Other forms of growth

Have another look at the model for growth of the U.S.-population. Whether human population dynamics adhere to exponential growth is disputed. In 1960, the Austrian Cybernetician Heinz von Foerster and his colleagues P.M. Mora and L.W. Amiot suggested that - according to available historical data – the development of world population is not proceeding exponentially but rather hyperbolically. In regard to the decrease in doubling time, H.v. Foerster jokingly predicted that population growth would become infinite by Friday, November 13, 2026 – his 115th birthday anniversary. On this day doubling time should become zero. In regard to population growth, the corresponding equation has been called “Doomsday Equation”. Meanwhile several indices suggest that world population – at least up to about 1970 – indeed grows faster than exponentially. One option to model this would be to consider second-order (or squared) exponential growth where \(\gamma\) is proportional to the square of \(N_t\).


Below is a plot representing the growth of world population between 10000BC and 2000AC:

Hyperbolic growth

Foerster, Hv./ Mora, P. M. / Amiot, L. W. (1960): Doomsday: Friday, 13 November, A.D. 2026. At this date human population will approach infinity if it grows as it has grown in the last two millenia. Science 132 (3436): 1291–1295.

Limited or density dependent growth

Whatever the actual dynamics of human population growth, a realistic assumption for any population is that it might exert a self-delimiting influence on its own dynamics, since its growth rate depends on its size. Think of a population of bacteria in a sucrose solution. If population size is small, resources are ample and growth can proceed unhampered. Once the population reaches a certain size however, the bacteria might start to compete for sucrose. Further growth is hampered and might even reach zero at some point. In this case, growth is inhibited by the density of the population, it is inhibited by a limit, in this case the capacity of the resource as a nutrition for bacteria. This limit capacity is called carrying capacity of its environment.

The growth of the bacteria population hence would proceed in respect to how close its size \(N\) is already to the carrying capacity \(K\). Mathematically this can be expressed as \(\gamma\) being proportional to the difference of \(N_t\) and the carrying capacity \(K\)


or recursively: \(N_{t+1}=N_t+\gamma*(K-N_t)*dt\)

Depicted with the modeling platform VENSIM (which is introduced here):

Limited growth Vensim-input
Limited growth
Limited growth