The authors have declared that no competing interests exist.

On the basis of Hardy – Weinberg law the problem of migration from the genetic point of view is considered. It is proved the linear differential equation of migratory process of a panmictic population. The phase of the solution of this equation is investigated. On the basis of the carried out analysis the dependence of migration velocity of a population on average time of alternation of generations is found. It is shown that migration of primitive people from Africa to Europe needed alternation the several hundred generations. The dependence of migration velocity of a population on the average area developed by a population for year is investigated. Lacks of the carried out analysis owing to absence of the account of natural selection and inbreeding are marked.

Populations during the vital activity carry out various motions, i.e. migrate in searches of the best life places, at excess of individuals which can exist in the given region, migrate in order to prevent collisions with other populations, for a new territories gain, etc. All history of a human population development is connected to migration of its parts in various regions of Globe.

Migratory processes have the laws and genetic consequences. At migration of populations there is a genes flow

Not concerning the concrete reasons of migration we shall analyse genic changes in a migrating population under various conditions.

Let's consider the elementary case of migration of the panmictic populations.

Moving of the population individuals is not equivalent to moving of genome. But individuals of the population are carriers of genome therefore moving of the population individuals and moving of genome are closely connected with each other.

Let’s assume that at the migration of the population keeps it panmictic character. Though such assumption is the big idealization but for the initial analysis it is allowable especially if the moving part of the population is great enough i.e. contains a plenty of family trees.

For the solving of our problem it is necessary to have, first of all, the base equation which describes the genome of the moving population. We shall consider requirements which are necessary such equation to satisfy.

First of all, this equation should include Hardy - Weinberg law i.e. to pass in Hardy - Weinberg law at absence of a population movement.

Second, the equation should reflect indifferent character of genome balance for moving panmictic populations.

Thirdly, since movement of a population represents some wave process including elements of individual diffusion in space it is possible to assume that the equation for moving genome should have diffusive-wave character.

For a basis we shall accept the differential equation reflecting balance of genome of the motionless panmictic population or Hardy - Weinberg balance

…………… (1)

where q_{f} there is frequency of the some recessive allele of two-allele systems inthe

Taking into account required diffusive-wave character of the equation we shall write down the base equation for genome of the moving population as:

…………. (2)

Where ^{*}^{* }- factor of diffusion of the population individuals, and hence of genome in space. This factor is multiplied for normalizing time

Let's note that the equation (2) simultaneously reflects indifferent character of the genome balance of the moving population since satisfies to the solving _{f}= const.

The equation (2) allows analyze moving of the population and it genome in the certain direction

……………. (3)

where Δq_{f} there is the Laplacian of the functions q_{f}.

With the purpose of the solving finding of the equation (2) we shall make replacement of the variable under the formula:

……………. (4)

Where u(n,X) there is new variable dependent on dimensionless time and coordinate of moving.

Substituting (4) in (2), we shall receive the equation:

……………. (5)

The result of the solution of Cauchy problem for the given equation has rather complex character for the analysis. For example, for initial conditions: at n = 0 the function q_{f}_{f}_{0} = u_{0 }=const, and

(initial constant on time value of recessive allele frequency), and also in spatial area - ∞ < X < ∞ the solution of the equation (5) looks like

……………. (6)

where_{n} and x-√D_{n}, ξ variable of integration, I_{1}(z)- modified Bessel function of the first order.

Function f(x-√D_{n}_{)} means a wave of the population genome, and hence the wave of the population spread to the right, and f(x+√D_{n}_{)} - to the left.

Let’s consider in more detail the form of a migrating population wave. It is obvious that approximation of uniform movement in this case is rather rough. The population goes then after a while stops and develops the certain territory. Then, after an exhaustion of resources the movement renews, etc. On

Each man is the carrier of all genome. Therefore examined allele frequency we shall assume an identical at each individual of a population q_{f}_{f}_{ 0}

For our purposes is allowable to use the exponential kind of function

In the formula (7) we shall be interested the parameters of the wave phase

Substituting (7) in (4) we shall pass to former function q_{f}:

……….(9)

Once again we shall note that is real any change of function q_{f} is not present since each individual carries a full genome. Function (9) reflects the conditional change shown on

In the formula (9) k = _{f}_{0 }=u_{0}= const is used also.

First of all we shall find speed of the wave front propagation of population genome. If the population at the movement keeps the panmictic character the Hardy - Weinberg balance is indifferent and should be observed q_{f}=q_{f}_{0 }= const .

Hence, according to (9) the phase of the wave is equal:

………….(10)

Speed of the wave front movement of population genome according to (10),

…………..(11)

Expression (11) has completely general character which has been not connected to the concrete form of a population front wave (9).

Substituting in (11) parameter ω from (8) and taking into account ^{*}

…………..(12)

Let's note that in (12) the condition is observed: at D^{*} = 0 the population is motionless V = 0.

Using length of the population wave front under the formula

…………..(13)

It is interesting to analyse on extremum the function of speed

Simple transformations of derivatives ∂V/∂T and ∂V/∂λ, and equating of these derivatives with zero result in a trivial conclusion ^{*}= 0 i.e. a motionless population.

On

At calculation the λ = 0.4km length of the population front wave was accepted factor of diffusion D^{*} = 10Km^{2}/year.

Calculation shows that with increase in the period of a generation alternation the speed of the population and it genome movement to become less. It is correct since younger individuals are more dynamical i.e. if the generations is more often are alternated the population faster move. At the period of the generation alternation T ≈ 25 years the calculated speed of genome movement, and hence the population is equal to V ≈ 0.63 km/year

It is obvious that this speed basically is caused by a gradual exhaustion of resources on a way of the population movement.

It is supposed, that occurrence Homo sapiens has taken place in the central Africa about 300 thousand years ago. About 40 thousand years ago Homo sapiens migrated to Europe. Believing, that during migration from Africa to Europe,

On

At the analysis of the equation (13) there is a question on sense of diffusion factor ^{*}. This parameter reflects the area developed by the population within year at each stage of movement. Actually this parameter reflects the resources necessary for vital activity and movement of the population. We shall notice that the parameter ^{* }

On ^{*} is shown.

The found dependence is rather strong. It shows that the big resource area the population can develop for the year at the movement the faster it move.

The basic lack of the carried out analysis of the moving population is the assumption about it panmictic character. Actually the migration of the human population part in various directions occurred rather small groups, tribes or separate communities of tribes. At these tribes there was be relative small number of family trees therefore consanguineous mating, i.e. inbreeding has been widespread. But research of process of a population migration with the account of inbreeding and selection