Ethnos as Unit of Society Model

Introduction

The society is a compound object. The people are an element of this object. The society is evolved by fixed rules. But these rules do not found. The finding of these rules is a goal of modern science. The mathematics may help social science in modern investigation. The formalised language of mathematics helps the science to be free of subjective. The modern methods of modeling permit to create and to investigate the various models. The evolution of society is a mysterious phenomenon. The creation model of society is a very difficult task. But we must try.

Some attempts are made in papers [GUT 97], [KOR 97]. The model of this paper is an attempt to create a new model. The distinction of this model is an area modeling, to register the situation of element and the bound interaction.

1. Biosphere Modeling

The society is a multilevel system. I can mark the four levels. Any level is a system of the human actions. These levels are Biosphere, Ethnosphere, Sociosphere and Psychosphere.

Vladimir Vernadsky described the level of Biosphere. His theory was created in beginning of 20th century. Biosphere theory was a push to the creation new approaches in the research of Earth and society. This theory notes the unity of living nature. The biomass is a main concept of this theory. The biomass forms the any part of Biosphere. A people, flora, fauna and all of living are the biomass. The birth, growth, feeding and death are the processes of the biomass change. The feeding process is process of the biomass increase by one and the biomass decrease by other. On the basis of these suggestions has made modern research. The model "predator-victim" by Lotki-Volterra based on these suggestions too. The system of equations describes these model is

here x is the number of victims (or its biomass), y is the number of predator (or its biomass), a is a speed of victim birth, b is a speed of victim decrease in result of predator feeding, d is a speed of predator death, h is a speed of predator increase in result of feeding.

Because x and y are the incessant images than named these function as a number of animals it is impossible. The number of animals doesn't incessant function. Better to name they as a biomass of animals.

I give the phase portrait of equation system (figure 1).

Figure 1. Phase portrait of system "predator-victim"

On the figure the horizontal axe is the victim biomass, the vertical axe is the predator biomass. As you see from the figure 1 the solution of equation system has the cycled type. That answers the description of nature processes. The increase of predator biomass brings to the decrease of victim biomass (curve AB, see the figure). The predators destroy the victims. Afterwards the stock of food is exhausted and the predators die by starving (curve BC). The biomass of predators sharply decreases. When the biomass of predators decreased the beginning to growth the biomass of victims (curve CA). The big number of victims brings to increase the predators. The cycle has closed. But the cycle can destroy if the biomass one of them has approach to the zero.

The model "predator-victim" is a simple example of the biosphere part model. But the approaches of Volterra are used in different models.

The model of Biosphere must describe all elements of the feeding chain. As well as the change of the oxygen concentration, CO2 concentration, temperature of air, temperature of water, moving of the water mass and more other factors. Many ones tried to create the model of Biosphere. The model of Rome Club and other research centers. I want to mark one of them. This is a model by Krapivin, Svirizhev, Tarko described in the monograph [KRA 82]. This model describes by 24 differential equations. Any equation is the evolution model of one subsystem. These subsystems are biomass of forests, biomass of grass and agriculture, biomass of people, biomass of animals, humus (mineral sediment), pollution, biomass of fitoplankton (the green plant of ocean), biomass of nekton (the animals of ocean), biomass of detrit (organic sediment), concentration of oxygen, concentration of carbon. As well as the water streams is included in the model. The water of ocean, water of atmosphere, water on the landscape surface, soil water, biological water and water of ice. I give one equation (subsystem of fitoplankton biomass)

This example demonstrates the difficulty of the model. In this model we can view the elements of the Lotki-Volterra equation. Here RF describes process of birth, MF process of die, Rr feeding process. But on the process of birth is influenced many factors. There are the density of population, luminosity of water surface, concentration of carbon, concentration of pollution and change of temperature. Other equations have the qsimilar type.

I tried to make the computer simulation of this model. I made the software that allowed the simulation. I select the value of parameters. The number of parameters exceeded the three hundred. It was the hard work. The result of simulation gives in the article [KOR 99-2]. The description of the software was in the article [KOR 99]. I give one graph getting in the simulation (figure 2).

