A majorizing forecast of the earth population in a reduced resource-limited model

¹ The first President of Russia Boris Yeltsin Urals Federal University, 620002 Yekaterinburg, Mira, 19, Russia
² Ural State Forestry Engineering University, 620100 Yekaterinburg, Sibirsky Trakt, 37, Russia

^* Corresponding author: mрk46@mail.ru

Abstract

Three two-parameter phenomenological models of population growth dynamics of the Earth’s population N leading to a conclusion about the existence of its limiting number N_max are briefly described. It is noted that the Ferhulst model, which assumes external resource constraints of exponential growth and leads to a logistic dependence N (t), has a maximum growth rate dN/dt at N = N_max/2 and is adequate to asexual method of reproduction. Two other models are associated with sexual reproduction, however, in Kapitsa model, it is believed that N growth restriction is due only to internal causes (the “demographic imperative” of Kapitsa). The authors develop a model in which resource constraints are reflected by the square of Verhulst multiplier. It is significant that in such a model (as in Kapitsa model) there is not only a maximum growth rate, but also a maximum relative growth rate (1/N) dN/dt at N ≈ N_max /2.7 for Kapitsa model and N = N_max/3 in resource limited model. These features make it possible to estimate the value of N_max using statistical data on N(t) dependence. The analysis of transition from “microscopic” to macroscopic description indicates the possibility of transition to a reduced model in which for (1/N) dN/dt the square of Verhulst multiplier is preserved, and data processing allows using the value N_max = 14.7 billion. For this value N_max after calibration of the model parameters according to the beginning of 2018 a population forecast was made. Comparison with two forecasts based on UN data showed that calculations of the reduced model majorize them, exceeding, for example, in 2050 by 1.4%. The effect of explicit concern of the rate of population decline due to mortality, leading to decrease in the Nmax estimates, is briefly discussed.