Growth With a Ceiling

You'll learnto read the carrying capacity off either standard form, to put the fastest growth and the inflection point together at half capacity, and to say what a population does in the long run without solving anything.

Exponential growth has no ceiling, and nothing real behaves that way for long. The logistic model keeps the exponential idea and multiplies it by a second factor that closes the growth down as the quantity fills the space available. Almost everything a question asks about it can then be read off without solving anything. Set the right side to zero and the two equilibria fall out, whichever of the two standard dresses the equation is wearing. The rate is a downward parabola in the population, so it peaks halfway between those equilibria — which is also where the solution curve stops bending upward and starts bending down. Growth is slowest near the carrying capacity, not fastest, and the population approaches it without ever arriving.

8 STEPS · 6 QUICK CHECKS · HALF CAPACITY IS WHERE EVERYTHING HAPPENS · v1

dP/dt = P(1 − P/500) = 0.002P(500 − P) ⇒ equilibria 0 and 500 ⇒ fastest at 250, rate 125

Before you start

What you're looking at

Two kinds of panel. One plots the population against time, so its horizontal axis is a clock. The other plots the rate against the population, so its horizontal axis is a headcount. Each says which underneath itself, and keeping the two apart is most of this topic.

The question

Where does the growth stop, where is it fastest, and what does the curve do near the ceiling? All three are answered by the two factors on the right side, with no integration anywhere.

Watch for

The teal dashed lines. They carry the two equilibria and the one landmark between them, half capacity — and that single height is the answer to three separate exam questions.

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