Timeless, Generational Mathematical Model of the Influence of Age Structure on the Change in the Numerical Composition of living Systems

In the present work, on the basis of the arithmetic progression of the age structure of the population, a generational, visual method is proposed for predicting the preservation, growth or decrease in the numerical composition of the population, including the human population of any region. Unlike the existing ones, this mathematical approach does not operate on the lifetime of individuals and therefore is universal, that is, applicable to any living systems. It is shown that the average number of individuals born in this generation per each heterogeneous couple is a criterion for the survival of a population the human population. However, this value depends on the age structure of simultaneously living generations. Our mathematical approach does not require any preliminary assumptions.


Introduction
Typically, a mathematical prediction requires an author's hypothesis about the future, explaining the mathematical result based on clear physical assumptions. In the developed accepted forecasts of the world population, groups of methods are used that involve certain assumptions [1][2][3][4][5][6]. So, extrapolation methods are performed under the assumption of the invariance of the absolute population growth or growth rate. Analytical methods are since functions are selected based on past demographic dynamics that describe it most closely [7][8][9][10][11][12][13][14]. The component method (age shifting) is based on the use of the demographic balance equation and allows one to obtain not only the total population, but also its distribution by sex and age [15,16]. However, to use the equation of demographic balance, knowledge of the differential rates of birth and death is required.
When using existing mathematical models of demographic dynamics, difficulties also arise in describing the real non-monotony and "jumps" of the dynamic temporal process of population changes. These features of real dynamics are determined by possible specific historically determined interactions of populations with each other, their psychological and biological responses to the effects of various environmental factors of anthropogenic and natural nature [8,17]. Ultimately, they can lead to fundamental deviations from "fate" predicted by evolutionary mathematical models that predict stabilization and sustainable development in optimistic scenarios. Therefore, based on inevitable assumptions and mathematical difficul-

Research Method and Results
The dynamics of the population, including a population explosion or stabilization, is determined in any region of residence by the age structure of the population. Typical options for changing the population will be calculated on the basis of the following procedure.
The total number of people born in ( 1) K + generations can be written in arithmetic progression Here is the number of people of the initial (zero) generation ( We confine ourselves to four generations, i.e., 0, k = , 1 , 2 3 . In this case we get Suppose that when great-grandchildren are born, then there is no longer a fraction of the great-grandfathers and the proportion of grandfathers. Then, the number of dead individuals (3) 0, 1 ( , , ) d N g f F F will be determined by the expression The number of individuals remaining alive will be equal The relationship we are interested in We require that r=1. Then from (6) we obtain: (1 ).
Consider examples of population stabilization that correspond to the "old" population. Let us accept the following conditions most  For example, if under the previous conditions there will be 3 children in a family, then the population over 4 generations will increase by 2 times (Figure 2). The average number of children per each living heterosexual couple is a criterion for the survival of the population in any region of the world.

Conclusion
In the generational mathematical model proposed above, the analysis of the dynamics of the number of individuals does not require knowledge of their lifetime. Therefore, it is applicable for any biological system that has the property of reproduction. The age structure of a specific totality of individuals is the basis of the model. It can be used, in particular, to assess the nature of changes in the human population in real time and increase the reliability of demographic forecasts, regardless of the time and influence factors.