The authors have declared that no competing interests exist.

On the basis of Hardy – Weinberg’s law the problem of inbreeding in a family tree and a population was investigated. With use of an inbreeding factor are received the discrete equation for a family tree and differential equation for a population. The numerical solution of the differential equation for a population was found and analyzed at various values of the inbreeding factor. Migration of inbred population is investigated in view of natural selection. It was shown that velocity of migration falls with increase of the inbreeding factor. Interrelation of the recessive allele frequency at woman for a migrating population with inbreeding factor and standard parameter of selection was found.

Problem of inbreeding is one of the major problems of genetics. Since times of the human occurrence the population size was very small. People lived separate tribes. Therefore, consanguineous mating or inbreeding has been submitted enough. In process of the population size increase the intensity of such mating was reduced. But till now the consanguineous mating are present at small communities.

Inbreeding are widely submitted at animals. Very much frequently the man uses consanguineous mating of animals at artificial selection to receive useful properties of animals: speed of run, force, fleshiness, a plenty of milk, a wool, etc.

Let's consider principles of inbreeding in a family trees and a population.

There are family trees and populations where the principle panmictic mating is essentially broken. Usually it is geographically or socially relatively isolated small communities. In these communities the inbreeding i.e. consanguineous mating of various orders (a parent - a child, sibs, cousins, uncle - niece, etc.), can be wide-spread.

The problem of inbreeding influence on the population is the important problem of genetics

In most cases the inbreeding in a family tree take into account on three generations. It is the most distant relatives who are taken into account at consanguineous mating there is second cousins.

The inbreeding is taken into account with the help of inbreeding factor.

The inbreeding factor is the probability two alleles are identical by origin.

Let's consider the method of factor inbreeding factor calculation in a family tree offered by Wright

For the inbreeding factor finding it is necessary:

to find for the marrying descendants all ways in a family tree which through the common ancestor connect these descendants.

to count up number of steps (alternations of generations) on each way.

Then the inbreeding factor is equal:

where_{1}_{2}_{i}- number of steps on each way.

Once again we shall note that the number of addends in the sum (1) is equal to number of ways which through the common ancestors connect marrying.

Let's consider concrete cases of family trees,

The marriage a father - daughter is incest,

F_{t} = 1/2.2^{-1} = 1/4

There is marriage a brother - sister (sibs),

Let each descendant who has married connects with partner two ways with two steps on the each way, i.e.

F_{t} = 1/2.(2^{-2} + 2^{-2}) = 1/4

The marriage between cousins is shown on

Each descendant who has married connects with partner the 2 ways with 4 steps on the each way, i.e.

F_{t} = 1/2.(2^{-4} + 2^{-4}) = 1/16

The formula (1) is easy for understanding from the following logic.

Obviously there 1/2 genes of the child is the common with each parent, 1/4 - with the grandfather and grandmother, 1/8 - with the great-grandfather and great-grandmother, etc. If from the woman to any ancestor leads the way with ^{-}^{η}. If from the man to any ancestor leads the way with μ steps the share of the common genes is equal ^{-}^{μ}. At the marriage of the man and woman with the common ancestor the share of the common genes with this ancestor will be ^{-}^{η}^{-}^{μ}^{-}^{m} where m = ^{-m} shows probability that the gene casually chosen at the woman will be identical by origin to the gene casually chosen at the man. The same size gives the contribution to the inbreeding factor _{t} one way through the common ancestor from the woman up to the man marrying.

For example, if the common ancestor will be away from marrying on ^{-(5+5)} = 1/2048 This size at the analysis of mating can be neglected. (

The kind of consanguineous marriage | Inbreeding factor _{t} |

Second cousins | 1/32 |

Cousin uncle - niece | 1/32 |

Third cousins | 1/64 |

The inbreeding factor _{t} is the probability of two alleles in posterity are identical by origin i.e. occur from same gene of ancestor _{t}.

Let's find total probability of the recessive homozygote

This formula follows from the following logic of mating: the origin of identical alleles in homologous autosomes is non-consanguineous (the probability of this hypothesis is P(H_{1}) = 1– _{t} )also arises homozygote_{1}) = q^{2}) or origin of identical alleles is consanguineous (the probability of this hypothesis is P(H_{2}) = F_{t} ) and arises homozygote_{2}) = q). The probability

Hence, the total probability of the recessive homozygote occurrence is equal:

where it is taken into account

Similarly there find total probability of the dominant homozygote

Thus, according to (3) and (4) total frequency of homozygotes (dominant and recessive) in inbred family tree increases on 2_{t}_{t }= 2pq(1-_{t}) since the sum of the all zygote frequencies is equal 1.