Figure 2. Evolution of fitoplankton and nekton biomass (from [KOR 99-2])

On the figure the horizontal axe is the time of evolution, the vertical axe is the biomass.

2. Sociosphere Modeling

The Sociosphere is a system of people action on the social level. In this field I include the government system, political and economical systems. This is a level of social interactive of people. We can argue about the type of model which can described this level. But one confirm that this level exists.

A modeling approach of sociosphere is contained in the work [GUT 97], [LAP 99]. This approach based on the society theory by T. Parsons. The model consists of the four subsystems: societal community (black on figure), system of maintenance of ethnical samples (blue), political system (cyan) and economical system (green). Any subsystem is described by differential equation. The example of simulation I give below (figure 3).

Figure 3. Sociosphere Evolution

On figure 3 indicated the graphs of evolution of social subsystems. The horizontal axe is the time. On the vertical axe we measure the power of subsystems.

The particularity of social processes reveals in the cycle change of economical and political power. In the sociology mark the cycle evolution of economy and policy. This is reminding the model "predator-victim". The political system is a predator and the economical system is a victim. The policy oppresses the economy. May be this is a trivial view on this theme but this way very well confirms in Russia. The economy doesn't will develop while the policy system became the less active system. When the political conflicts have abated and the experiments under economy have stopped we will have strong economy.

I understand that this modeling approach on social system doesn't unique. The Luhmann's theory of society is the more modern theory than Parson's theory. But I don't know the models of society created on the basis of Luhmann's theory. Though the concept of autopoesis is the tempting prospect in the developing of social modeling. The selforganization of the social subjects is the basis for new models. This is such model which can reproduced itself. We can use the theory of bifurcation for construct new model.

3. Ethnosphere Modeling

An ethnosphere is a level of people interaction that one is a member of own ethnos. An ethnos is an organisation level between a biosphere and a sociosphere. This level is an ability of people to divide on own and alien. The division is made by complimentary sensation and behaviour stereotype. The ethnos members never recognise the members of other ethnoses. The majority of warfare is arisen between hostile ethnoses. The armistice of ethnoses does not last long time.

The drive energy is a force that controls to ethnos. The drive energy is a surplus biochemical energy that depresses a self-preservation instinct and promotes to accomplish an overeffort. The people is separate into three classes: passionary, subpassionary and harmony people. The harmony people are people that drive energy approaches to zero. The drive energy of subpassionary is less than a zero. The self-preservation instinct predominates by them. The drive energy of passionary is more than a zero. Its self-preservation instinct depressed and they ready to do the super-effort. This people are great commanders as like Alexander Makedonsky, Chingiz Khan, Napoleon Bonaparte and Alexander Suvorov. They are at war always. The sense of rivalry put in they by nature.

The ethnos is not only a people. The ethnos is a dynamic system that includes people, organization, landscape, culture, art, science and technology. These components define an originality of ethnos. One has a strong power and modern technologies but other has an original culture and does not have powered state. At that these components evolute. The change is arised under influence of passionaries. As in a military as in a science and culture exists the passionaries who develop they. One can call the many names of famous scientists and we can regard they as passionaries.

Dr. A. K. Guts created the model of the ethnosphere [GUT 97]. This model is a system of differential equation consisted from 7 equations. The 7 equations are the 7 subsystems: passionary, subpassionary, harmonic people, organization, science and technology, culture and art, landscape. Any subsystem has the reserve of drive energy. All subsystems interaction with themselves and the drive energy is flowing from one to other. I give the one equation

here Pi is energy of passionaries, the summands with '+' describes the inflow of energy and summands with '-' describes the outflow of energy. The summands is the different factors of energy change. The other equations have similar type.

The index of ethnos state is a passionary intensity. The passionary intensity is an average of ethnos drive energy. We define it as a sum of drive energy of all ethnos members divided on total quantity of people. On the figure 4 you can see the graphs of ethnos component evolution and passionary intensity.

