ODE-03

Logistic

Saturating growth.

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ODElogistic

Governing equation

y=K/(1+(K/y_0-1)e^{-rt})

$t \in \left[0, 30\right]$

where

K: K (—)
y_0: y0 (—)
r: r (1/s)
t: t (s)
y: y (—)

The figure, part by part

Each numbered marker wears the color of its card. Hover a card to light the marker. Tap a marker to jump to that card. The slider walks a red point along the curve; Animate drawing traces it like a pen.

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①

Governing law — red

y=K/(1+(K/y_0-1)e^{-rt})

$t \in \left[0, 30\right]$

$t = 0$ $y = 1$

Along the curve

②

Domain of the variable — blue

t_{\min} = 0,\quad t_{\max} = 30

$t \in \left[0, 30\right]$

$t = 0$ $t = 0$

Along the curve

③

Response curve — green

y=K/(1+(K/y_0-1)e^{-rt})

$t \in \left[0, 30\right]$

$t = 0$ $y = 1$

Along the curve

④

Rate of change — amber

\frac{\mathrm{d}y}{\mathrm{d}t}

$t \in \left[0, 30\right]$

$t = 0.189$ $y^{\prime} = 0.382$

Along the curve

Lecture brief

Historical brief

Exponential decay, the harmonic oscillator, SIR epidemics, Lotka–Volterra and Bessel are the first solvable ODEs of nature and populations. The lab is rate, period and phase portrait in closed form. This sheet (ODE-03 — Logistic) is the form associated with logistic. Working symbols: $K$ , $y_0$ , $r$ , $t$ $\rightarrow$ $y$ . Saturating growth.

Purpose

Purpose: compute

y

from

K

y_0

r

t

in Differential equations via

y=K/(1+(K/y_0-1)e^{-rt})

Saturating growth. Use it when a real differential equations question must be answered in SI before a code check.

Live realistic example

In symbols

Live case. Given

K = 10.000\,\mathrm{—}

y_0 = 1.000\,\mathrm{—}

r = 0.400\,\mathrm{1/s}

t = 5.000\,\mathrm{s}

, the governing relation

y=K/(1+(K/y_0-1)e^{-rt})

yields

y = 4.509\,\mathrm{—}

. Saturating growth. Move a slider: the numbers are this situation, not a canned story.

Calculator

Inputs

K K10 —y0 y_01 —r r0.4 1/st t5 s

Outputs

y y4.509 —

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Narration of this film

Saturating growth.

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