icon-komplexe-Zahl Math Tutorial: Logistic function


The model of exponential growth extends the logistic growth of a limited resource. The solution of the differential equation describing an S-shaped curve, a sigmoid. In the center of the development, the population is growing the fastest, until it is slowed by the limited resources.

Turning point

At the turning point of the logistic growth function value equal to half the saturation limit.

tW = lnGy0-1 k G

ytW = G2

Maximum growth rate

The maximum growth rate is achieved in the turning point.

ytW = kG24

Application Examples

  • Growth of populations with limited resources

Differential equation of logistic growth

The logistic growth is described by a differential equation with constant factors k and G.

yt= d y d t =kyG-y

Differential equation of logistic growth


Separation of variables


Integration gives


Dissolving and replacing the initial condition y 0 yields the solution of the logistic differential equation

Calculator for the model of exponential growth



Increment direction field
Range of values ​​of the axes
t-min= t-max=
y-min= y-max=
Range of values ​​of the parameters
k-min= k-max=
Current value of the parameter
G= k=
Current value of the initial values