Figure 4. Ethnos Evolution

The curve of passionary intensity is an invariant for any ethnos. It defines the phases of ethnogenesis. Any ethnos passes all phase of ethnogenesis in defined order. These phases are a rise, akmatic phase (overheat), crisis, inertial phase and obscuration (collapse). Any phase last about 200-300 years. The maximum duration of ethnos life equals about 1500 years. However, no one has lived for last phase. Some of they perish in intermediate phase. The cause of destruction is inner and outer conflicts. The rivalry of tribes, classes, sects and castes is an inner disturber in ethnos. The rivalry between ethnoses is the control factor of ethnos influence distribution. In order to pass the simulation under such compoud object as ethnosphere we construct the model of ethnos fields. The model demonstrates a space influence of some ethnoses. We construct the ethnos drive energy field based on some rule of energy dynamic.

4. Mathematical Model of Ethnos Fields

The drive energy of ethnos is a sum of drive energy of ethnos member. The density of drive energy is a ratio of ethnos drive energy to area of ethnos landscape. The four rules defines the density change:

1. induction
2. loss
3. dissipation
4. shift

The drive energy has an induction that is the passionary infect the harmony people located near it. In addition, passionary sign has transmited to descendant. The children of the passionary possess a passionary sign too.

The drive energy is lost in the warfare and on the transformation of landscape. The warfare starts on the bound between ethnoses in consequence of discord. The discord arises on the division of territory. The ethnogenesis period define the loss of energy on the landscape transformation. For the first period the transformation is a maximum, therefore the loss is a maximum too.

We divide the ethnoses into two types: braked and evolved. The braked ethnos saves own territory and exists on the own biohoure and evolved ethnos captures the new territory always and changes the biohoure. The member of this ethnos moves on the landscape. One may tell that the drive energy dissipated by the landscape. The energy dissipates in all direction is unevenly. In first direction a move is quickly, but in other direction one is slowly. The communication influences on the movement speed.

The ethnos moves the troops to the bound of territory in the warfare. The passionaries head the troops. Therefore, the drive energy shifts to the warfare direction.

Given below the model describes the dynamic of ethnos field. The ethnos field is the field arisen on the territory as a result of distribution of ethnos drive energy. Here we consider the energy of ethnos as the full energy of all subsystems of ethnos. It is the average value of drive energy in any point of territory. We average the energy by all subsystem and does not by territory that we see in previous paragraph. We see on the distribution of drive energy on the territory. This process describes the increase of state size and interaction of the ethnoses (wars and other territory conflicts).

Let G subset RxR is an area of ethnogenesis evolution. The curve $\Gamma$ is a bound of G. The constant T is an evolution time. The number of ethnoses be k. The function ui(x,y,t): G x [0,T] to R+, i=1..k (i is a number of ethnos). Then the ethnos drive energy density ui satisfies the differential equation

here $\alpha_i (x,y,t)$ is a function defined the energy capacity of landscape, $\epsilon_i (x,y,t)$ is an energy conductivity of lanscape, the vector $\nabla\phi_i (x,y,t)$ (gradient of function) is a direction vector of shift, $\beta_i(x,y,t)$ is an intensity of an induction and a loss of drive energy, $\gamma_{ij}$ is a coefficient of loss in the warfare, $\Delta u_i$ is a Laplas operator, (.,.) is scalar multiplication of two vectors.

The first summand $[\epsilon_i \Delta u_i + (\nabla \epsilon_i ,\nabla u_i)]$ describes the process of energy distribution. The Laplas operator usualy describe such processes. Thus in the mathematical physics it describes the heating in enveronment and diffusion in air. Exist more other examples of using this operator. The function $\epsilon_i (x, y, t)$ describes the speed of distribution. It can change in space and in time. The summand $(\nabla \epsilon_i ,\nabla u_i)$ appeares under conversation of line integral to double integral from Green formula.

The second summand $[(\nabla\phi_i, \nabla u_i) + \Delta\phi_i u_i]$ describes the energy movement under effect of outer forces. In this, the function $\phi_i (x, y, t)$ describes the energy movement direction field. In any point of G it defines the direction in that the energy will move in this moment of time. However, in next moment the direction will have to change. This summand describes the ethnos capability in energy concentration and shift it in necessery direction.

The next summand $\beta_i u_i$ describe the induction process. This process is a line function by energy density $u_i$. The function $\beta_i (x, y, t)$ may be as more than a zero as less. If value less a zero than the summand describe the energy loss process.

