Exponential growth

The graph illustrates how an exponential growth surpasses both linear and cubic growths

Exponential growth (including exponential decay) occurs when the growth rate of a mathematical function is proportional to the function's current value. In the case of a discrete domain of definition with equal intervals it is also called geometric growth or geometric decay (the function values form a geometric progression).

With exponential growth of a positive value its rate of increase steadily increases, or in the case of exponential decay, its rate of decrease steadily decreases.

Exponential growth is said to follow an exponential law; the simple-exponential growth model is known as the Malthusian growth model. For any exponentially growing quantity, the larger the quantity gets, the faster it grows. An alternative saying is 'The rate of growth is directly proportional to the present size'. The relationship between the size of the dependent variable and its rate of growth is governed by a strict law of the simplest kind: direct proportion. It is proved in calculus that this law requires that the quantity is given by the exponential function, if we use the correct time scale. This explains the name.

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Examples

Exponential increases are promised to appear in each new level of a starting member's downline as each subsequent member recruits more people.

Basic formula

A quantity x depends exponentially on time t if

x(t)=a\cdot b^{t/\tau}\,

where the constant a is the initial value of x,

x(0)=a\, ,

and the constant b is a positive growth factor, and τ is the time required for x to increase by a factor of b:

x(t+\tau)=x(t)\cdot b\, .

If τ > 0 and b > 1, then x has exponential growth. If τ < 0 and b > 1, or τ > 0 and 0 < b < 1, then x has exponential decay.

Example: If a species of bacteria doubles every ten minutes, starting out with only one bacterium, how many bacteria would be present after one hour? The question implies a = 1, b = 2 and τ = 10 min.

x(t)=a\cdot b^{t/\tau}=1\cdot 2^{t/(10~\mathrm{min})}
x(1~\mathrm{hr})= 1 \cdot 2^6 =64.

After one hour, or six ten-minute intervals, there would be sixty-four bacteria.

Many pairs (b, τ) of a dimensionless non-negative number b and an amount of time τ (a physical quantity which can be expressed as the product of a number of units and a unit of time) represent the same growth rate, with τ proportional to log b. For any fixed b not equal to 1 (e.g. e or 2), the growth rate is given by the non-zero time τ. For any non-zero time τ the growth rate is given by the dimensionless positive number b.

Thus the law of exponential growth can be written in different but mathematically equivalent forms, by using a different base. The most common forms are the following:

x(t) = x_0\cdot e^{kt} = x_0\cdot e^{t/\tau} = x_0 \cdot 2^{t/T} = x_0\cdot \left( 1 + \frac{r}{100} \right)^{t/p},

where x0 expresses the initial quantity x(0).

Parameters (negative in the case of exponential decay):

The quantities k, \tau, and T, and for a given p also r, have a one-to-one connection given by the following equation (which can be derived by taking the natural logarithm of the above):

k = \frac{1}{\tau} = \frac{\ln 2}{T} = \frac{\ln \left( 1 + \frac{r}{100} \right)}{p}\,

where k = 0 corresponds to r = 0 and to \tau and T being infinite.

If p is the unit of time the quotient t/p is simply the number of units of time. Using the notation t for the (dimensionless) number of units of time rather than the time itself, t/p can be replaced by t, but for uniformity this has been avoided here. In this case the division by p in the last formula is not a numerical division either, but converts a dimensionless number to the correct quantity including unit.

A popular approximated method for calculating the doubling time from the growth rate is the rule of 70, i.e. T \simeq 70 / r (or better: T \simeq 70 / r + 0.03).

Differential equation

The exponential function x(t) = x0 ekt has initial value x0 and satisfies the differential equation:

\!\, \frac{dx}{dt} = kx.

Thus growth is proportial to the value. In the special case k = 0 the function is constant. If the function is constantly zero any growth rate trivially applies.

Starting from the differential equation, it is solved by the method of separation of variables. Formally multiply by dt/x, and integrate to obtain

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

Carrying out the integrations,

\log |x| = kt + \text{constant}\,

So the logarithm grows linearly, and the solution is found.

See logistic function for a simple correction of this growth model where k is not constant.

Other growth rates

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_{t\rightarrow\infty} {t^\alpha \over ae^t} =0.

There is a whole hierarchy of conceivable growth rates that are slower than exponential and faster than linear (in the long run). See Degree of a polynomial#The degree computed from the function values.

Growth rates may also be faster than exponential.

In the above differential equation, if k < 0, then the quantity experiences exponential decay.

Limitations of models

Exponential growth models of physical phenomena only apply within limited regions, as unbounded growth is not physically realistic. Although growth may initially be exponential, the modelled phenomena will eventually enter a region in which previously ignored negative feedback factors become significant (leading to a logistic growth model) or other underlying assumptions of the exponential growth model, such as continuity or instantaneous feedback, break down.

Further information: Limits to Growth, Malthusian catastrophe

Exponential stories

The surprising characteristics of exponential growth have fascinated people through the ages.

Rice on a chessboard

A courtier presented the Persian king with a beautiful, hand-made chessboard. The king asked what he would like in return for his gift and the courtier surprised the king by asking for one grain of rice on the first square, two grains on the second, four grains on the third etc. The king readily agreed and asked for the rice to be brought. All went well at first, but the requirement for 2 n − 1 grains on the nth square demanded over a million grains on the 21st square, more than a million million (aka trillion) on the 41st and there simply was not enough rice in the whole world for the final squares. (From Meadows et al. 1972, p.29 via Porritt 2005)

For variation of this see Second Half of the Chessboard in reference to the point where an exponentially growing factor begins to have a significant economic impact on an organization's overall business strategy.

The water lily

French children are told a story in which they imagine having a pond with water lily leaves floating on the surface. The lily population doubles in size every day and if left unchecked will smother the pond in 30 days, killing all the other living things in the water. Day after day the plant seems small and so it is decided to leave it to grow until it half-covers the pond, before cutting it back. They are then asked, on what day that will occur. This is revealed to be the 29th day, and then there will be just one day to save the pond. (From Meadows et al. 1972, p.29 via Porritt 2005)

See also

References

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