Logistic curve

Parametric equations

The logistic curve, in continuous time, is growth under a limited resource. Early on it is close to exponential growth. It bends in the middle. It levels off at the carrying capacity $L$. This is the smooth S-shaped solution, not the bifurcation diagram of the discrete map $x_{n+1}=r{x}_n(1-x_n)$.

The differential equation is

The inflection is at $y=L/2$, when $t=\frac{\ln a}{k}$. The bands on the figure read Slow, Rapid, and Saturated.

Interaction

• Carrying capacity L, Growth rate k, and Initial offset a: set the saturation level, the growth rate, and the initial shift
• Show dy/dt: draw $\frac{dy}{dt}$ under the curve. The three bands stay either way
• Compare Ce^kt: overlay the early piece $C{e}^{kt}$. If it leaves the range the figure can compare, the caption says so
• Reset: return the sliders to the initial parameters

What to notice

• The marked inflection is half the carrying capacity, and it is where the drawn $\frac{dy}{dt}$ is largest.
• The dashed $C{e}^{kt}$ stays with the curve only while the height is still small. After that the curve bends toward $L$.
• Slow, Rapid, and Saturated are three parts of one smooth solution. They are not the period-doubling of the discrete map.

Related

• Exponential growth and decay
• Logistic map bifurcation diagram

Further reading

• Logistic function (Wikipedia)
• Sigmoid function (Wikipedia)