Exponential growth

Exponential growth (also called unlimited or free growth ) describes a mathematical model for a growth process in which the stock size is always multiplied by the same factor in the same time steps . The value of the stock size can either increase ( exponential increase ) or decrease ( exponential decay or exponential decrease ) over time. Such a course can be clearly indicated in the case of an exponential increase due to the doubling time and in the case of an exponential decrease due to the half-life . In contrast to linear or polynomial growth, exponential growth causes significantly larger changes even with initially only small changes, so that exponential growth from a certain point in time exceeds any linear or polynomial growth by orders of magnitude . Because of this, the impact of exponential growth can easily be underestimated.

Function of exponential growth

Exponential growth

Exponential decay

In the case of a growth function, the population size is dependent on time . She is of the form ${\displaystyle B(t)}$ ${\displaystyle t}$

${\displaystyle B(t)=A\cdot b^{t}}$with and ${\displaystyle A>0}$ ${\displaystyle b>0}$

or equivalent to . ${\displaystyle B(t)=A\cdot e^{\lambda t}}$ ${\displaystyle \lambda =\ln(b)}$

Because is the initial stock at the time . ${\displaystyle B(0)=A}$ ${\displaystyle A}$ ${\displaystyle t=0}$

If , then , it is an exponential increase. The doubling time (also called double value time and generation time in biology ) is then . ${\displaystyle b>1}$ ${\displaystyle \lambda >0}$ ${\displaystyle T_{2}={\frac {\ln(2)}{\lambda }}}$

With and therefore one speaks of an exponential decrease. The half-life is then . ${\displaystyle b<1}$ ${\displaystyle \lambda <0}$ ${\displaystyle T_{0,5}={\frac {\ln(0{,}5)}{\lambda }}}$

In general, a duplication factor is the duplication time . Conversely, the multiplication factor is calculated . ${\displaystyle v}$ ${\displaystyle T_{v}={\frac {\ln(v)}{\lambda }}}$ ${\displaystyle v={\frac {B(t+T_{v})}{B(t)}}=b^{T_{v}}=e^{\lambda {\dot {T}}_{v}}>0}$

Example 1: compound interest with an interest rate of 8% pa

${\displaystyle K(t)=A\cdot 1{,}08^{t}}$

This means the capital accumulated after years in €. ${\displaystyle K(t)}$ ${\displaystyle t}$

With an initial capital of € 100 . After 9 years the capital is due ${\displaystyle A=100}$

${\displaystyle K(9)=100\cdot 1{,}08^{9}=199{,}90}$

increased to € 199.90, so it has almost doubled.

Example 2: epidemic

In one country, the number of infected people doubles every 3 days. Has one z. For example, at time 0 a number of 1000 infected people, after 3 days 2000, after 6 days 4000 infected, etc. The number of infected initially grew exponentially and can then be described by the following function:

${\displaystyle I(t)=1000\cdot b^{t}}$with ( is the number of days) ${\displaystyle b={\sqrt[{3}]{2}}}$ ${\displaystyle t}$

After 27 days it's already

${\displaystyle I(27)=1000\cdot ({\sqrt[{3}]{2}})^{27}=512.000}$

Infected. With unchecked growth and a limited population, an epidemic leads to logistic growth .

Example 3: radioactive decay

Cesium-137, a product of nuclear fission, has a half-life of 30 years. Its decay function is therefore

${\displaystyle C(t)=A\cdot b^{t}}$with ( is the number of years) ${\displaystyle b=2^{-{\frac {1}{30}}}}$ ${\displaystyle t}$

After 90 years there is because of

${\displaystyle C(90)=A\cdot ({2^{-{\frac {1}{30}}}})^{90}={\frac {A}{8}}}$

still the original amount of cesium present . ${\displaystyle 1/8=12{,}5\ \%}$ ${\displaystyle A=C(0)}$

In Examples 1 and 2 it is an exponential increase and in Example 3 it is an exponential decrease.

properties

Model description

Different types of growth

exponential growth

linear growth

cubic growth

The picture on the right shows as an example that in the long term the existence (as well as the growth rate) of a positive exponential process is always greater than with linear , cubic growth or generally with all growth processes that can be described by completely rational functions .

