The end of hyperbolic growth in human population and CO2 emissions

Abstract

Using current empirical data from 10,000 BCE to 2023 CE, we re-examine a hyperbolic pattern of human population growth, which was identified by von Foerster et al. in 1960 with a predicted singularity in 2026. We find that human population initially grew exponentially in time as $N\left(t\right)\propto e^{t/T}$ with $T=2080$ years. This growth then gradually evolved to be super-exponential with a form similar to the Bose function in statistical physics. Around 1700, population growth further accelerated, entering the hyperbolic regime as $N\left(t\right)\propto {(t_s-t)}^{-1}$ with the extrapolated singularity year $t_s=2030$, which is close to the prediction by von Foerster et al. We attribute the switch from the super-exponential to the hyperbolic regime to the onset of the Industrial Revolution and the transition to massive use of fossil fuels. This claim is supported by a linear relation that we find between the increase in the atmospheric CO${}_2$ level and population from 1700 to 2000. In the 21st century, we observe that the inverse population curve $1/N\left(t\right)$ deviates from a straight line and follows a pattern of “avoided crossing” described by the square root of the Lorentzian function. Thus, instead of a singularity, we predict a peak in human population at $t_s=2030$ of the time width $\tau =32$ years. We also find that the increase in CO${}_2$ level since 1700 is well fitted by $\mathrm{arccot}[(t_s-t)/{\tau }_F]$ with ${\tau }_F$ = 40 years, which implies a peak in the annual CO${}_2$ emissions at the same year $t_s=2030$.