# Logistic Growth 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.

Figure: The figure shows a logistic growth curve and its derivative as dotted curve. The maximal growth is indivated by the red dot. The vectors show the direction field of the growth model.

## Logistic Growth Formula

Differential equation of logistic growth:

$y′t=kyG-y$

$G:Growth maximum value$ $k:Logistic growth rate$

With the growth function for the inital values t0 = 0 and y0 = y(0)

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

With the growth function for the general inital values t0 and y0 = y(t0)

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

Turning point of the logistic growth function:

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

$tW =t0+ lnGy0-1 k G$

$ytW = G2$

Maximum growth rate:

The maximum growth rate is achieved at the turning point.

$y′tW = kG24$

Application Examples

• Growth of populations with limited resources

• Logistic regression

• Neural networks

• Modeling of a pandemic

## 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

$y=G1+e-kGt-t0Gy0-1$

Dissolving and replacing the initial condition t0, y0 yields the solution of the logistic differential equation

## Calculator for the logistc growth function

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### Screenshot of the logistc growth graph

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## Releated sites

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