In view of inbreeding factor and also (3) and (4) the Hardy - Weinberg law in the following generation of inbred family tree for an autosomal genome needs to be written down the distributions of zygote genotypes as

Where _{n} is the dominant alleles _{n} - the recessive alleles _{n} + _{n} = 1.

The ratio (5) refers to also Wright’s ratio _{t }_{n}_{n}and reduction of the autosomal heterozygotes fraction in (1-_{1}) ones in comparison with a panmictic population.

But already through one generation the Hardy – Weinberg balance is restored that specifies the small importance of the separate consanguineous mating for a family tree:

Obviously, for a population as whole the separate act of the consanguineous mating will have still smaller importance than for a family tree.

The same effect of variation of zygote genotypes frequency is observed and for sex-linked genes.

Distribution of the genotypes linked tothe

The frequency of a dominant alleles_{m} and at the women _{f}. For recessive alleles_{m} and _{f} .

At mating there is a ratio of genotypes at the women according to product (_{f} + _{f}) (_{m} +_{m}).

Let's find the total probability of the recessive homozygote _{1}) = _{mn}_{fn}. For the consanguineous alleles the conditional probability of a genotype_{2}) = _{f}_{(n+1)}. At daughters, i.e. in generation

Hence, the total probability of the recessive homozygote occurrence is equal:

where _{mn} is the dominant alleles _{fn} - at women. For the recessive alleles _{mn} and _{fn} . Besides it is used _{mn} =1 - _{mn} and _{fn} = 1 - _{fn}.

Similarly finding the total probability of the dominant homozygote occurrence with use:

and a heterozygote P(Aa) = 1– P(aa) - P(AA) we shall find distribution of the genotypes in

It is simultaneously demonstrated that as well as in autosomes the inbred additives in dominant and recessive homozygote frequencies in the sex-linked chromosomes are identical.

Other way of the ratio (10) finding will consist in the following.

Let's find first the total probability of a heterozygote _{1}) = 1 - _{t} ) also arises heterozygote_{1}) = _{mn}_{fn} + _{fn}_{mn}, see the formula (7)) or origin of alleles is consanguineous (the probability of this hypothesis is P(H_{2}) = _{t}) and arises heterozygote_{2}) = 0 ). Last statement is defined by that at consanguineous origin similar alleles in homologous

Hence, using the formula of total probability we find total probability of the heterozygote

Reduction of a heterozygotes

where reduction of the heterozygotes _{t} (_{mn}_{fn} + _{fn}_{mn}) Using (11) and (12) to update (7) for cases of inbreeding we find the distribution of genotypes (10) in

In following generation the frequency of the recessive alleles

That fully complies with same frequency at absence of inbreeding, i.e. there is restoration of the Hardy – Weinberg balance or is more exact approach to the balance.

The ratio (7) and (13) show as a whole the allele frequencies in a family tree for the account of inbreeding do not vary. There is only an alleles redistribution. A part of alleles leave the heterozygotes and pass to the homozygotes.

In the inbred populations so-called the genetic load frequently connected to harmful mutations only recessive alleles passed to offspring is accumulates.

It is connected to increase in the recessive homozygotes

The concept of the genetic load in the population was used for the first time the known geneticist Nobel laureate H. J. Muller

The consanguineous mating in a population increase homozygotes frequency of harmful mutant alleles. Therefore, it is supposed that inbreeding is one of principal causes of occurrence and accumulation of the genetic load.

The genetic load can be expressed in lethal equivalents. The lethal equivalent is, for example, one lethal mutation resulting in death of an individual in all cases or two lethal mutations each of which resulting in death of the individuals in 50% cases, etc.

At research of the genetic load in a family tree the action of selection we shall not take into account to not complicate the analysis. It is allowable since selection submits to other laws rather than occurrence of mutations.

According to the genetic load theory the formula (10) needs to be transformed

where

Let's notice that the ratio (14) concerns to a family tree as separate usually small making population.

Using (14) we shall find the recessive allele

In the following generation there is no restoration of the Hardy - Weinberg balance, i.e. accumulation of the genetic load in a family tree, and hence and in a population is observed.

Let's enter the designation _{t}_{t}s - factor of inbreeding for the population. We shall note that if the heterozygotes are resistant to mutations owing to inbreeding _{t}. Thus the formula (15) can be written down as:

Lack of the formula (16) is absence normalization on the general fraction of genes less than 1 the rest after lethal mutations. However the standard way of normalization results in too complex formulas which do not allow proceed to the analysis of the population.

For updating the formula (16) it is used the following logic. In the condition of balance of the family tree with inbreeding the formula (13) for calculation of the recessive allele

Taking into account (13), and also F<<1 and small size of the second addend in the right part (16) it is possible to assume that for normalization (16) it is necessary to divide the right part into value 1 - F .

