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.

Differential equation of logistic growth:

$$y\prime \left(t\right)=k\phantom{\rule{0.3em}{0ex}}y\phantom{\rule{0.3em}{0ex}}\left(G-y\right)$$

$$G\phantom{\rule{0.3em}{0ex}}:\phantom{\rule{0.3em}{0ex}}\text{Growth maximum value}$$ $$k\phantom{\rule{0.3em}{0ex}}:\phantom{\rule{0.3em}{0ex}}\mathrm{Logistic\; growth\; rate}$$

With the growth function for the inital values t_{0} = 0 and y_{0} = y(0)

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

With the growth function for the general inital values t_{0} and y_{0} = y(t_{0})

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

Turning point of the logistic growth function:

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

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

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

Maximum growth rate:

The maximum growth rate is achieved at the turning point.

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

Application Examples

Growth of populations with limited resources

Logistic regression

Neural networks

Modeling of a pandemic

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

$$y\prime \left(t\right)=\frac{dy}{dt}=ky\left(G-y\right)$$

Differential equation of logistic growth

$$kdt=\frac{1}{y\left(G-y\right)}dy$$

Separation of variables

$$kGt+C=\mathrm{ln}\frac{y}{G-y}$$

Integration gives

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

Dissolving and replacing the initial condition t_{0}, y_{0} yields the solution of the logistic differential equation

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