Logistic Growth

Logistic growth is always explained with the help of the logistic curve. Logistic curve can be shown in form of sigmoid curve. Logistic growth models the S-shaped curve of growth of  set P, where P might be referred as  population. At the initial stage of the growth the graph is in exponential then saturation begins and the growth slows and finally at maturity the growth stops.

I like to share this Logistic Regression Model with you all through my article.

Learn the definition of Logistic growth:

learn the Logistic Growth formula:

P (t) = 1/ (1+e –t)

where P is the population and t is considered as a time.

The S-curve is obtained if the range of the time over the real numbers from −∞ to +∞. In practice, due to the nature of the exponential function e−t, it is then enough to calculate time t over a small range of real numbers like [−7, +7].

Learn the derivative which is most commonly used:

d/dt P(t) = P(t).(1-P(t))

The computation of function P is

1-P(t) = P(-t)


Learn the logistic differential equation:

learn the logistic growth function, it can be used to calculate the first order nonlinear differential equation.

d/dt P(t) =P(t)(1- P(t))

Here P is a variable with respect to time t and by applying the condition P(0) we can get ½ .

One may readily find the solution to be

P(t) = et / (e t + e c )

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Decide the constant of integration e c = 1 gives the other well-known form of the definition of the logistic curve

P (t) = e t / (e t + 1) = 1 / (1 + e –t)

The logistic curve demonstrates the exponential increase for negative t, which can slow down the curve to linear growth. It is in the slope of 1/4 at t = 0, then it approach exponentially decaying gap at y=1

The relationship among the logistic sigmoid function and the hyperbolic tangent is given by,

2P(t) = 1 + tanh (t/2)