Growth (math)

Growth is the increase in a certain measure of the time course . The opposite of growth is shrinkage, i.e. the decrease in a measured variable - sometimes also referred to as decay . In this context, the term negative growth, which is derived from mathematical modeling and is often misunderstood in everyday language, is often used as a counterpart to (positive) growth.

Mathematical description

Growth is the behavior of a system measured variable over time. First, the value of this variable is determined at a certain point in time . The value of this variable is determined again at a later point in time , this time . ${\displaystyle t_{1}}$ ${\displaystyle W_{1}}$ ${\displaystyle t_{2}}$ ${\displaystyle W_{2}}$

If this second value is greater than the first , then we speak of positive growth. This corresponds to common usage. ${\displaystyle W_{2}>W_{1}}$

However, if the second value is smaller than the first, then one speaks of negative growth, i.e. shrinkage or decrease or decay. ${\displaystyle W_{2}<W_{1}}$

Mathematically speaking, this is comparable to the concept of monotony as a property of the growth function, depending on whether it increases or decreases (strictly) monotonically.

In the case one speaks of zero growth. Then the function is "constant" at this point. ${\displaystyle W_{2}=W_{1}}$

Representation of growth behavior

If there are a large number of measuring points, these are shown in a diagram for illustration. The variables of interest are often not measured continuously, but are only available at equidistant points in time. Strictly speaking, the actual behavior between the discrete, sampled values ​​is not known and can only be interpolated if the sampling is sufficiently close . The following representations can be distinguished:

• the recursion formula that produces discrete values ​​of the growth function.
• the explicit, continuous growth function , which reflects the stock at any point in time in its domain . ${\displaystyle B(t)}$ ${\displaystyle t}$
• the differential equation , which in the form used here represents the first derivative of the growth function.

Essential terms

The following terms are relevant in connection with growth functions:

"Opening balance" (also "starting value") ${\displaystyle B(0)}$

This indicates the value of the stock at the beginning of the era, i.e. H. for recursive representation and for discrete representation . From a function-theoretical point of view, this is the point of intersection of the growth function with the y-axis (ordinate) - also referred to as the y-axis intercept . ${\displaystyle B}$ ${\displaystyle t_{0}}$ ${\displaystyle t=0}$

"Growth constant" ${\displaystyle k}$

This value provides information about the degree of growth, contributes to the characterization of the respective growth model and therefore has a different shape. Its sign has an impact on whether it is positive or negative growth. Furthermore, the higher the value of the rate of change, the steeper the growth function.

Growth rate ${\displaystyle B'(t)}$

In the case of growth processes, the current rate of change ( growth rate ) is called this. It can be determined with the help of the first derivative of the growth function. ${\displaystyle B'(t)}$ ${\displaystyle B(t)}$

Maximum or minimum growth rate

The maximum or minimum growth rate is given by the extreme point of the growth rate. This also represents a turning point in the function, i.e. that is, the curvature of the function changes here. With linear growth there is no extreme point. Here the growth rate is constant.

Half-life or doubling time ${\displaystyle t_{H}}$ ${\displaystyle t_{V}}$

The length of time after which the opening inventory has halved or doubled. ${\displaystyle B(0)}$

Mathematical growth models

In mathematics, there are essentially four models of growth: linear, exponential, restricted and logistic growth.

Linear growth

Example of graphs for linear positive negative and zero growth.

Growth is called linear if the rate of change is constant . ${\displaystyle k}$

In the case of linear growth, the following applies to the stock according to time steps: ${\displaystyle B(t)}$ ${\displaystyle t}$

• recursive representation:

${\displaystyle B_{t+1}=B_{t}+k}$with rate of change and opening inventory ${\displaystyle k}$ ${\displaystyle B_{0}}$

• explicit representation (growth function):

${\displaystyle B(t)=k\cdot t+B(0)}$ with opening balance ${\displaystyle B(0)}$

• Differential equation:

${\displaystyle B'(t)=k}$

Exponential growth

→ Main article : Exponential growth

Growth is called exponential if the rate of change is not constant but proportional to the stock . ${\displaystyle k\cdot B(t)}$ ${\displaystyle B(t)}$

• recursive representation:

${\displaystyle B_{t+1}=B_{t}+k\cdot B_{t}=(1+k)\cdot B_{t}=a\cdot B_{t}}$

( called the growth factor) ${\displaystyle a=1+k}$

• explicit representation (growth function):

