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-curve

Logistic Growth Formula

Differential equation of logistic growth:

yt=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=G1+e-kGtGy0-1

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

y=G1+e-kGt-t0Gy0-1

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.

ytW = kG24

Application Examples

Differential equation of logistic growth

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

yt= 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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Initial values

t0=
y0=

Parameter value

G=
k=

Axes ranges

t-min=
t-max=
y-min=
y-max=

Parameter ranges

k-min=
k-max=
G-min=
G-max=

Screenshot of the logistc growth graph

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