Verhulst-type equation and the universal pattern for global population growth

Abstract

The global population [Formula: see text] (growth from 10,000 BCE to 2023) is discussed in frames of the Verhulst-type scaling, recalling the sustainable development concept. The analysis focuses on the per capita global population growth rate, for which the analytic counterpart is considered:[Formula: see text]. The focused insight reveals two near- linear domains for [Formula: see text] changes: from ~ 700 CE till ~1968 and from ~1968 till 2023. It can be considered a reference pattern for long-term global population changes. For models recalling the Verhulst-type scaling, such analysis indicates that a single pair of growth rate and system resource coefficients [Formula: see text] should describe the rise in the global population. However, the Verhulst relation with such effective parameters does not describe [Formula: see text] changes, which raises the question of whether it is adequate to describe global population changes. Notably is the new way of data preparation, based on their collections from various sources and numerical filtering to obtain a 'smooth' optimal set. The changes of [Formula: see text] were analyzed via the 'reversed protocol' analysis, in comparison to the standard pattern, namely: (i) first, the linearized, distortions-sensitive transformation of [Formula: see text] data is carried out; it indicates domains where the validated application of a given scaling equation is possible and yields optimal values of relevant parameters, (ii) the final fitting via the selected scaling equation is carried out for identified domains, and using obtained optimal values of parameters. The analysis reveals links between [Formula: see text] local 'disturbations' and some historical and prehistorical reference events, showing their global scale impacts.

Fig 1. The plot showing global population

$\mathbf{P}(\mathbf{t})$ changes, in a semi-logarithmic scale, from 10,000 BCE to 2023. It is based on the data given in the S1 Appendix. The inset focuses on the Industrial Revolutions [–8] epoch. The arrows indicate some characteristic dates/periods manifested in the plot.

Fig 2. Changes of the per capita relative world population growth

${\mathit{G}}_{\mathit{P}}\left(\mathit{P}\right)$ determined by the derivative analysis defined by Eq. (6) and based on data shown in Fig 1 and collected in the S1 Appendix. The crossover between the two emerging domains is shown. The extrapolation to ${\mathit{G}}_{\mathit{P}}\left({\mathit{P}}_{max}\right)\mathbf{=}\mathbf{0}$ indicates the onset of the stationary phase, which can be associated with the maximal population. Note the ’squeeze/compression’ of the first 10 millennia of global population growth caused by the scale applied. Table I and its caption give parameters related to linear domains (in green and blue) and for the polynomial portrayal (in violet).

Fig 3. The log-log scale presentation of the per capita growth of the global population

${\mathit{G}}_{\mathit{P}}\left(\mathit{P}\right)$ data, shown in the linear scale in Fig 2. Emerging relevant historical domains are indicated. It is visible that the hypothetical 1^st linear domain visible in Fig 2 can be considered only from early Medieval times.

Fig 4. Changes in the global population from

~1940 till 2023. The parameterization is related to the empowered exponential Super-Malthus Eq. (10) [84], with the parameters given in the plot. The inset recalls data from Fig 2 but is presented in a semi-log scale: ${\mathit{G}}_{\mathit{P}}\mathbf{=}\frac{\left(\frac{\mathbf{d}\mathbf{P}\left(\mathit{t}\right)}{\mathit{P}\left(\mathit{t}\right)}\right)}{\mathbf{d}\mathbf{t}}\mathbf{=}\frac{\mathbf{d}\mathbf{l}\mathbf{n}\mathbf{P}\left(\mathit{t}\right)}{\mathbf{d}\mathbf{t}}$ (Eq. (5)). Emerging characteristic time-related events are indicated.