Biological Sciences

Ecology

Intermediate

Logistic Growth Equation

\frac{dN}{dt} = rN\left(1 - \frac{N}{K}\right)

Population grows exponentially at first, then slows as it approaches carrying capacity K.

By Pierre-François Verhulst

Biological Sciences

1838 · Pierre-François Verhulst

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Why it matters: Standard model for limited resources—from bacteria to markets.

Discoverers: Pierre-François Verhulst (1838)

What does it mean?

Population grows exponentially at first, then slows as it approaches carrying capacity K.

Why should I care?

Standard model for limited resources—from bacteria to markets.

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Pierre-François Verhulst

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Derivation

Variables & Units

Symbol	Name	Unit	Meaning
$N$	Population	—	Population size
$r$	Growth rate	—	Intrinsic growth rate
$K$	Carrying capacity	—	Maximum sustainable population
$t$	Time	—	Time

Worked Example

N=K/2 is maximum growth rate point (inflection of S-curve).

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Sources & further reading

Search Wikipedia Search Wolfram MathWorld Open in Wolfram|Alpha Watch explainers on YouTube

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