Exponential growth

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The graph illustrates how exponential growth (green) surpasses both linear (red) and cubic (blue) growth.
  Exponential growth
  Linear growth
  Cubic growth

Exponential growth occurs when the growth rate of the value of a mathematical function is proportional to the function's current value. Exponential decay occurs in the same way when the growth rate is negative. 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+r)^t

where x0 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.

Examples[edit]

Bacteria exhibit exponential growth under optimal conditions.

Basic formula[edit]

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\, ,

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

x(t+\tau)=a \cdot b^{\frac{t+\tau}{\tau}} = a \cdot b^{\frac{t}{\tau}} \cdot b^{\frac{\tau}{\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 (bτ) 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, τ, 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 τ 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[edit]

If a variable x exhibits exponential growth according to x(t)=x_0(1+r)^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 + t \cdot \log (1+r).

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[edit]

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 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 + \text{constant}\, .

Incorporating the initial value gives:

\ln x =  kt + \ln a\,
\Rightarrow x =  ae^{kt}\,

The solution also applies for a ≤ 0 where the logarithm is not defined.

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

Difference equation[edit]

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[edit]

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[edit]

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.

Exponential stories[edit]

Rice on a chessboard[edit]

According to an old 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)[5]

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.

Water lily[edit]

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 half-coverage 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)[5]

See also[edit]

References[edit]

  1. ^ 2010 Census Data. “U.S. Census Bureau.” 20 Dec. 2012. Internet Archive: http://web.archive.org/web/20121220035511/http://2010.census.gov/2010census/data/index.php
  2. ^ Bob Yirka (2013-04-18), Researchers use Moore's Law to calculate that life began before Earth existed, phys.org, retrieved 2013-04-22 
  3. ^ Larger than Life Indeed. Economic Times. April 2013. Retrieved 2013-04-23. 
  4. ^ Sublette, Carey. "Introduction to Nuclear Weapon Physics and Design". Nuclear Weapons Archive. Retrieved 2009-05-26. 
  5. ^ a b Porritt, Jonathan (2005). Capitalism: as if the world matters. London: Earthscan. p. 49. ISBN 1-84407-192-8. 

Sources[edit]

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