Hyperbolic growth
When a quantity grows towards a singularity under a finite variation it is said to undergo hyperbolic growth. This growth is created by non-linear positive feedback mechanisms. Hyperbolic growth is highly nonlinear and it is a stronger form of growth than exponential growth.
Certain mathematical models suggest that the world population undergoes hyperbolic growth. However, since the world population cannot truly become infinite within some finite amount of time such models should only be seen as approximations that may be valid for certain time periods. Of course, the same is true for the exponential growth models that also may be valid for certain time periods only.
Another example of hyperbolic growth can be found in queuing theory: the average waiting time of randomly arriving customers grows hyperbolically as a function of the average load ratio of the server. The singularity in this case occurs when the average amount of work arriving to the server equals the server's processing capacity. If the processing needs exceed the server's capacity, then there is no well-defined average waiting time, as the queue can grow without bound. (A practical implication of this particular example is that for highly loaded queuing systems the average waiting time can be extremely sensitive to the processing capacity.)
References
- Alexander V. Markov, and Andrey V. Korotayev (2007). "Phanerozoic marine biodiversity follows a hyperbolic trend". Palaeoworld. Volume 16. Issue 4. Pages 311-318.
- Kremer, Michael. 1993. "Population Growth and Technological Change: One Million B.C. to 1990," The Quarterly Journal of Economics 108(3): 681-716.