Therefore, the corrected formula (16) looks like:

Accuracy of the formula (17) can be estimated as follows. Using the formula similar (13) for frequency of dominant allele_{f(n+1) }_{f}_{(n+1) }_{fn}_{fn}_{mn}_{mn}_{mn}_{fn} + _{mn}_{fn}) F/(1 - F) i.e. concerns to higher (second) order of smallness on allele frequencies.

The formula (17) allows find the recessive allele

Let’s carry out the analysis of the inbreeding action on a population.

At research of inbreeding in populations the action of selection we shall not take into account (it will be taken into account later) to not complicate the analysis. It is allowable since selection submits to other laws rather than occurrence of mutations owing to inbreeding.

Using a standard technique of transition from a family tree to a population

Taking into account that the man inheritsthe _{mn} = _{f}_{(n-1}_{)} it is possible to transform (17) to kind:

where

The differential equation for change of frequency _{fn} of recessive allele

where the independent variable

Let's pass in (19) to finite-differential form:

Let's collect the like terms:

Let's multiply (21) on -2 besides we shall replace

that is allowable for continuous in time of the generation alternation scale i.e. for a population:

Let's identify (22) and (18). In result we have system of three algebraic equations:

The solution of the given system looks like:

that similarly the equilibrium panmictic population,

Hence, the differential equation (19) can be copied as:

The number of generation for the population does not play a role - the continuous time scale therefore, an index

The found nonlinear differential equation (28) is the Hardy-Weinberg law for a population with the account of inbreeding.

At F = 0 (inbreeding is absent) the equation (28) it will be transformed to the kind that corresponds to the equilibrium panmictic population:

We use in (28) new independent variable . t = n/Δn

we shall find

For the new variable

where it is designated

The nonlinear differential equation (29) can be solved only numerically.

Initial conditions first of all are necessary for the solution of the differential equation (29): initial frequency _{f}_{0} of the recessive allele _{f}/ dt)_{0} or (_{f}/dn)_{0}.

Initial frequency _{f}_{0} depends on time of the beginning of the population analysis. Therefore, without reduction of the analysis generality at t = 0 or n = 0 we shall accept _{f0} = 0.6

Various variants of numerical calculations under the formula (29) show that the result of calculation at small initial speed of the frequency increase _{f} practically does not depend on this speed. Therefore, was accepted. (_{f}/ dt)_{0} = (_{f}/dn)_{0} = 0 The step of the variable

On

From graphs it is visible that at presence of inbreeding in the population there is continuous increase of the recessive allele _{f} in which can arise lethal adverse (at association in homozygotes) mutations i.e. there is the negative genetic load in the population. The growth of the recessive allele frequency received in calculation is observed due to growth of the recessive homozygotes _{f} on all zygotes in a population (

Traditionally

S = exp (A+BF) ……..(30)

Unfortunately the data on experimental sizes of parameters

More detailed data are available on the population of Japan. In different prefectures the ratio B/A = 15.2-(-5). Negative value B/A means that children’s death rate in consanguineous marriages was lower than in non consanguineous marriages. Average sizes on the country A = 0.1036, B = 0.67 and B/A = 6.7

It is possible to assume that there is some threshold of a recessive homozygous genes allowable frequency compatible with a survival of individual.

Taking into account values

gives the fraction of lethal recessive alleles at association theirs in homozygous

_{f} ≈ 0.316 _{f}_{ max }= 1-_{f }=0.684.

On

The received result shows that accumulation of the genetic load it is process very slow. Achievement of the threshold frequency for Japan to be carried out only in 167 generations or 4175 years at time of the one generation alternation T = 25 years.

In ^{-5}. Increase of the mutant alleles number was compensated by selection against recessive homozygotes with parameter of selection 0.5.

Therefore, most likely that limiting values of the genetic load in a human population of many countries especially taking into account small sizes of inbreeding factor (Canada F = 45.10^{-5}, France F = 23.10^{-5}, Germany F = 19.10^{-5}, Italy F = 16.10^{-5}

In our opinion with help of the offered method it is possible to examine not only harmful for the population genetic load related with inbreeding but also useful and indifferent genetic load (or attributes).

At early stages of mankind development the role of inbreeding was more in connection with smaller number of people and isolation of separate groups. Therefore, accumulation of the genetic load was more intensively (the inbreeding factor was much more). Primeval people after the going out from Africa and moving in northern Europe began to live in conditions of smaller light exposure including in the ultra-violet part of spectrum. In result the skin with originally dark pigmentation started to have white color. It promoted occurrence of vitamin D which deficiency results in rickets

To the indifferent genetic load in population it is possible to ascribe occurrence such recessive attribute as blond hair, for example, in northern part of Europe where apparently at early stages the inbreeding has been especially widespread. Color of hair does not play a role in survival of population. Fixed of this attribute was carried out by the sexual selection i.e. marriage preferences of people in region.

