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

## 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.

$${t}_{W}=\frac{ln\left(\frac{G}{{y}_{0}}-1\right)}{kG}$$

$$y\left({t}_{W}\right)=\frac{G}{2}$$

## Maximum growth rate

The maximum growth rate is achieved in the turning point.

$$y\prime \left({t}_{W}\right)=\frac{k{G}^{2}}{4}$$

## Application Examples

Growth of populations with limited resources