Logistic growth: how certain processes proceed

Processes such as the adoption of a fashion trend or a technology in society, the growth of sales of a specific product or service or the sales of a new music album often follow a so-called logistic growth pattern:

The S-shaped growth curve indicates distinct growth phases: At first, slow, creeping growth which gradually changes into fast, exponential growth which then gradually slows down until it reaches a plateau beyond which it grows no longer.

This pattern can be captured by the so-called logistic function:

\[ f(x) = \frac{L}{1 + e^{-k(x-x_0)}} \]

where:
• L: the value at which the process 'tops of.'
• k: the logistic growth rate.
• x₀: value of the midpoint in time.
• e: the base of the natural logarithm.

Note that if L or k is chosen negative, the function displays negative growth (decline).

If both L and k are chosen negative, growth is positive.

The derivative of f(x) is:

f'(x) = f(x) × (1 - f(x))

It indicates the speed at which the process grows (declines).

Below (left) we have plotted the function for L = 1, k = 1 and x₀ = 0.

On the right we have plotted the derivative showing the speed of growth at all values of x. You can play with this. Fill in your own values for L, k and x₀, hit Plot! and the browser will plot the associated graphs^*.

L:      k:      x0: minimum x:      maximum x:      Plot!

^*Depending on your choices of L, k and x₀ you may have to adjust the values for minimum x and/or maximum x in order to the plot or to see the entire growth process.