However it is traditionally accepted to examine accumulation of the genetic load in the population are determined by harmful mutations. Some fears of H. J. Muller are connected to this process about danger of the future biological degeneration of mankind

As it was already specified above in primitive populations, for example, leaving of Africa and moving aside northern Europe influence of inbreeding was important. Therefore, we shall consider influence of inbreeding on a migrating population.

The way of the inbreeding account for a motionless population has been analyzed in paragraph 3, equation (28). Adding in the equation (28) for allele frequency _{f} motionless inbred populations the item reflecting movement of a population

The found equation is the nonlinear differential equation of the second order with square-law nonlinearity. _{f} =

Let’s lower the order of the equation (32) having designated

We shall receive Abel’s differential equation:

Where ω/k = V there is speed of the population movement. Let’s note that the received differential equation does not depend on size Δn.

The analytical solution of the equation (33) does not exist. However the greatest interest has dependence of the population movement speed

Let's consider the right part of the equation (33) which defines the inbreeding in a population. The inbreeding factor _{f} = _{f0} = _{f} grows, _{f} growth. Therefore, the inbreeding factor in numerator of the equation (33) right part should be kept at any transformations.

Taking into account the given position we shall transform the right part (33) as follows:

All parameters connected to the wave phase are present only at the second addend in brackets of the right part (34). Therefore, it is possible to conclude that for the account of the inbreeding factor influence on the moving population speed it is necessary to multiply the speed of the population wave in

Using result received in

On ^{2}/year, T≈25 years.

The received result,

At F = 1 according to (35) the population stops. The reason of this stop there is fast and full degeneration of the population. If to address to the equation (28) for the motionless population (or (31) at D = 0) at F _{f} growth in the left part of the equation (28). But this speed is limited by the biologo-reproductive opportunities of the population. Therefore, in a reality achievement F = 1 does not occur because the population quickly degenerates.

During migration of populations owing to small number of individuals the role of consanguineous mating was great. The found differential equation (31) of the moving inbred populations allows to analyse reduction in speed

The population is in process of migration usually enough long time. During migration of a population there is an alternation of generations and natural selection operates.

Using (31) and result received in

The received nonlinear equation is difficult for the analysis. However, taking into account cubic nonlinearity of the differential equation of the second order it is possible to assume presence of solutions as solitary population waves

However, the equation (36) allows receive very important analytical result. For this purpose it is used wave substitution ς = 1/Δn^{2} (kX - ωn) where

Let’s find the particular solution of the equation (37). We shall equate the right part of this equation to zero

In result we have:

Obviously, value of frequency _{f} (or _{f} = 1 – _{f} = 2F/(1-F)s ) in the formula (38) satisfies to the equations (36) and (37).

The received expression (38) can be interpreted as follows. If in inbred populations which began to migrate there is some frequency of a recessive allele in_{f}. Formula (38) is correct and for motionless populations.

We assume the frequency of the recessive allele is equal _{f} = 0.6. If to use inbreeding factor F = 19.10^{-5}, for example, for Germany ^{-4}.

The problem of interaction of populations is very important. Obviously, it is connected to a finding of the multiwave solutions of the equation (36) that is extremely complex mathematical problem. Population waves can cooperate with each other in the various ways.

Solitons - it is stable nonlinear solitary waves which at interaction with other local perturbations or with each other show particle like properties. For example, they restore the initial form, i.e. interactions similarly to absolutely elastic particles, for example, to absolutely elastic spheres. As well as spheres the solitons, for example, can be reflected from a barrier. The marked properties of solitons are caused they has the strict balance of the nonlinear processes resulting in increase of a steepness of a wave forward front and dispersive processes lead to destruction a solitary wave, i.e. to reduction of a front steepness. The balance of these processes leads to stability of soliton, i.e. invariability of its form.

We shall note that the third degree of nonlinearity of the differential equation (36) is sufficient for occurrence of a soliton

As a whole, the further analysis of interaction of populations has no genetic-mathematical character and passes to area of the social phenomena.

Inbreeding in a family tree and a population it is possible to take into account with the help of an inbreeding factor which is probability that two alleles at various individuals are identical by origin.

The Hardy – Weinberg’s law has a various kind for inbred a family tree and populations. In a family tree the inbreeding factor is included into the discrete equation, and populations in the differential equation. The numerical solution of this differential equation is found.

Found the distribution of genotypes in

It is shown that speed of a migrating population at increase of inbreeding factor is reduced.

Interrelation of the recessive allele frequency at woman for a migrating population with inbreeding factor and standard parameter of selection was found.