The authors have declared that no competing interests exist.

The possibility of applying the kinetic theory of aging of biological species published earlier by the authors of this work to assess and predict changes in the number of specific populations is evaluated. The populations of the USA, China and Russia, as well as the population of mice observed in the experiment "mouse paradise" of the American scientist John Calhoun are considered. To this end, a historically consistent analysis of the main previously proposed multi-scenario mathematical models describing demographic data and predicting the dynamics of the population was performed. The results of these models show a decrease in the population growth rate, a tendency toward a limit with an increase in historical time, the achievement of such a limit in some developed countries with a relatively high level of social security, a subsequent decrease in the number and further uncertainty of the final population outlook in the distant future. In addition, these models made it possible to establish that the observed population growth in developed countries is unambiguously accompanied by its aging - a relative predominant increase in the number of elderly people compared to the number of the younger generation (people are aging, the population of countries is aging). In this work, the assumption was made and confirmed that the dynamics of the aging of the population of the countries of the World corresponds to the dynamics of aging of a person of one generation and is mathematically described by the differential equation of the kinetic theory of aging of living systems of the same type with close values of the parameters. The biophysical meaning of the parameters of the kinetic equation reflects G. Selye's concept of the determining role of stress in human life and populations. An analysis of the changes in the numbers of the considered populations of humans and mice at various stages of their development is qualitatively commented on from the standpoint of comparative tension according to G. Selye. To assess the degree of aging of a biological object of one population in kinetic theory, the probability of death during life is selected as an indicator of aging. In this work, the probability of reaching the maximum population size was chosen as an indicator of the aging of a biological object of various populations. The published literature predicts various options for changing the population after reaching a maximum - maintaining the reached maximum level and decreasing to a certain limit, less than the maximum achieved. In this paper, based on an analysis of its results and an analogy with the complete degeneration of mice in the “mouse paradise” experiment, a conclusion is drawn about a hypothetically possible third variant of the limiting decrease in the population - its complete degeneration.

Essentially, all models are wrong, but some are useful.

Dr. George E. P. Box

The mathematical description and forecasting of population dynamics of the countries of the World is considered in numerous scientific publications. Malthus model of exponential population growth was the result of the assertion that population increases in geometric progression

However, in period 60-90 years of the 20^{th} century, population growth began to decelerate (demographic transition period), hyperbolic growth stopped, and hypothesis appeared about the limit of population growth of humanity. To describe new demographic trends, Kremer introduced in his model of hyperbolic growth additional function of per capita product, that equilibrium value determines equilibrium population size, according to his concept of technological development

S. Kapitsa put forth markedly different concept stated that the change in population over millennia is determined by biological factor, namely by dominant feature of human psychology and information interaction of members of society, and that change is not related to other factors of environment (the principle of demographic imperative) _{1} = 2000 years corresponding to the middle of demographic transition period. Asymptotic stabilization of the population of the Earth corresponds to 12 billion people, while 90% of maximum population, equals to about 11 billion, is expected by 2150.

One of the modern approaches to evaluating trends in demography is solving partial differential equation concerned demographic balance of birth and death rates

In recent decades, reducing the birth rate, which overlaps the simultaneous reducing mortality, has become the prevailing trend in economically developed countries leading to decrease in the growth rate of the population and to change its age composition towards aging the population. Aging process is characterized by increase in the relative share of the elderly population. Part of population aged 60 and over has increased from 8% in year 1950 to 12.3% in year 2015,. by year 2030, it will be 16.5%, by year 2050, it will reach 21.5% of the total World’s population

Calculation results on 4 options made by experts of HSE Institute for Demography published in Bulletin "Population and Society” No. 371-372 (2009) show wide interval in predicted size of the World population in 2060 ranging from about 27 billion people down to 6 billion people. Currently, total population is approximately equal to 7.7 billion people.

Thus, in highly advanced countries, aging of population and even depopulation has been recorded. However, key challenge remains not clear – what will be the end of this process? Bright hypothetical illustration of negative forecast concerned possible future of mankind may be the results of experiments with mice conducted repeatedly by American scientist John Calhoun in conditions when mice were provided with full availability of space, food, water, favorable physical environmental factors and high hygiene in their crate

Anyway, mathematical forecast requires not only, and rather, not so much a formal adequacy of the model to previous experimental data but of author's hypothesis about the future, which predicting mathematical result on the base of clear physical idea.