${\displaystyle B(t)=B(0)\cdot a^{t}=B(0)\cdot (1+k)^{t}=B(0)\cdot e^{{\widehat {k}}\cdot t}\;\textstyle \quad {\text{mit}}\quad {\widehat {k}}=\ln(1+k)}$

The variable with dimension one denotes the duration of the process divided by the time span in which the inventory changes by the factor (e.g. half-life or generation time). ${\displaystyle t}$ ${\displaystyle B}$ ${\displaystyle a}$

• Differential equation:

${\displaystyle B'(t)={\widehat {k}}\cdot B(t)}$

Limited growth

→ Main article : Restricted growth

Growth is called limited with the limit (also capacity (limit) or saturation (limit / value)) if the rate of change or not constant, but proportional to the saturation deficit (also residual stock) . The limited growth is sometimes also referred to as limited growth and can be described by an exponential function. ${\displaystyle S}$ ${\displaystyle B(t+1)-B(t)}$ ${\displaystyle B'(t)}$ ${\displaystyle S-B(t)}$

• Recursive representation:

${\displaystyle B_{t+1}=B_{t}+k\cdot (S-B_{t})=(1-k)\cdot B_{t}+k\cdot S=q\cdot B_{t}+c}$

• Explicit representation (growth function):

${\displaystyle B(t)=S-(S-B(0))\cdot q^{t}=S-(S-B(0))\cdot (1-k)^{t}=S-(S-B(0))\cdot e^{-{\widehat {k}}\cdot t}\;\textstyle \quad {\text{mit}}\quad {\widehat {k}}=-\ln(1-k)}$

• Differential equation:

${\displaystyle B'(t)={\widehat {k}}\cdot (S-B(t))={\widehat {k}}\cdot S-{\widehat {k}}\cdot B(t)={\widehat {c}}-{\widehat {k}}\cdot B(t)}$

Logistic growth

A growth is called logistically with the limit if the rate of change or is not constant, but proportional to the product of the stock and the saturation deficit . Logistic growth is basically composed of exponential and limited growth. The trend reversal ( turning point of the growth function), which marks the transition from exponential to limited growth, is characteristic of this type of growth . ${\displaystyle S}$ ${\displaystyle B(t+1)-B(t)}$ ${\displaystyle B'(t)}$ ${\displaystyle B(t)\cdot (S-B(t))}$

• Recursive representation:

${\displaystyle B_{t+1}={\frac {B_{t}\cdot S}{B_{t}+(S-B_{t})\cdot e^{-k\cdot S}}}}$

• Explicit representation (growth function):

${\displaystyle B(t)={\frac {B(0)\cdot S}{B(0)+(S-B(0))\cdot e^{-k\cdot S\cdot t}}}=S\cdot {\frac {1}{1+e^{-k\cdot S\cdot t}\left({\frac {S}{B(0)}}-1\right)}}}$

• Differential equation:

${\displaystyle B'(t)=k\cdot B(t)\cdot (S-B(t))=k\cdot S\cdot B(t)-k\cdot B(t)^{2}}$

See also : Logistic function  and population dynamics

• Further recursive representation using the explicit Euler method :

${\displaystyle B_{t+1}=B_{t}+k\cdot B_{t}\cdot (S-B_{t})=(1+k\cdot S)\cdot B_{t}-k\cdot B_{t}^{2}}$

This representation does not give the exact solution of the logistic differential equation, since only an approximation is used here for the derivation of the growth function. This variant is also known as quadratic recursion .

See also : Logistic equation

Further forms of growth

Different types of growth

Exponential growth

Linear growth

Cubic growth

In addition to the classic growth models, there are other forms that are suitable for describing complex growth processes.

Growth according to a power law

Growth processes can also be represented using power functions . This also includes cubic growth , as it is partly used to model the development of animal populations.

Growth function: ${\displaystyle B(t)=B(0)\cdot t^{n},\qquad n\in \mathbb {R} }$

The parameter influences the growth rate of the stand. ${\displaystyle n}$

Poisoned growth

→ Main article : Poisoned growth

Here the free growth of a stock or a population is inhibited by means of a substance that slows the growth of the population. This inhibitor is added to the population at regular intervals and is toxic to the population. The previously undisturbed positive growth is reversed into a negative process that ultimately results in the extinction of the stock. The stock is reduced roughly proportionally to the amount of the inhibitor. Examples of this can be found u. a. in pharmacokinetics as in the toxic effect of an antibiotic on a bacterial culture. ${\displaystyle B}$

Growth function: ${\displaystyle B(t)=B(0)\cdot e^{k\cdot t-{\frac {1}{2}}\cdot c\cdot t^{2}}}$

This describes the growth constant of the stand, while the inhibitor-specific parameter indicates the strength of the poisonous effect on the stand. ${\displaystyle k}$ ${\displaystyle c}$

This form of growth also includes growth with self-poisoning. Here the inhibitor is not added externally, but is produced by the stock or the population itself, such as harmful metabolic residues. This property is u. a. specifically exploited when brewing beer .

