Hyperbolic growth

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The reciprocal function, exhibiting hyperbolic growth.

When a quantity grows towards a singularity under a finite variation it is said to undergo hyperbolic growth.^{[citation needed]} More precisely, the reciprocal function $1 / x$ has a hyperbola as a graph, and has a singularity at 0, meaning that the limit as $x \to 0$ is infinity: any similar graph is said to exhibit hyperbolic growth.

[edit] Description

If the output of a function is inversely proportional to its input, or inversely proportional to the difference from a given value $x 0$ , the function will exhibit hyperbolic growth, with a singularity at $x 0$ .

Hyperbolic growth can also be created by certain non-linear positive feedback mechanisms.

[edit] Comparisons with other growth

Like exponential growth and logistic growth, hyperbolic growth is highly nonlinear, but differs in important respects. These functions can be confused, as exponential growth, hyperbolic growth, and the first half of logistic growth are convex functions; however their asymptotic behavior (behavior as input gets large) differs dramatically:

logistic growth is constrained (has a finite limit, even as time goes to infinity),
exponential growth grows to infinity as time goes to infinity (but is always finite for finite time),
hyperbolic growth has a singularity in finite time (grows to infinity at a finite time).

[edit] Applications

[edit] Population

Certain mathematical models suggest that until the early 1970s the world population underwent hyperbolic growth (see, e.g., Introduction to Social Macrodynamics by Andrey Korotayev et al.). Other models suggest exponential growth, logistic growth, or other functions.

[edit] Queuing theory

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.

[edit] Enzyme kinetics

A further practical example of hyperbolic growth can be found in enzyme kinetics. When the rate of reaction (termed velocity) between an enzyme and substrate is plotted against various concentrations of the substrate, a hyperbolic plot is obtained for many simpler systems. When this happens, the enzyme is said to follow Michaelis-Menten kinetics.