exponential growth

In mathematics, a quantity that grows exponentially is one that grows at a rate proportional to its size. Anything that grows by the same percentage every year (or every month, day, hour etc.) is growing exponentially. For example, if the average number of offspring of each individual (or couple) in a population remains constant, the rate of growth is proportional to the number of individuals. Such an exponentially growing population grows three times as fast in individuals per year when there are six million individuals, as it does when there are two million. Bank accounts with fixed-rate compound interest grow exponentially provided there are no deposits, withdrawals or service charges.

Misusing the term

The phrase exponential growth is incorrectly used by persons not versed in quantitative matters to mean merely surprisingly fast growth (a potential malapropism). Exponential here means something only in relative terms and it merely implies a proportionality of rate of growth to size (and certainly not that the proportion is large). In fact, a population can grow exponentially but at a very slow absolute rate (while the population is small, for instance), and can grow surprisingly fast without growing exponentially. Indeed, the logistic function grows approximately exponentially when it is growing very slowly, but nowhere near exponentially when it is growing fastest.

Technical details

Let x be a quantity growing exponentially with respect to time t. By definition, the rate of change dx/dt obeys the differential equation:

\!\, \frac{dx}{dt} = k x

where k > 0 is the constant of proportionality (the average number of offspring per individual in the case of the population). (See logistic function for a simple correction of this growth model where k is not constant). The solution to this equation is the exponential function

\!\, x(t)=x_0 e^{kt}

-- hence the name exponential growth. The constant

\!\, x_0

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

\lim_{x\rightarrow\infty} {x^\alpha \over Ce^x} =0

There is a whole hierarchy of conceivable growth rates that are slower than exponential and faster than linear (in the long run). Growth rates may also be faster than exponential. The linear and exponential models are merely simple candidates but are those of greatest occurrence in nature. In the above differential equation, if k < 0, then the quantity experiences exponential decay.

Examples of exponential growth

Biology.

Bacteria in a culture dish will grow exponentially until the available food is exhausted.
A new virus (SARS, West Nile, smallpox) of sufficient infectivity (k > 0) will spread exponentially. Each infected person can infect multiple new people.
Human population.

Engineering

Processing power of computers. See also Moore's law.
Charging and discharging of capacitors and changes in current in inductors are also exponential growth and decay phenomena. Engineers use a rule of five time constants to estimate when a steady state has been reached.

Investing.

The effect of compound interest over many years has a substantial effect on savings and a person's ability to retire. See also rule of 72

Physics

Chain reaction (e.g in the explosion of atomic bomb). Each uranium atom that undergoes fission produces multiple neutrons, each of which can be absorbed by adjacent uranium atoms, causing them to fission in turn. If the probability of neutron absorption exceeds the probability of neutron escape (a function of the shape and mass of the uranium), k > 0 and so the production rate of neutrons and induced uranium fissions increases exponentially.**Newton's law of cooling $T=Ae^{-kt}\,$ where T is temperature, t is time, and, A and k > 0 are constants.

Science

Science develops exponentially.