Exponential growth

From Wikinfo

In mathematics, a quantity grows exponentially when it grows at a rate proportional to its value. For example, if the average number of offsprings of each individual (or couple) in a population remains constant, the growth is proportional to the number of individuals. Such an exponentially growing population grows, in individuals per year, three times as fast when there are six million individuals, as it does when there are two millions. Also, a snowball rolling downhill grows exponentially with time since when it is twice as big it gathers snow twice as fast.

If we call x this quantity, the rate of change dx/dt' obeys by definition the differential equation:

<math>dx/dt=\alpha x</math>

where α is the constant of proportionality (the average number of offspring in the case of the population), it is strictly positive for a growth while it is negative for a decay. (See the logistic map for a simple correction of this model growth where α is not constant). The solution to this equation is the exponential function x(t)=exp(αt), whence the name of the associated growth.

The phrase exponential growth is also a misnomer used by persons unaware of quantitative matters to mean merely surprisingly fast growth. In fact, a population can grow exponentially but at a very slow rate (as the fission process in a nuclear power plant), and can grow surprisingly fast without growing exponentially.

In the long run, exponential growth of any kind will however overtake linear growth of any kind (the basis of the Malthusian catastrophe) as well as any polynomial growth, i.e., for all α:

<math>\lim_{x\rightarrow\infty} {x^\alpha \over e^x} =0</math>

There is a whole hierarchy of conceivable growth laws that are sub-exponential and also super-linear, and of course faster than exponential growth is also possible. The linear and exponential models are merely simple candidates but are those of greatest occurence in nature.

Examples of Exponential Growth

• Investing. The effect of compound interest over many years has a substantial effect on savings and a person's ability to retire.
• Biology.
  • Bacteria in a culture dish will grow exponentially until the available food is exhausted.
  • A new virus (SARS, West Nile, smallpox) will spread exponentially. Each infected person can infect multiple new people.
  • Human population.
• An atomic bomb. Each uranium atom that undergoes fission produces neutrons, which in turn split more uranium atoms. If the mass of uranium is sufficent, the number of neturons increases exponentially.
• A nuclear power plant. Same as above but this time the fission process (also called divergence of the reactor) is controled so that the growth, while exponential, is very slow.

See also: Bacterial growth, Logistic curve, Arthrobacter, Exponential algorithm, Exponential function, Asymptotic notation

References

• Adapted from the Wikipedia article, "Exponential growth" http://www.wikipedia.org/wiki/Exponential_growth August 17, 2003