Exponential growth

(Also known as unlimited or free growth ) Exponential growth presents a mathematical model for a growth process in which the population size at equal time steps forever changed by the same factor. The value of stock size over time can either increase ( exponential increase ) or decrease ( exponential decay ).

  • 2.1 Natural Sciences
  • 2.2 Economic and Financial Affairs
  • 2.3 technology
  • 2.4 Mathematics
  • 2.5 Limits of the model

Properties

Model Description

  • Exponential growth
  • Linear growth
  • Cubic growth

In the model of exponential growth, the change in the stock size ( discrete case ) or (continuous case ) is not constant but proportional to the inventory. The new stock value is obtained for positive growth by the old value is added a fixed multiple of the old value and is subtracted for negative growth.

The accompanying chart shows that in the long run, the growth rate of the positive exponential process in this model is larger than in growth processes, which can be described by rational integral functions, such as the linear and cubic growth.

For the exponential decrease in the x-axis is the asymptote of the growth function. The stock size approaches to zero, disappears mathematically but not completely. In application references such as biology are the stock sizes usually an integer, so that very small values ​​ultimately have no meaning and the stock size die out in practical terms.

Key concepts and notation

  • Denotes the time.
  • Is the observed stock size.
  • Marks the beginning inventory ( initial condition ) at the time.
  • Represents the growth rate and is often expressed in percent.
  • Let ( also called multiplication factor ) of the stock-specific growth factor. It is calculated as the ratio of inventory values ​​of two consecutive time points.
  • Is the constant growth or decay constant. Because it provides a measure of the strength of the positive and negative growth. On the basis of its sign can be deduced whether an exponential increase or decrease is present. This constant has the dimension [ 1: time ].
  • Shows the growth rate and the instantaneous rate of growth.
  • Denotes the half-life or doubling time (also double -life and in biology called generation time ) and hence the period in which the stock size has been halved or doubled with respect to the initial stock.
  • Says the Euler number.
  • Refers to the natural logarithm.

Differential equation

Differential equations ( ODE ) are used for description of continuous ( steady ) growth models.

The ODE for the exponential process is:

This is a homogeneous linear differential equation with constant coefficients and can be achieved for example by means of the method " variable separation".

Explicit representation (growth function)

The special solution of the differential equation describing the explicit representation of the growth process and also forms the growth function. This represents an exponential

It is for the exponential growth:

With

For and, therefore, positive growth is described. Here the population size grows - in mathematical terms - to infinity. The graph of the function increases monotonically and a left turn. The growth is progressive. The growth rate decreases with time.

For the exponential decay follows:

With

For an exponential decay and, consequently, is described. The graph of the function falls monotonically and a left turn. The growth is gradually decreased. The growth rate decreases with time. The x-axis (abscissa) forming the asymptote of the function.

Growth rate

This can be easily derived from the ODE:

Recursive representation

To represent the discrete growth model as a recursive representation used from differences derived consequences. This is a time difference represents in a sequence of equidistant time points and refers to the corresponding component sizes. Mathematically, one can distinguish between an exact and approximate discretization.

Exact discretization

For the exponential increase applies:

For the exponential decay follows:

Approximate discretization

The following approximation is obtained by applying the explicit Euler method:

For the exponential increase:

For the exponential decay:

Here, it should be noted that

By a series expansion of the exponential function can be shown that the exact and approximate recursive representation matches higher than first order terms except for.

Half- and doubling time

The half- or doubling time depends follows directly related to the size:

Exponential growth:

Exponential decay:

The doubling time of an exponentially growing size can be combined with roll over. In the counter goes to this rule of thumb, as applies. Compare with the 72 rule.

Correspondingly obtained.

Examples

Natural sciences

  • Growth of populations
  • Radioactive decay
  • Lambert- Beer 's law
  • Fomenting of an oscillator

Economy and Finance

  • Compound interest
  • Snowballing

Technology

  • Each folds, the thickness of paper or foil doubled. In this way, thin films with a simple calipers can measure. The mylar film on the picture is a 5 -fold wrinkles out of 25 = 32 layers of foil having a thickness of 480 microns together. A film is thus about 15 microns thick. After 10 -fold wrinkles, the situation would already be 15 mm thick, after 10 folds more than 15.7 m. Since in the same exponentially reduces the stacking surface, paper can be in a standard paper size no more than seven times to beat.

Mathematics

  • Chessboard with a grain of wheat

Limitations of the model

The model approach to exponential growth encounters in reality to its limits - especially in the economic field.

" Exponential growth is not realistic " as a long- term trend, the economist Norbert Reuter. He argues that the growth rates are falling in more developed societies due to cyclical influences. Indicator is the gross domestic product (GDP). With regard to statistical data it can be concluded that an exponential economic growth is more typical for the beginning years of an industrial economy, but is converted into a linear growth at a certain level when significant development processes have been completed. Thus, if a further exponential growth extrapolated, a discrepancy occurs between the growth expectations and the actual course. This relates to the national debt. By computationally false expectation that the national debt could be limited by economic growth, the threshold for new debt decreases. However, If the expected growth of, creating a deficit that restricts the future act of a State. Because of compound interest is the risk that the national debt is growing exponentially.

Another aspect is that the need not to skyrocket, but experiences a saturation effect, which also can not be compensated by appropriate economic policies. In the same vein going considerations in relation to biological contexts, for example by competition for food or space. Based on the world's population deals with this, the debate about the ecological footprint - in other words to the carrying capacity of the earth with the relatively small consumption of renewable resources relative to the total consumption of resources. In this regard, neglects the exponential growth model also demographic trends such as the relationship between fertility and mortality rates, and the ratio between female and male population.

Growth models that take into account the saturation effect, are the limited growth and logistic growth, while the model of growth and growth-inhibiting factors poisoned in the process included in the calculation.

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