In the exponential growth model , the change ( discrete case) or ( continuous case) in the stock size is proportional to the stock. In the discrete case, the new inventory value is subject to a positive growth by the old value by a constant greater than 1 is multiplied is, and is smaller in negative growth with a positive constant is multiplied as the first ${\displaystyle B_{n+1}-B_{n}}$ ${\displaystyle B'(t)}$

With the exponential decrease , the x-axis forms the asymptote of the graph of the growth function. The stock size approaches zero but does not go away. In application cases such as B. in biology, the stock sizes are often whole numbers , so that very small values ​​ultimately have no meaning and the stock is practically extinct.

Differential equation

Differential equations (DGL) serve to describe continuous ( steady ) growth models.

The DGL for the exponential process is:

${\displaystyle {B}'(t)={\frac {\mathrm {d} B}{\mathrm {d} t}}=\lambda \cdot B(t)}$

This is a linear homogeneous differential equation with constant coefficients and can be solved, for example, using the " variable separation " method.

The growth rate can be derived from the DGL: . ${\displaystyle B'(t)=\lambda \cdot B(t)=\lambda \cdot B(0)\cdot \mathrm {e} ^{\lambda t}}$

Discrete growth model

To illustrate the discrete growth model in a recursive form derived from differences are consequences . The time difference in an equidistant sequence of times denotes for ; and means the corresponding stock sizes. ${\displaystyle \Delta t}$ ${\displaystyle t_{n}=n\Delta t}$ ${\displaystyle n=0,1,2,\dotsc }$ ${\displaystyle B_{n}}$

In recursive form, discrete-time exponential growth (increase and decrease) is achieved

${\displaystyle B_{n+1}=B_{n}\cdot b}$

described. The growth factor is identical to that in the continuous-time case. ${\displaystyle b=1+p}$

The stock size follows from the formulas for continuous growth with the substitutions , and to ${\displaystyle B_{n}}$ ${\displaystyle t=n\Delta t}$ ${\displaystyle T_{b}=\Delta t}$ ${\displaystyle \lambda ={\frac {\ln b}{\Delta t}}}$

${\displaystyle B_{n}=B_{0}b^{n}=B_{0}\mathrm {e} ^{n\ln b}}$.

Dissolution according to time

The time span in which an exponentially developing stock changes by the factor is to be determined. The growth equation is given with the multiplication factor and the multiplication time. From follows ${\displaystyle t_{f}}$ ${\displaystyle f=B(t_{f})/B(0)}$ ${\displaystyle b}$ ${\displaystyle T_{b}}$ ${\displaystyle b^{\frac {t_{f}}{T_{b}}}=f}$

${\displaystyle {\frac {t_{f}}{T_{b}}}=\log _{\,b}f={\frac {\ln f}{\ln b}}}$.

Example: For near one , approximately applies . A doubling ( ) therefore takes time . ${\displaystyle b=1+p}$ ${\displaystyle \ln b=p}$ ${\displaystyle f=2}$ ${\displaystyle t_{f}\approx T_{b}{\frac {0{,}7}{p}}}$

Examples, general and detailed

Natural sciences

Bacterial growth in E. coli . The generation time is around 20 minutes.

Population growth

The growth of microorganisms such as bacteria and viruses , cancer cells and also the world population can theoretically increase exponentially without limiting factors (e.g. competitors , (eating) enemies or pathogens, finite food sources). However, this is usually only a theoretical example. The growth z. B. of bacteria is normally described by a logistic function , which, however, is very similar to an exponential function at the beginning.

Radioactive decay

The number of nuclear decays in a quantity of radioactive material decreases almost exponentially over time (see also the law of decay ). In equally long time intervals, the same fraction of the amount still present at the beginning of the interval always decays.

Lambert-Beer law

If a monochromatic (single-colored) light beam with a certain incident intensity travels through an absorbing , homogeneous medium (e.g. dye) of a certain layer thickness, the intensity of the emerging beam can be represented by an exponential decay process. The intensity of the outgoing beam is proportional to the intensity of the incoming beam. This is closely related to the so-called absorption law for, for example, X-rays .

Exponential increase in amplitude after switching on an oscillator , until the limit sets in

Fanning an oscillator

The time-linear amplitude change when an oscillator starts to oscillate corresponds to a time-exponential increase in amplitude of a real oscillator with parameter resonance .

