# The mathematical model of population dynamics

*Alexander A Victorov
Department Of Radiology, Burnazyan Federal Medical And Biophysical Center, Zhivopisnaya Str., Moscow, Russian Federation

##### *Corresponding Author:
Alexander A Victorov
Department Of Radiology, Burnazyan Federal Medical And Biophysical Center, Zhivopisnaya Str., Moscow, Russian Federation
Email:a-victorov@mail.ru

Published on: 2019-11-29

#### Abstract

We show applicability of kinetic theory of living systems aging to describe population dynamics of the countries of the World by examples of the USA, China and Russia. Proposed mathematical model is proven also by changes of mice population in “mouse paradise” experiment by American scientist John Calhoun. Proposed approach allows to describe the stages and features of that dynamics: population growth in the United States, growth and possible decrease in population in China, loss of part of the population of the Russian Empire and the USSR due to two World wars and the collapse of the USSR and biological degradation of “mouse paradise” until complete mice population extinction. Obtained results suggest that a human, population of the countries of the World, whole humanity and other biological species are developing and aging at the same time: under the influence of always existing tension in society (stressing) and according to the same pattern corresponding to the mathematical model proposed here.

#### Keywords

Model, kinetic theory, dynamics, population, tension, living system

#### Introduction

The mathematical description and forecasting of population dynamics of the countries of the World is considered in numerous scientific publications, the most significant, in our opinion, are [1-14]. Malthus model of exponential population growth [1] was the result of the assertion that population increases in geometric progression. Verhulst[15] examined population dynamics under the initial assumptions that the rate of population reproduction is proportional to its current base and available resource pool. The author called the solution of the corresponding differential equation as logistic curve. In accordance with empirical “law” of Forester [11], the hyperbolic growth of Earth’s population observed over several millennia is described by function with a singularity point - a point in time when the function goes to infinity. Moreover, according to calculations of Horner [12], it corresponds to year 2025, and according to Forester - November 13, 2026.

However, in period 60-90 years of the 20th 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 [13] 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 [2,3] 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). S. Kapitsa modified Forester model, excluding the singularity point, and obtained the equation for dependence of population on reduced time τ in the form of the inverse trigonometric function arcctg(τ). The reduced time τ includes time t1= 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 [7,10]. The most popular databases on demographic prospects of 226 countries and regions of the World are the United States Sensus Bureau (International Database) and the United Nations Population Department (UNPD). UNPD database contains statistics and provides projections of global population changes for the period up to year 2150 - the Global Review and Inventory of Population Policies (GRIPP), and, coordinates the Population Information Network (POPIN) [15-20]. Forecast calculations methods are based on mortality table. Four groups of forecasting methods are used: extrapolation methods, economic and mathematical methods, classification by year of birth and cohort component method, and methods of expert estimates. Unlike extrapolation and analytical methods, cohort component method (classification by year of birth) based on use of demographic balance equation allows to get not only the total population, but also its distribution by sex and age. In practice, several variants of demographic forecasts are always developed. So, e.g., there are three options of forecast by Russian Statistics Agency (ROSSTAT), and by UNPD - eight. Since the future is not exactly known today, the forecast is multi-scenario and is determined, first, by used ideology of forecast model.

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 [19, 20]. The accuracy of population estimates in the census is 5%, and in the long-term forecast, tens and hundreds of % [16,21]. For example, according to the forecasts of demographers from the Expert Council under Russian Government, based on the report of the Higher School of Economics (HSE) “Population of Russia” of 2012, the population of Russia was estimated on 6 scenarios, and by optimistic scenario, in year 2060, population of Russia may be close to 150 million people, and pessimistic - near 70 million people. According to the UN forecast [19, 20], presented by 9 variants, the population of Russia at the same time was estimated from about 152 million to 110 million. At present, population of Russia is about 146.8 million.

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 [23]. The result of same-type experiments was population extinction after a rapid increase in size, hundreds of times higher than the initial cohort (several initial heterosexual couples), due to the gradual stratification of mice society into separate non-interacting and aggressive caste clans, cannibalism, the termination of mating of the opposite sexes, homosexuality, desire for self-sufficiency.

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 [22] to describe changes in the population of different countries of the World.