Growth of consequences

Processes that run in discrete time are described by sequences . The growth rate of a sequence is the equivalence class of the sequence with regard to the following equivalence relation : Two sequences and are called equivalent if there is a constant such that and holds for all . ${\displaystyle (a_{n})_{n\in \mathbb {N} }}$ ${\displaystyle (a_{n})_{n\in \mathbb {Z} }}$ ${\displaystyle (b_{n})_{n\in \mathbb {Z} }}$ ${\displaystyle C\in \mathbb {N} }$ ${\displaystyle a_{n}\leq b_{Cn}}$ ${\displaystyle b_{n}\leq a_{Cn}}$ ${\displaystyle n\in \mathbb {N} }$

In particular, a sequence is said to have polynomial growth of degree when it is equivalent to , and a sequence is said to have exponential growth when it is equivalent to . ${\displaystyle d}$ ${\displaystyle a_{n}=n^{d}}$ ${\displaystyle a_{n}=e^{n}}$

Properties of growth processes

On the one hand, growth can be characterized qualitatively on the basis of its time course, as shown in the diagram. The unit of the measured variable of the respective growth is used for quantitative description.

Boundary behavior

The course of growth processes can be limited (restricted) or unlimited (unlimited). In relation to the mathematical growth models, the exponential and linear growth can be assigned to an unlimited process, whereby this is more of a theoretical construct of mathematics. In principle, all real growth processes are subject to a restriction, since the resources from which growth is fed are not unlimited or growth is limited in some other way before the resources are exhausted and strives for a dynamic equilibrium (for example in the predator-prey system ). However, a limited growth process does not necessarily lead to a reversal of growth, but allows long-term positive growth within its growth limits during the life of a system. The classic example is entropy in closed systems. The maximum entropy of the system is the growth limit here.

Curvature behavior

Example of graphs for exponential positive or negative or accelerated or retarded growth

On the one hand, linear processes or exponential processes can be distinguished here. On the other hand, degressive (delayed) and progressive (accelerated) growth processes can be classified here, whereby growth itself can in turn be positive or negative. This essentially relates to the change in the rate of growth. The radioactive decay is an example of exponential, delayed, negative growth.

continuity

This property relates to the type of measurement and the property of the measurement data. The measurement takes place either continuously over the entire period or discontinuously only at certain times. Measurement data can in principle be continuous or discrete .

• Continuous measurement data have an infinite number of results. They can be as measured variables with a technical device to measure and partially as a physical quantity called that certain units have. Examples are length of the rail network (including in kilometers), area of ​​the rainforest (including in square meters), volume of milk produced (including in liters), mass (including in kilograms), time (including in seconds). While the physical dimension of the measurand is clearly defined, the specific unit of measurement can vary, e.g. B. Length in centimeters or kilometers.
• Discrete measurement data only have a countable number of results. They can be counted and are available in a specific, absolute number or quantity as a whole number. Examples are the number of a country's population, unemployment figures, and sales of a product.
• The dimensionless measurement data include relative values ​​(ratio numbers) such as u. a. the business key figure , return on sales and the intelligence quotient , complexity or capacity . See Internet , information flood

Real growth processes can have an apparent continuity. The increase in length in humans during the growth period occurs, for example, in spurts.

periodicity

→ Main article : Periodicity

Furthermore, a characteristic of growth processes is whether the measured variable increases or decreases over the entire course of time (monotonously) or is subject to a periodicity or fluctuation, but a general tendency, a trend , to increase or decrease , as is the case with the economy, for example the case is. Fluctuations can u. a. divide into:

• Periodic fluctuations (for example, in systems with feedback ) can be undamped, dampened or rocking.
• Aperiodic fluctuations, so-called fluctuations , can be random or chaotic .

application areas

biology

→ Main article : Growth (biology)

In biology, growth is a condition for life and is defined in principle as the increase in size of an organism due to the formation of new body mass. The theory of evolution ( Darwin ) examines the development of biological species and presents them as a result of growth of the species (overproduction of offspring) and selection . In physiology , growth can be determined by the difference between the structure of substances ( anabolism ) and the degradation of View metabolic products ( catabolism ). Growth comes about through:

• Hyperplasia : The number of cells in an organism increases, particularly through cell division.
• Hypertrophy : The size of the cells themselves increases.
• The extracellular matrix grows in size.

