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

The global population P(t) (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:GP(P)=(dP(t)P(t))dt=dlnP(t)dt. The focused insight reveals two near- linear domains for GP(P) 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 (r,s) should describe the rise in the global population. However, the Verhulst relation with such effective parameters does not describe P(t) 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 P(t) were analyzed via the ‘reversed protocol’ analysis, in comparison to the standard pattern, namely: (i) first, the linearized, distortions-sensitive transformation of P(t) 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 GP(P) local ‘disturbations’ and some historical and prehistorical reference events, showing their global scale impacts.