Math Shorts - Introduction to Differential Equations by Metin Bektas

Author:Metin Bektas [Bektas, Metin]
Language: eng
Format: epub
Published: 2014-07-28T00:00:00+00:00

Now let's modify the fraction:

And use the reciprocals:

The next step is multiplying both sides by K/N0 - 1 and adding one:

Multiply both sides by N0 and make use of the reciprocals:

This is the solution, solved for N, in all its glory. It allows us to calculate the number of animals N(t) after time t, given the constant r and the capacity K. Note that the initial and limiting conditions check out. For t = 0 we get:

And if we let t go to infinity we get:

Verhulst's equation is not just useful for population growth. One interesting application was discovered by the Russian linguist Piotrowski, who noticed that quantifiable changes in language generally follow logistic growth. For example, the frequency of usage of new words initially grows in an exponential fashion, before flattening out and approaching a limiting value, all along following the logistic growth curve with remarkable accuracy.

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