Exponential growth

Exponential growth (including exponential decay when the growth rate is negative) occurs when the growth rate of the value 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).

The formula for exponential growth of a variable x at the (positive or negative) growth rate r, as time t goes on in discrete intervals (that is, at integer times 0, 1, 2, 3, ...), is

x_t = x_0(1%2Br)^t

where x_0 is the value of x at time 0. For example, with a growth rate of r = 5% = 0.05, going from any integer value of time to the next integer causes x at the second time to be 1.05 times (i.e., 5% larger than) what it was at the previous time.

The exponential growth model is also known as the Malthusian growth model. US scholar Albert Bartlett pointed out the difficulty to grasp ramifications of exponential growth, stating: "The greatest shortcoming of the human race is our inability to understand the exponential function."[1]

Contents

Examples

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%2B\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^{(60\text{ min})/(10\text{ min})}
x(1\text{ 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 %2B \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 %2B \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.

Reformulation as log-linear growth

If a variable x exhibits exponential growth according to x_t = x_0(1%2Br)^t, then the log (to any base) of x grows linearly over time, as can be seen by taking logarithms of both sides of the exponential growth equation:

\log x_t = \log x_0 %2B t \cdot \log (1%2Br).

This allows an exponentially growing variable to be modeled with a log-linear model. For example, if one wishes to empirically estimate the growth rate from intertemporal data on x, one can linearly regress log x on t.

Differential equation

The exponential function \scriptstyle x(t)=ae^{kt} satisfies the linear differential equation:

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

saying that the growth rate of x at time t is proportional to the value of x(t), and it has the initial value

x(0)=a.\,

For \scriptstyle a>0 the differential equation is solved by the method of separation of variables:

\frac{dx}{dt} = kx
\Rightarrow \frac{dx}{x} = k\, dt
\Rightarrow \int \frac{dx}{x} = \int k \, dt
\Rightarrow \ln x = kt %2B \text{constant}\, .

Incorporating the initial value gives:

\ln x = kt %2B \ln a\,
\Rightarrow x = ae^{kt}\,

The solution also applies for \scriptstyle a\le0 where the logarithm is not defined.

For a nonlinear variation of this growth model see logistic function.

Difference equation

The difference equation

x_t = a \cdot x_{t-1}

has solution

x_t = x_0 \cdot a^t,

showing that x experiences exponential growth.

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, Apparent infection rate

Exponential stories

Rice on a chessboard

See also: Wheat and chessboard problem

According to legend, vizier Sissa Ben Dahir presented an Indian King Sharim 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 Swirski, 2006)

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

  1. ^ Bartlett, Albert (2004). The Essential Exponential! For the Future of Our Planet. Center for Science, Mathematics and Computer Education, University of Nebraska-Lincoln. ISBN 0-9758973-0-6. http://scimath.unl.edu/exp/expmain.html. Retrieved 2011-06-15.  See also the author's video lecture: Arithmetic, Population and Energy.
  2. ^ 2010 Census Data. “U.S. Census Bureau.” 12 Nov. 2011. http://2010.census.gov/2010census/data/index.php
  3. ^ Sublette, Carey. "Introduction to Nuclear Weapon Physics and Design". Nuclear Weapons Archive. http://nuclearweaponarchive.org/Nwfaq/Nfaq2.html. Retrieved 2009-05-26. 

Sources

External links