The end of hyperbolic growth in human population and CO2 emissions

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)∝et/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)∝(ts−t)−1 with the extrapolated singularity year ts=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 CO2 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 ts=2030 of the time width τ=32 years. We also find that the increase in CO2 level since 1700 is well fitted by arccot[(ts−t)/τF] with τF = 40 years, which implies a peak in the annual CO2 emissions at the same year ts=2030.