Economy and finance

compound interest

The interest is added to a capital over a certain period of time and earns interest. This leads to an exponential growth in capital. The compound interest formula is , with the interest rate and the initial capital being (see also interest calculation , Josephspfennig - here a penny is invested in the year zero). ${\displaystyle K}$ ${\displaystyle K(t)=K(0)\cdot (1+i)^{t}}$ ${\displaystyle i}$ ${\displaystyle K(0)}$

In the case of a savings book with 5% interest per year, the doubling time is included according to the rule of thumb above . ${\displaystyle {\tfrac {70}{5}}\approx {\text{14 Jahren}}}$

Pyramid scheme

These are business models in which the number of participants grows exponentially. Each employee has to recruit a certain number of additional employees, who in turn are supposed to recruit this number, and so on. Gift circles and chain letters work according to the same principle .

technology

5-fold folded mylar sheet

wrinkles

With each fold, the thickness of the paper or film doubles. In this way, thin foils can be measured with a simple caliper . The Mylar film in the picture consists of 2 ⁵  = 32 layers of film after being folded 5 times , which together have a thickness of 480 µm. A film is therefore approx. 15 µm thick. After folding it 10 times, the layer would already be 15 mm thick, after another 10 folds it would be more than 15.7 m. Since the stacking surface exponentially reduced, can be paper in a commercially available paper size barely beat than seven times.

mathematics

Chess board with a grain of wheat

The story , according to the intended Brahmin Sissa ibn Dahir , a game that now form the chess is known for the Indian rulers Shihram have invented to illustrate his tyrannical rule, which plunged the people into misery and distress, and to keep him . He was granted a free wish for this. Sissa wanted the following: On the first field of a chessboard he wanted a grain of wheat (depending on the literature also a grain of rice ), on the second field double, i.e. two grains, on the third again double the amount, i.e. four and so on. The king laughed and gave him a sack of grain. He then asked the ruler to have his mathematicians determine the exact amount, as one sack was not quite enough. The calculation showed: In the last (64th) field there would be 2 ⁶³ ≈ 9.22 · 10 ¹⁸ grains, i.e. more than 9 trillion grains. More than any grain in the world. The increase in the number of grains can be understood as exponential growth using an exponential function of base 2.

music

The frequency ratio of intervals grows exponentially.

example

interval	size	Frequency ratio
0  octaves ( Prime )	0	01
1 octave	1200  cents	02
2 octaves	2400 cents	04th
3 octaves	3600 cents	08th
4 octaves	4800 cents	16
• • •

The intervals are an additively ordered group. The frequency ratio of a sum is the product of the frequency ratios.

example

 Quinte = 702 Cent (Frequenzverhältnis 3/2)
 Quarte = 498 Cent (Frequenzverhältnis 4/3)
 Quinte + Quarte = 702 Cent + 498 Cent = 1200 Cent = Oktave (Frequenzverhältnis 3/2•4/3 = 2)

Limits of the model

The model approach to exponential growth has its limits in reality - especially in the economic area.

“Exponential growth is not realistic” as a long-term trend , according to economist Norbert Reuter . He argues that the growth rates in more developed societies decline due to cyclical influences. The indicator for this is the gross domestic product (GDP). Looking at statistical data, it can be deduced that exponential economic growth is more typical of the early years of an industrial economy , but that at a certain level, when essential development processes have been completed, it changes into linear growth . If a further exponential growth is extrapolated , a discrepancy occurs between the growth expectation and the actual course. Among other things, this affects the national debt . The computationally incorrect expectation that national debt could be limited by economic growth, however, only lowers the threshold for new debt. If, however, the expected growth fails to materialize, a deficit arises that limits the future ability of a state to act. Because of the interest and compound interest, there is a risk that the national debt will grow exponentially.

Another aspect is that the demand does not increase immeasurably, but experiences a saturation effect, which cannot be compensated by appropriate economic policy . Considerations with regard to biological relationships, for example through competition for food or space, go in the same direction. In relation to the world population, this thematizes the debate about the ecological footprint - that is, about the carrying capacity of the earth with the relatively small consumption of regenerative resources in relation to the total consumption of resources . Here, the exponential growth model also neglects demographic developments such as the relationship between birth and death rates and the relationship between the female and male population.

