The End of Hyperbolic Growth in Human Population and CO$_2$ 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 $N(t)\propto e^{t/T}$ 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 $N(t)\propto(t_s-t)^{-1}$ 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 $1/N(t)$ 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 ${\rm arccot}[(t_s-t)/\tau_F]$ with $\tau_F$=40 years, which implies a peak in the annual CO$_2$ emissions at the same year $t_s$=2030.