Logistic growth: carrying capacity and fastest growth

Warm-up

Three quick questions from earlier lessons.

Discover

A population starts at 10 and follows a logistic model with carrying capacity $k$ . Move $k$ .

Logistic curves starting at 10 with carrying capacity k: slow growth, then fast growth in the middle, then levelling off at k.

Where on the curve is the population growing fastest?

Halfway to $k$
Just before it reaches $k$

Key idea

Growth slows as resources run out.

Logistic model:

\dfrac{dP}{dt} = rP\left(1 - \dfrac{P}{K}\right)

. Equilibria

P = 0

and

P = K

(the carrying capacity). For

0 < P_0

,

P(t) \to K

. Growth is fastest at

P = \tfrac{K}{2}

.

Worked example

$\dfrac{dP}{dt} = 0.4P\left(1 - \dfrac{P}{500}\right)$ .

Carrying capacity

K = 500

Set the rate to 0.

Fastest growth at

P = 250

Half of K.

Maximum rate

50

per unit time

0.4 × 250 × ½.

Your turn, guided

Your turn: $\dfrac{dP}{dt} = 0.2P\left(1 - \dfrac{P}{800}\right)$ . Fastest growth at…

P = 400

K/2.

Practice ★

A population follows $\dfrac{dP}{dt} = \dfrac{ 7 }{10}P\left(1 - \dfrac{P}{ 558 }\right)$ . What is its carrying capacity?
A population follows $\dfrac{dP}{dt} = \dfrac{ 9 }{10}P\left(1 - \dfrac{P}{ 428 }\right)$ . What is its carrying capacity?

$\dfrac{dP}{dt} = \dfrac{ 3 }{10}P\left(1 - \dfrac{P}{ 280 }\right)$ . For which population $P$ is it growing fastest?
$\dfrac{dP}{dt} = \dfrac{ 9 }{10}P\left(1 - \dfrac{P}{ 590 }\right)$ . For which population $P$ is it growing fastest?

Watch out!

Practice ★★

$\dfrac{dP}{dt} = 0.2P\left(1 - \dfrac{P}{ 750 }\right)$ with $P(0) > 0$ . Tick all the true statements.
1. $P = 0$ and $P = 750$ are equilibrium solutions
2. If $P(0) > 750$ , then $P$ decreases towards 750
3. $\displaystyle\lim_{t \to \infty} P(t) = 750$
4. Starting below 750, $P$ eventually exceeds 750
5. $\displaystyle\lim_{t \to \infty} P(t) = 0.2$

Boss problem. $\dfrac{dP}{dt} = \dfrac{ 2 }{10}P\left(1 - \dfrac{P}{ 388 }\right)$ .

Exit ticket

Three questions to check you have got it.

$\dfrac{dP}{dt} = \dfrac{ 3 }{10}P\left(1 - \dfrac{P}{ 280 }\right)$ . For which population $P$ is it growing fastest?
$\dfrac{dP}{dt} = \dfrac{ 9 }{10}P\left(1 - \dfrac{P}{ 590 }\right)$ . For which population $P$ is it growing fastest?
$\dfrac{dP}{dt} = \dfrac{ 1 }{10}P\left(1 - \dfrac{P}{ 342 }\right)$ . For which population $P$ is it growing fastest?

Summary card

Logistic growth

$P' = rP(1 - P/K)$ ; equilibria $0$ and $K$ .
$P \to K$ ; fastest growth at $K/2$ .

Watch out: $r$ is a rate, $K$ is a population.

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