The Logistic Equation

Growth that runs into a limit.

12 problems · about 24 minutes · F-LE.A.1, F-IF.B.4

NameDate

Figure — use these to answer the problems

A logistic curve rises steeply then flattens. This curve shows the same flattening: fast at first, then levelling against a ceiling.

y = 2 ln x + 0

the steepness2

the time shown2

Figure 1

Straightforward practice. Get the method working first.

5 problems

A carrying capacity of 4000. At what population is growth fastest?
Answer
2
Why does logistic growth level off?
- The bracket shrinks toward zero as the population nears the capacity.
- The population suddenly stops reproducing.
A carrying capacity of 1000. At what population is growth fastest?
Answer
dP/dt = 0.3P(1 − P/240). What is the carrying capacity?
Answer
A population exactly at the carrying capacity. What is dP/dt?
Answer

The same ideas with more to keep track of.

3 problems

A population of 0. What is dP/dt?
Answer
A logistic model with capacity 900. What is the long-run population from any positive start?
Answer
A population well above the carrying capacity. Does it rise or fall? Enter 1 for rise, 2 for fall.
Answer

Mixed problems. Work out what kind of question it is before you start.

4 problems

9
Order the stages of a logistic population starting small.
Write 1 to 4 in the boxes to put these in order.
- Nearly exponential growth while the population is small
- Fastest growth at half the carrying capacity
- Growth slowing as the ceiling nears
- Levelling off at the carrying capacity
A carrying capacity of 50. At what population is growth fastest?
Answer
The Capacity
dP/dt = 0.2P(1 − P/1500). What is the carrying capacity?
Answer
The Peak
A carrying capacity of 600. At what population is growth fastest?
Answer