Exponential growth

Overview
Exponential growth (or geometric growth) occurs when the growth rate of a mathematical function is proportional to the function's current value. Such 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 proportional to the state of growth'. 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.

Basic formula
A quantity x depends exponentially of time t if
 * $$x(t)=a\cdot b^t\,$$

where the constant a is the initial value of x,
 * $$x(0)=a\, ,$$

and the constant b is a positive growth factor
 * $$x(t+1)=x(t)\cdot b\, .$$

If b is greater than 1, then x has exponential growth. If b is less than 1, then x has exponential decay. If b is equal to 1, then x is constant.

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?
 * $$x=a\cdot b^t=1\cdot 2^6=64\,$$

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

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


 * $$ \!\, \frac{dx}{dt} = \log b \cdot x= kx$$

where log b = k ≠ 0 is the rate of growth. (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 ('e' being a mathematical constant). The constant $$\!\, x_0$$ is the initial value of the quantity x.

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). Growth rates may also be faster than exponential. The linear and exponential models are not 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.

Characteristic quantities of exponential growth
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,$$

where as in the example above x0 expresses the initial quantity x(0).

The quantity k is called the growth constant; the quantity r is the percent increase per unit time; $$\tau$$ is the  e-folding time; and T is the doubling time. Indicating one of these four equivalent quantities automatically permits calculating the three others, which are connected by the following equation (which can be derived by taking the natural logarithm of the above):


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

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: $$T \simeq 70 / r + 0.03$$).

Limitations of exponential models
An important point about exponential growth is that even when it seems slow on the short run, it becomes impressively fast on the long run, with the initial quantity doubling at the doubling time, then doubling again and again. For instance, a population growth rate of 2% per year may seem small, but it actually implies doubling after 35 years, doubling again after another 35 years (i.e. becoming 4 times the initial population). This implies that both the observed quantity, and its time derivative will become several orders of magnitude larger than what was initially meant by the person who conceived the growth model. Because of this, some effects not initially taken into account will distort the growth law, usually moderating it as for instance in the logistic law. Exponential growth of a quantity placed in the real world (i.e. not in the abstract world of mathematics) is a model valid for a temporary period of time only.

For this reason, the exponential growth model is at times challenged on the ground that it is valid for the short term only, i.e. nothing can grow indefinitely. For instance, a population in a closed environment cannot continue growing if it eats up all the available food and resources; industry cannot continue pumping carbon from the underground into the atmosphere beyond the limits connected with oil reservoirs and the consequences of climate change. Problems of this kind exist for every mathematical representation of the real world, but are specially felt for exponential growth, since with this model growth accelerates as variables increase in a positive feedback, to a point where human response time to inconvenience can be insufficient. On these points, see also the Exponential stories below.

Examples of exponential growth

 * Biology.
 * Microorganisms in a culture dish will grow exponentially, after the first microorganism appears and a lag phase, and until an essential nutrient is exhausted.
 * A virus (SARS, West Nile, smallpox) of sufficient infectivity (k > 0) will spread exponentially at first, if no artificial immunization is available. Each infected person can infect multiple new people.
 * Human population, if the number of births and deaths per person per year were to remain at current levels (but also see logistic growth).
 * Many responses of living beings to stimuli, including human perception, are logarithmic responses, which are the inverse of exponential responses; the loudness and frequency of sound are perceived logarithmically, even with very faint stimulus, within the limits of perception. This is the reason that exponentially increasing the brightness of visual stimuli is perceived by humans as a linear increase, rather than an exponential increase. This has survival value. Generally it is important for the organisms to respond to stimuli in a wide range of levels, from very low levels, to very high levels, while the accuracy of the estimation of differences at high levels of stimulus is much less important for survival.
 * Computer technology
 * Processing power of computers. See also Moore's law and technological singularity (under exponential growth, there are no singularities. The singularity here is a metaphor.).
 * In computational complexity theory, computer algorithms of exponential complexity require an exponentially increasing amount of resources (e.g. time, computer memory) for only a constant increase in problem size. So for an algorithm of time complexity 2^x, if a problem of size x=10 requires 10 seconds to complete, and a problem of size x=11 requires 20 seconds, then a problem of size x=12 will require 40 seconds. This kind of algorithm typically becomes unusable at very small problem sizes, often between 30 and 100 items (most computer algorithms need to be able to solve much larger problems, up to tens of thousands or even millions of items in reasonable times, something that would be physically impossible with an exponential algorithm). Also, the effects of Moore's Law do not help the situation much because doubling processor speed merely allows you to increase the problem size by a constant. E.g. if a slow processor can solve problems of size x in time t, then a processor twice as fast could only solve problems of size x+constant in the same time t. So exponentially complex algorithms are most often impractical, and the search for more efficient algorithms is one of the central goals of computer science.
 * Internet traffic growth.
 * Investment. 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
 * Avalanche breakdown within a dielectric material. A free electron becomes sufficiently accelerated by an externally applied electrical field that it frees up additional electrons as it collides with atoms or molecules of the dielectric media. These secondary electrons also are accelerated, creating larger numbers of free electrons. The resulting exponential growth of electrons and ions may rapidly lead to complete dielectric breakdown of the material.
 * Nuclear chain reaction (the concept behind nuclear weapons). Each uranium nucleus 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, in an uncontrolled reaction.


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

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 $$n$$th square demanded over a million grains on the 21st square, more than a quadrillion 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)