See also : stunted growth , atrophy, and bacterial growth

A sufficient supply of food is a condition for (positive) growth. Biological growth is controlled by hormones. In vertebrates, the hormones somatropin and thyroxine are mainly responsible for this, whereas in plants it is the phytohormones . The science that deals with human growth is called anthropometry . In humans and other individuals, physical growth occurs only in childhood up to approximately puberty . The growth rate of various parts of the body and organs is not 1: 1, which sometimes leads to a shift in proportions ( allometric growth ). In adulthood ( adults ) one speaks of homeostasis when anabolic and catabolic processes are in balance. If too much energy is supplied, it is stored in the form of fat, so that the size of the living body continues to increase (see obesity as a civilization disease ).

Economy or economy

Population development of Augsburg as an example of real growth that comes close to the course of the logistic function.

→ Main article : Economic growth

In the broadest sense, growth describes the increase or decrease in an economic variable over time. On the one hand, this relates to the increase in the number of individuals such as B. the population development and on the other hand on the increase in the economic performance of an economy . Economic growth in the narrower sense means the change in the gross domestic product (GDP) from one period to the next, i.e. the increase in domestic production. This can be done either by improving the utilization of existing production capacities or by adding new production capacities. Economic growth can be viewed from a quantitative point of view (e.g. the purely quantitative increase in GDP) and from a qualitative point of view (e.g. an increase in the quality of life), although the measurement of the latter is difficult. Since 1967, economic growth has also been politically anchored in the law for the promotion of stability and economic growth . Furthermore, economic theories deal with the investigation of change processes in diverse economic developments. In particular, it is about the question of which prerequisites determine the respective growth.

See also : Equilibrium Growth Strategy  and Cobb-Douglas Function

ecology

In the course of the discussion about environmental protection and sustainability , the consideration of growth and decrease processes, in particular of civilization- ecological and human-ecological variables , is increasing . This serves in particular to work out alternative courses of action in dealing with the environment. Examples are the consideration of the greenhouse effect , changes in scarce resources such as water, and an increase in environmental pollution .

See also

• Diffusion limited aggregation
• Growth theory
• Growth law
• Limits to growth
• Growth (group theory)

literature

• Rupert Riedl , Manuela Delpos (ed.): The causes of growth. Kremayr & Scheriau Verlag, Vienna 1996, ISBN 3-218-00628-7 .
• Johannes M. Waidfeld: Growth, the error of prosperity, a social consideration. Fischer & Fischer Medien, Frankfurt am Main 2005, ISBN 3-89950-076-8 .
• 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 , pp. 273-281.
• Joachim Engel: Application-Oriented Mathematics. From data to function. Springer Verlag, Heidelberg 2010, ISBN 978-3-540-89086-7 .
• Walter Seifritz: Growth, Feedback and Chaos. An introduction to the world of nonlinearity and chaos. Hanser Verlag, Munich 1987, ISBN 3-446-15105-2 .

Web links

Wiktionary: growth  - explanations of meanings, word origins, synonyms, translations

Wiktionary: grow  - explanations of meanings, word origins, synonyms, translations

Wikiquote: Growth  Quotes

• Analysis of growth processes
• Linear, exponential, constrained and logistic growth
• Steven Bärwolf: Theory topic 6. Types of growth. At: Exwauzer.Cynecto.de. Retrieved January 12, 2013.

Individual evidence

1. ^ ^{A b} 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. 275.
2. ↑ Eric W. Weisstein : Logistic Map . In: MathWorld (English).
3. ^ Wolfgang Niemeier: compensation calculation. An introduction for students and practitioners of surveying and geographic information. De Gruyter Verlag, Berlin 2002, ISBN 3-11-014080-2 , p. 2.
4. ↑ growth. On: Wissen.de. Retrieved January 15, 2013.
5. ↑ growth. On : wirtschaftslexikon.gabler.de. Retrieved January 15, 2013.