Logistic function

The logistic distribution characterized a one-dimensional continuous probability distribution, and is a functional representation of saturation processes from the class of the so-called sigmoid indefinite temporal extent.

Even into the 20th century, the logarithm with the Italian name of the logistic curve ( curva Logistica ) was occupied occasionally. Today the name is unique to the S function.

Description

The logistic function, as it results from the discrete logistic equation describes the relationship between the passage of time and growth, for example, an ideal bacterial population. For this purpose, the model of exponential growth is modified by a consumable resource with the growth - the idea is so about a bacteria breeding ground of limited size. In practice, the function does not start at 0, but at the initial time is already an initial value f (0) before.

Therefore applies to the bacteria example:

• The limited life space forms an upper bound for the number of bacteria G f (t).
• Bacterial growth f '(t ) is proportional to: the current component f (t)
• The remaining capacity G - f ( t)

This development therefore, by a differential equation of the form

Described by a proportionality constant. Solving this differential equation results in:

The graph of the function describes an S-shaped curve, a sigmoid. At the beginning of the growth is small because the population and, hence, the number of proliferating individuals is low. In the middle of the development (more precisely, at the turning point ), the population is growing most until it is slowed down by the exhaustible resources.

Other applications

The logistic equation describes a relationship very frequent and is widespread over the idea of describing a population of organisms also apply. Also, the life cycle of a product in the market can be simulated with the logistic function. Other applications include growth and decay processes in the language ( language change law, Piotrowski Act ) and the development in the acquisition of the native language ( language acquisition Act). One application of the logistic function in the SI model of mathematical epidemiology.

Solution of the differential equation

Denoting the values ​​of the sought solution, we obtain

The differential equation can be solved " separation of variables " with the procedure. We bring the variable to the left and the variable to the right.

To give the final equation for by a partial fraction decomposition or by a simple calculation. We bring the to the left side and by integration with a yet to be determined constants of integration:

As long as the values ​​are between 0 and what can be assumed because of the condition. It is the natural logarithm. The application of the exponential function on both sides results in

And subsequent reciprocal of education

We now take the one on the left side, then re-form the inverse, and finally obtain

And from this

To determine the constants of integration we put into the equation denoted by (*). The corresponding function value, and we find

Let's put this into the solution found (** ) and note so we come to the above alleged solution of the logistic differential equation:

At this functional equation we read off easily, that the values ​​are always between 0 and are, therefore, the solution to all of these. This can be confirmed in retrospect, of course, by substituting into the differential equation.

Calculation of the inflection point

To determine the inflection point of the solution function, we first determine by means of the product rule leads

And determining the zero of the second derivative:

So we know the function value at the turning point and find that the population in the turning point just exceeds half the saturation limit. To determine we use the solution formula and calculate as follows:

Thus, the turning point is completely determined and there is only this one. Substituting into the first derivative the maximum growth rate:

Further illustrations

Or also: