The term logistic growth goes back to the Belgian mathematician Pierre François Verhulst (1804-1849), who in 1837 formulated the so-called logistic equation to describe demographic developments.
The purpose of the program is to perform a curve fit to the logistic function ƒ(t) for a series of measured values.
In addition to the measured values, an estimated upper limit of the function values (saturation limit) is entered. It is also possible to enter a factor (dark figure) by which the measured values are multiplied, assuming that the actual values are higher than the measured values.
The value pairs can either be entered in the table or imported or exported from a file in CSV format (comma separated values).
To import or export data from a CSV file, open the local menu by double-clicking with the right mouse button on the table.
The menu items Cut Data , Copy Data and Paste Data correspond to the Cut , Copy and Paste functions of word processing and use the clipboard as well.
Example 1: Hop growth
Hops, which are required for the production of beer, are a fast growing creeper. In order to make statements about hop growth, the following series of measurements was taken during an investigation.
|Time t (in weeks)||0||2||4||6||8||10||12||16|
|Height (in m)||0.6||1.2||2.0||3.3||4.1||5.0||5.5||5.8|
(Example from Lambacher Schweizer: Leistungskurs Analysis, Edition Baden-Wuerttemberg, 1st edition 2000)
Data from: "hopfenwachstum.csv" Saturation limit: 6 Dark figure: 1 4,0189 ƒ(x) = ———————————————— 0,66981 + 5,3302 · e^(-0,35622·t) Inflection point W(5,8226/3) Maximum growth rate ƒ'(xw) = 0,53433 8 Values Coeff.of determin. = 0,99383916 Correlation coeff. = 0,99691482 Standard deviation = 0,16172584
As in the book, the value 6 was assumed as the saturation limit. Nevertheless, the results differ from those of the template. However, no curve fitting is carried out there, but the proportionality factor k is calculated using an arbitrary data pair, for example with ( 12 | 5.5 ). The data for this example are stored in the file hopfenwachstum.csv .