# Math Tutorial: Logistic function

## Model

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.

$y′tW = kG24$

## 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.

$y′t= d y d t =kyG-y$

Differential equation of logistic growth

$kdt=1yG-ydy$

Separation of variables

$kGt+C=lnyG-y$

Integration gives

$y=G11+ekGtGy0-1$

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

## Calculator for the model of exponential growth

#### Zoom

$y=G\frac{1}{1+{e}^{kGt}\left(\frac{G}{{y}_{0}}-1\right)}$

step=
t-min= t-max=
y-min= y-max=
k-min= k-max=
G= k=
y0=