The goal of research is to evaluate the possibility of applying mathematical model of aging of various biological species to describe changes in the population of different countries of the World

Kinetic theory of aging of living systems (LS) considers the human life cycle as time interval during which a biological aging process occurs

∂

Where D is cumulative function of mortality (CFM) of the living system; τ = C · t is dimensionless time (0≤τ≤1); t is calendar time; C is constant with dimension inverse to calendar time; μ is dimensionless parameter illustrating the "tension" of the system; k is constant that accounting for adaptation of the system; θ is parameter that taking into account the change in the aging rate with increasing in age of the system. The probability of human death (Dτ), as indicator of the degree of human aging to age τ, is determined statistically by the ratio of the number of people dying in a certain time interval τ to the total number of people of a given generation (generation size). Dimensionless time τ is the ratio of the calendar age t to the life expectancy t_{mb}, i.e., τ = t / t_{mb}. Term "tension" means tension, stress, pressure, load. The term stress was introduced into biology by Canadian physiologist G. Selye in 1936 ^{nd} signal system (word power). The indicator of tension S = μ/(1-θ·D), as the level of the general adaptive syndrome of the body under influence of stressors, increases with increasing in body age D(τ).

Kinetic equation (1) allows obtaining various integral and differential probabilistic indicators of LS aging in dimensionless units: CFM of LS D(τ) = ʃ (∂D/∂τ) dτ - probability of system death over a period not exceeding t; ¶D/¶t - probability density of death; mortality rate (∂D/∂t)/(1-D); life duration expectancy:

τ_{mb}

Eq.(1) contains 3 parameters, and each, in its own time interval, affects to the greatest extent on character of time dependence D(τ). These parameters are found as a result of analysis and processing of the experimental statistical dependences D(τ) in accordance with kinetic Eq. (1). Equation is solved numerically to select suitable dimensionless model parameters by comparing with available experimental data. Based on the mathematical analysis of this equation, it follows that the average rate of change of D with time τ in the interval 0.07≤D≤0.7 is determined mainly by parameter μ. Initial interval 0<D˂0.07 serves for “correction” of parameter k. In the final interval 1>D>0.7, parameter θ plays key role. Therefore, finding values of parameters begins in a first approximation with estimate of parameter μ using Gompertz distribution, when the experimental data D(τ) are approximated by solving equation (∂D/∂τ)/(1-D) ≈ Aexp(ατ) in the interval 0.07 ≤D≤0.7

A human is getting old. Population of the countries of the World is also aging. Obviously, indicators of aging, as also process of increase in age of LS, can be different. To estimate the degree of human aging in kinetic theory, the probability of death is chosen as such indicator. Population growth in highly industrial countries is unambiguously accompanied by its aging - a relative predominant growth in the number of elderly people. Therefore, we choose the ratio of current population size to the maximum possible as indicator of aging population of the countries of the World. The limit of this relationship with increase in calendar time is 1. According to scenarios of World forecasts, the maximum possible humanity size can be reached before year 2060, and maybe not. We introduce the dimensionless coordinates. Dimensionless time is defined as the ratio of calendar time from the beginning of the process to time interval corresponding to aging index reaching value of 1. Time at which selected aging index is much less than 1 is taken as the beginning of process. Let’s take the previous notations D and τ for these coordinates. These parameters will vary from 0 to 1 accounting for that D(0) is much less than 1. Aging index D(τ) will display the probability of reaching population its maximum value. Under these assumptions, the task of modeling the dynamics of population size is reduced to previous mathematical model of kinetic theory of LS aging – Eq. (1), but with different interpretation of the calculation results. In this case, the previous interpretation of model parameters (1) is retained. Current population N(τ) will be calculated as the product of D(τ) by expected maximum number N_{m}: N(τ) = D(τ)·N_{m}. If, after reaching the maximum, population begins to decrease, this means the transition of aging process to a new phase - the excess of mortality over fertility. For the mathematical description of this phase, function D will be considered as the probability of death of the population, varying from 0 to 1, with 1 corresponding to probability of reaching the maximum number of dead. The maximum number of dead can range from the initial maximum number of 1^{st} stage of aging (e.g., similar to the case of complete degeneration of mice paradise), to level lower than this value (e.g., as in the case of long-time demographic forecast of population size of China). Obviously, to describe the aging process at 2^{nd} stage, the use of the same kinetic Eq. (1) is also valid. Then the total population size at these two stages N_{m}(τ) is determined by difference between population size of 1^{st} stage N_{m1}×D_{1}(τ) and number of deaths of 2^{nd} stage N_{m2}×D_{2}(τ), i.e., N_{m}(τ) = N_{m1}×D_{1}(τ)-N_{m2}×D_{2}(τ). In general, this process can represent a continuous oscillation of growth and decrease waves with possible difference in amplitudes and duration. Each wave can be described by the same kinetic equation.

Let’s verify the feasibility of this model by comparing calculations with demographic data.