The summand $-(\Sum_j \gamma_{ij} u_j) u_i$ describe the process of inner and outer rivalry. The sum do by any ethnoses (j=1..k). The energy loses in warfare and conflicts. The loss is a proportional by energy density of i-th ethnos and j-th ethnos. The more density the more loss. In addition, the summand $\gamma_{ii} u_i^2$ describes the inner conflicts that the force is proportional by $u_i^2$.

The function $\alpha_i (x, y, t)$ describes the energy capacity of landscape. Therefore, the total energy reserve of ethnos on all territory we can calculate

here the integration do by all G. Considering such definition of function $\alpha_i$ we get the summand $-\partial\alpha_i\over\partial t u_i$ and denominator $\alpha_i$ in equation system (1).

The system (1) is a system of parabolic differential equations. This system describes a variation of ethnos drive energy for any ethnos in every point of G. Set the initial data and edge condition

The function $\psi_i (x,y)$ defines the first phase (t=0) of ethnos drive energy field. The condition (2') is necessery for correct problem formulation. It denotes the absence of flows over G.

The conclusion of this system made the author in [KOR 2000-1]. The existence and uniqueness of such equation system discuss in [KOR 2000-2].

5. Computer Simulation

The computer simulation was realized on the model of ethnos field. There were three ethnoses in the simulation: Western-Catholic, Slavic-Ortodox and Arabic-Muslim. Used the landscape map of Europe, Middle East and North Africa.

The author has created the software Terri to test this model. Every test is a solving differential equation system (1). The initial data (2) are set a drive energy impulse in the random point on the own landscape. The ethnos birth is set in the various moments of time in segment [0,400]. The solving has finished on 1000. The ethnoses passed the first periods of ethnogenesis. The example of software testing is on the figure 5.

Figure 5. Evolution of ethnos field

6. Conclusion

The series of tests is a pass of different version of global history. The statistical characteristics mark the most probability version. The computer model marks the most probability history hypothesis.

The bound between the ethnoses is a line on the landscape. Often this line is a physical barrier. It is a mountain or a strait. But the bound between Slavic and Western ethnoses floats from West to East.

The computer test in this paper is a first attempt to create the model of ethnos field. This model is needed to add the rule of increasing entropy, to correct the functions and to add the elements of ethnos system: an organisation, a culture and a science.

The model creation method of this paper illustrates a new approach in the social modeling.

References

1. [GUM 94] GUMILEV L.N., Ethnogenesis and Biosphere of Earth, Moscow, Tanais Di-Dik, 1994. (Russian)
2. [GUT 97] GUTS A.K., Global Ethnosociology, Omsk, Omsk University, 1997. (Russian)
3. [KOR 97] GUTS A.K., KOROBITSIN V.V., Computer Simulation of Ethnogenesis Processes, VINITI 24 sept. 1997, N2903-B97, Omsk State University, Omsk, 1997. 23 p. (Russian)
4. [KRA 82] KRAPIVIN V. F., SVIRIZHEV YU. M., TARKO A.M., Mathematical Modeling of Global Biosphere Processes, Moscow, Nauka, 1982.
5. [KOR 99] KOROBITSIN V.V., GUTS A.K., Software MEP for Simulation of Evolution and Social Processes, Vestnik of Omsk University, Omsk, no.2, 1997, p.23-25. (Russian)
6. [KOR 99-2] KOROBITSIN V. V., Computer Simulation of Biosphere, Mathematical Structures and Modeling, Omsk, no.3, 1999, p.96-108.
7. [KOR 2000-1] KOROBITSIN V. V., Model of Territorial Distribution of Drive Energy of Ethnos, Mathematical Structures and Modeling, Omsk, no.5, 2000, p.44-53.
8. [KOR 2000-2] KOROBITSIN V. V., Theorem Proof of Existence and Uniqeness for One Parabolic Equation, Mathematical Structures and Modeling, Omsk, no.5, 2000, p.54-60.
9. [LAP 99] LAPTEV A. A., Mathematical Modeling of Social Processes, Mathematical Structures and Modeling, Omsk, no.3, 1999, p.109-124.