Global population crisis scenarios predicted by the most general dynamic model

Abstract

We show that a simple nonlinear differential equation (originally studied in the physics of disordered systems) is able to mathematically describe the global population growth over the past 12000 years. Different regimes of population growth since the early Neolithic until today are shown to be all solutions to the same nonlinear differential equation in its various limits. These also include the well-known Malthus (exponential) and Verhulst (logistic) growth regimes, as well as von Foerster's ``doomsday'' formula. All these limits correspond to neglecting higher-order terms in a more general nonlinear dynamic model described by the proposed nonlinear differential equation. While the older models may provide valid fittings to limited time intervals in the global population growth curve in time, their clearly approximate nature prevents them from being predictive over longer periods of time. The proposed comprehensive solution of the proposed model is instead well suited to provide predictions for future scenarios. These include halving of the global population as early as 2064 due to resource depletion, if the effect of the Earth's limited carrying capacity were to set in today.

Figures (2)

• Figure 1: Evolution of the global population over the past 12000 years until now. Symbols refer to empirical measurements and continuous lines are the best fits obtained with the functional forms (i)-(iv) discussed in the text. Simple exponential growth is given by straight lines, Verhulst logistic plateaus appear as flat horizontal lines. The inset shows the stretched-exponential (SEF) and compressed-exponential (CEF) growth regimes, which are indicated with the corresponding functional forms. All these best-fitting functions, including CEF and SEF, are particular solutions to the Eq. \ref{['TZ']} proposed in this paper. Adapted from Ref. Sojecka2024.
• Figure 2: Extrapolations of the empirical trends of global population growth in Fig. \ref{['fig1']}. Here, $y\equiv y/y(0)$ and $t\equiv t/\tau$ are non-dimensionalized variables, where $\tau$ is the characteristic time used in the best fittings Sojecka2024 of the various regimes shown in Fig. \ref{['fig1']}. Hence, $t=0$ in each plot corresponds to the year at which the corresponding best-fitting regime in Fig. \ref{['fig1']} begins. In panels (a)-(b)-(c), the solid lines represent the best SEF or CEF fittings of the empirical data according to Ref. Sojecka2024 (see Fig. \ref{['fig1']}), while the dashed lines are numerical solutions to Eq. \ref{['TZ']}. Panel (a): the solid line is the SEF $y=\exp(t)^\beta$ with $\beta=0.85$, as a function of dimensionless time $t\equiv t/\tau$, where $\tau =72.7$ yrs; the dashed line is the numerical solution to Eq. \ref{['TZ']} with $K=-0.032$. Panel (b): the solid line is the CEF $y=\exp(t)^\beta$ with $\beta=1.40$ as a function of dimensionless time $t\equiv t/\tau$, where $\tau =160$ yrs; the blue dashed line and orange dashed line are the numerical solutions to Eq. \ref{['TZ']} with $K=0.04$ and $K=0.062117398$ (best fitting), respectively. Panel (c): the solid line is the CEF $y=\exp(t)^\beta$ with $\beta=1.52$ as a function of dimensionless time $t\equiv t/\tau$, where $\tau =58.6$ yrs; the blue dashed line and orange dashed line are the numerical solutions to Eq. \ref{['TZ']} with $K=0.04$ and $K=0.07227915$ (best fitting), respectively. Panel (d): projection into the future, where the solid line is just the continuation of the SEF with $\beta=0.85$ and $\tau =72.7$ (current global population trend, see Fig. \ref{['fig1']}), with no influence from the limited carrying capacity of the Earth; the dashed line is the prediction based on Eq. \ref{['TZ']} under the hypothesis that the influence of the Earth's limited carrying capacity were to set in as of today.