Growth models that take into account the saturation effect are constrained growth and logistic growth , while the poisoned growth model also includes growth inhibiting factors in the process.

literature

• Joachim Engel: Application-Oriented Mathematics: From Data to Function . An introduction to mathematical modeling for student teachers. Springer Verlag , Heidelberg 2010, ISBN 978-3-540-89086-7 , p. 150-153 .
• Hermann Haarmann, Hans Wolpers: Mathematics for obtaining the general university entrance qualification . Non-technical subjects. 2nd Edition. Merkur Verlag, Rinteln 2012, ISBN 978-3-8120-0062-8 , p. 272-274 .
• Klaus Schilling: Analysis: Qualification phase: core curriculum for vocational high school . Eins Verlag, Cologne 2012, ISBN 978-3-427-07770-1 , pp. 249-257 .
• Walter Seifritz: Growth, Feedback and Chaos: An Introduction to the World of Nonlinearity and Chaos . Hansen Verlag, Munich 1987, ISBN 3-446-15105-2 , p. 9-18 .

Web links

• Eric W. Weisstein : Exponential Growth . In: MathWorld (English).

Individual evidence

1. ↑ Our model assumes an infinite number. With a population of 80 million, the number of infected people calculated according to the model of logistic growth after 27 days is approx. 310,000.
2. ^ Begon, M. Mortimer, M. Thompson, DJ: Populationsökologie. Spectrum, Heidelberg 1997.
3. ^ Stefan Keppeler: Mathematics 1 for biologists, geoscientists and geoecologists: exponential functions and logarithms. (PDF; 454 kB) (No longer available online.) November 5, 2008, p. 9 , archived from the original on February 1, 2014 ; accessed on March 28, 2013 . 
4. ^ Stefan Keppeler: Mathematics 1 for biologists, geoscientists and geoecologists: exponential functions and logarithms. (PDF; 454 kB) (No longer available online.) November 5, 2008, p. 9 , archived from the original on February 1, 2014 ; accessed on March 28, 2013 . 
5. ^ Bouger-Lambert law. Retrieved March 28, 2013 . 
6. ↑ Valeriano Ferreras Paz: X-ray absorption. (PDF; 2.0 MB) Archived from the original on February 2, 2014 ; accessed on March 31, 2013 . 
7. ↑ Hans Dresig , II Vul'fson: Dynamics of Mechanisms . VEB Deutscher Verlag der Wissenschaften , Berlin 1989, ISBN 3-326-00361-7 , p. 198 ( full text ).
8. ↑ Dr. rer. nat. H. Schreier: Financial Mathematics. (PDF; 211 kB) pp. 9–11 , accessed on April 10, 2013 . 
9. ↑ Compound Interest and Exponential Growth. Archived from the original on August 30, 2012 ; Retrieved April 2, 2013 . 
10. ↑ Roland Spinola: Exponential growth - what is that. (PDF; 121 kB) Retrieved on April 13, 2013 . 
11. ↑ Roland Spinola: Exponential growth - what is that. (PDF; 121 kB) Retrieved on April 13, 2013 . 
12. ^ History. Retrieved March 23, 2014 . 
13. ↑ The chess board and the grains of rice. Archived from the original on October 4, 2013 ; Retrieved April 14, 2013 . 
14. ↑ Here, the base = 2 results for the power , i.e. a power of two , because the number of grains is doubled from field to field. The first doubling takes place from the first to the second field. Therefore, the sixty-fourth field results in 64 - 1 = 63 doublings. Therefore the exponent is equal to 63 here. On the 64th field there would be 2 ^{(64 - 1)} = 2 ⁶³ = 9.223.372.036.854.775.808 ≈ 9.22 · 10 ¹⁸ grains.
15. ↑ Hartmut Steiger: Exponential growth is not realistic. Retrieved April 16, 2013 . 
16. ^ Kai Bourcarde, Karsten Heinzmann: Normal case exponential growth - an international comparison. (PDF; 738 kB) p. 6 , accessed on April 16, 2013 . 
17. ↑ Kai Bourcarde: Linear economic growth - exponential national debt. (PDF; 345 kB) p. 4 , accessed on April 16, 2013 . 
18. ↑ Hartmut Steiger: Exponential growth is not realistic. Retrieved April 16, 2013 . 
19. ↑ Donella Meadows, Jorgen Randers, Dennis Meadows: Exponential growth as the driving force behind the transgression of ecological boundaries. Retrieved April 16, 2013 . 
20. ↑ Thomas Kämpe: World population. (PDF; 2.4 MB) (No longer available online.) Formerly in the original ; Retrieved April 16, 2013 .  ( Page no longer available , search in web archives )  Info: The link was automatically marked as defective. Please check the link according to the instructions and then remove this notice.