In accordance with study objects, the term living system (LS) is understood as population - totality of people of different generations living simultaneously on Earth or within a specific territory - continent, country, region, etc., as well as the group of animals discussed below. ^{th} column of _{i}(t) functions and N_{mi} parameters contribute. Intervals common for determining D_{i}(t) functions by time t are also indicated. All parameters of the approximating functions (columns 2-5 of) are found by the method indicated above for selecting scenarios that are closest to demographic data. Time intervals for parameters determination is indicated in column 6. The number of time segments of approximation for different countries is determined by the number of function jumps and/or by changing the sign of the first derivative, i.e., the presence of falling part of the function. The population of Russia itself, which by various names was consistently taken up different geographical territories, has suffered three jumps - in years 1914, 1941 and 1991, therefore, it is described by 4 segments of piecewise continuous functions. The population of China and mice paradise are approximated by two functions, due to the decrease in population after passing the maximum.

µ | q | k | N_{m}×10^{6} |
t | N(t), size of species | |

USA | 1,520 | 0,605 | 4,472 | 520 | 1700+400τ, year | N_{m}×D(t), years 1900-2050 |

China | 1,539 | 0,584 | 4,468 | N_{m1}=1570 |
1850+200τ, year | N_{m1}×D_{1}(t)-N_{m2}×D_{2}(t), years 1950-2050N_{m1}-N_{m2}×D_{2}(t), years 2050-2100 |

1,539 | 0,584 | 4,468 | N_{m2}=500 |
1930+200τ, year | ||

Russian Empire | 1,520 | 0,675 | 4,472 | 450 | 1670+400τ, year | N_{m}×D(t) 0,3≤τ≤0,61, years 1800-1914 |

RSFSR, USSR | 1,520 | 0,675 | 4,472 | 450 | 1670+400τ, year | N_{m}×^{D(t)} 0,62≤τ≤0,677, years1918-1941 |

USSR | 1,520 | 0,607 | 4,472 | 450 | 1670+400τ, year | N_{m}×D(t) 0,687≤τ≤0,802, years 1945-1991 |

Russia | 1,520 | 0,652 | 4,472 | 150 | 1740+300τ, year | N_{m}×D(t) 0,837≤τ≤0,923, years 1991-2018 |

Mice paradise | 1,555 | 0,550 | 4,468 | N_{m1}=2400 |
850×τ, days | N_{m1}×D_{1}(t)-N_{m2}D_{2}(t) |

1,565 | 0,510 | 4,468 | N_{m2}=2400 |
500+1200τ, days |

Data in

(

Parameters values μ and θ of various LSs used for producing graphs in

Live system name | μ | θ |

1. Russian Empire, RSFSR, USSR (years 1800-1941) | 1.520 | 0.675 |

2. Russia (years 1991-2018) | 1.520 | 0.652 |

3. LS |
1.520 | 0.632 |

4. USSR (years 1945-1991) | 1.520 | 0.607 |

5. USA (years 1900-2050) | 1.520 | 0.605 |

6. China (years 1950-2100) | 1.539 | 0.584 |

7. Mice (population increases) | 1.555 | 0.550 |

8. Mice (population decreases) | 1.565 | 0.510 |

Comparing population size changes trends show that the lowest tension (stress) is observed among the Chinese population, then with small maximum difference (5.5% at D = 1) - among the US population. The greatest tension is observed among Russian population in the interval from year 1800 to 1941, with maximum increase compared with China by 28% at D = 1. Tension among population of the USSR in period from year 1945 to 1991. significantly decreases and practically corresponds to the tension of the US population. After the collapse of the USSR in year 1991. and till year 2018. the tension among population of Russia again increases significantly, approaching stress of period 1800-1941 years, exceeding China by about 20% at D = 1. In Russian version, a decrease in the population in 1914-1918, 1941-1945 years and in year 1991 considered as function jumps in where function itself is not defined. Thus, the tension among population in the Russian Empire, the RSFSR, the USSR and the Russia is different at different stages of life.

It is also interesting to compare (

Even more interesting are the results of comparing the tension among mice paradise population (

Here we do not consider the detailed reasons for proximity and difference in tension of LSs when they are in different conditions, since this task is too complicated from the point of view of classification of stresses and stressors according to G. Selye’s theory and requires a special study. However, based on performed analysis, there is reason to believe that humans, humanity and other biological species are developing and aging at the same time: under the influence of always existing tension (stress) and according to one regularity corresponding to the mathematical model presented in article.

Note that sensitivity of LS to stressor intensity is different. For example, chronic X-ray radiation exposition of dogs within range of dose rate changes from 0 to 54 cGr /day results in change in parameter μ from 1.523 to 4.200, while the life expectancy of dogs decreases under effect of radiation from 192 months to 1 month

We consider that physical interpretation of essence of our mathematical model corresponds to the concept of G. Selye

1. The kinetic theory of aging of living systems can be used to describe the population dynamics of the countries of the World.

2. There is reason to believe that human, population of the countries of the World, humanity and other biological species are developing and aging at the same time: under the influence of always existing tension (stress) and according to the same regularity corresponding to the mathematical model presented above.

Alexander Alexandrovich Victorov, Viacheslav Alexandrovich Kholodnov. Analysis and Forecast Based on the Kinetic Equation for Changing the Numerical Composition of Living Systems