 ODE Logistic function

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

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

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