближении. Пулково. 1958, 88 с.)

19. Kozyrev, N.A. Causal mechanics and the possibilities of

an experimental study of the properties of time //

Istoriya i Metodologiya Estestvennykh Nauk, 2nd issue,

Fizika. M, MGU 1963, p. 95-113.

(Козырев Н.А. Причинная механика и возможности экспери-

ментальных исследований свойств времени // История и ме-

тодология естественных наук. Вып.2. Физика. М.: МГУ.

1963. с. 95- 113.)

20. Kozyrev, N.A. Man and Nature // N.A.Kozyrev's Archive,

Pulkovo 1975.

(Козырев Н.А. Человек и природа // Архив Н.А.Козырева.

Пулково. 1975.)

21. Kozyrev, N.A. Astronomical observations using the

physical properties of time // Flashing Stars. Yerevan

1977, p. 209-227.

(Козырев Н.А. Астрономические наблюдения посредством фи-

зических свойств времени // Вспыхивающие звезды. Ереван.

1977. с.209-227.)

22. Kozyrev, N.A. Time as a physical phenomenon //Modelling

and Forecasting in Bioecology. Riga, Latvian Univ. Press,

1982, p. 59-72.

(Козырев Н.А. Время как физическое явление // Моделирова-

ние и прогнозирование в биозкологии. Рига: ЛатГУ. 1982,

с.59-72.)

23. Le Temps et la pencee Physique contemporaine. Paris.1968.

24. Landau, L.D. and Lifshitz, E.M. Statistical physics. M.,

Nauka 1964, 567 pp.

(Ландау Л.Д., Лившиц Е.М. Статистическая физика. М.:Нау-

ка. 1964. 567 с.)

25. Levich, A.P. Structure of ecological communities. M.,

Moscow Univ. Press 1980, 181 pp.

(Левич А.П. Структура экологических сообществ. М.: Изд-во

МГУ. 1980, 181 c.)

26. Levich, A.P. Sets theory, the language of category theory

and their applications in theoretical biology. M., Moscow

Univ. Press 1982, p. 190.

(Левич А.П. Теория множеств, язык теории категорий и их

применение в теоретической биологии. М.: Изд-во МГУ.

1982, с.190.)

27. Levitch A.P. What are the Possible Theoretical Principles

in the Biology of Communities? // "Lectures in

theoretical biology". Tallinn: Valgus. 1988. 121-128.

28. Levich, A.P. The metabolic time of natural systems.//

Sistemnye Iissledovaniya (Annual), 1988. M., Nauka 1989,

p.309-325.

(Левич А.П. Метаболическое время естественных систем //

Системные исследования. Ежегодник, 1988. М.: Наука.

1989а, с.309-325.)

29. Levich, A.P. Time as variability of natural systems and a

way of its parametrization. VINITI deponent No. 7599-B89.

M., VINITI 1989, 101 pp.

(Левич А.П. Время как изменчивость естественных систем и

как способ ее параметризации. Рукопись депонирована ВИНИ-

ТИ. No 7599-В89. М.: ВИНИТИ, 1989б, 101 c.)

30. Levich, A.P. Entropy parametrization of natural systems'

variability.// Sistemnye Issledovaniya (Annual), 1990.

M., Nauka 1991.

(Левич А.П. Энтропийная параметризация изменчивости

естественных систем.// Системные исследования. Ежегодник

1990.М.: Наука. 1991.)

31. Levich, A.P., Revkova, N.V. and Bulgakov, N.G. The

"consumption-growth" process in micro-alga cultures and

the cells' need of mineral nutrition components.//

Ecological Forecast. M., Moscow Univ. Press, 1986, p.

132-139.

(Левич А.П., Ревкова Н.В., Булгаков Н.Г. Процесс "потреб-

ление-рост" в культурах микроводорослей и потребности

клеток в компонентах минерального питания // Экологи-

ческий прогноз. М.: Изд-во МГУ. 1986, с.132-139.)

32. Poincare M. "La Mesure de Temps" // Revue de Metaphysique

et de Morale. 1898. t. VI, p.1-13 (см. кн.: Принципы от-

носительности. М.:Атомиздат, 1973, с.12-21).

33. Prigogine I. From Being to Becoming. San Francisco. 1980.

34. Simakov, K.V. Theoretical foundations of geological time

division.// Geologiya i Geofizika, 1977, No.4, p. 49-57.

(Симаков К.В. Теоретические основы подразделения геологи-

ческого времени // Геология и геофизика. 1977. No 4,

с.49-57.)

35. Svirezhev, Yu.M. and Pasekov, V.P. Foundations of

mathematical genetics. M., Nauka 1982, 512 pp.

(Свирежев Ю.М., Пасеков В.П. Основы математической гене-

тики. М.:Наука. 1982. 512 c.)

36. Whitrow G.J. The Natural Philosophy of Time. London.1961.

37. Zotin, A.I., Alexeyeva, T.A. Rubner's constant as a

criterion of species' life duration // Fiziiol.Zh., 1984,

v.30, No.1, p.59-64.

(Зотин А.И., Алексеева Т.А. Константа Рубнера как крите-

рий видовой продолжительности жизни //Физиол. ж.1984.

Т.30. No 1. с.59-64.)