Cogito
Differential Equations · Chapter 2 · Lesson 3
The Logistic Equation
Growth that runs into a limit.
12 problems · about 24 minutes · F-LE.A.1, F-IF.B.4
What this lesson teaches
The student analyses logistic growth and identifies carrying capacity and inflection.
- Logistic growth is exponential growth with a brake that engages near the carrying capacity.
- Growth is fastest at exactly half the capacity, which is the inflection point.
- P = 0 is unstable and P = M is stable, so any positive start ends at the capacity.
Warm Up
Straightforward practice. Get the method working first.
5 problemsA carrying capacity of 4000. At what population is growth fastest?
Answer 2000
Why 2000.
Why does logistic growth level off?
Answer The bracket shrinks toward zero as the population nears the capacity.
Why The growth factor fades to nothing.
A carrying capacity of 1000. At what population is growth fastest?
Answer 500
Why Half.
dP/dt = 0.3P(1 − P/240). What is the carrying capacity?
Answer 240
Why The denominator.
A population exactly at the carrying capacity. What is dP/dt?
Answer 0
Why The bracket is zero.
Build It Up
The same ideas with more to keep track of.
3 problemsA population of 0. What is dP/dt?
Answer 0
Why Nothing to grow.
A logistic model with capacity 900. What is the long-run population from any positive start?
Answer 900
Why The stable equilibrium.
A population well above the carrying capacity. Does it rise or fall? Enter 1 for rise, 2 for fall.
Answer 2
Why The bracket turns negative.
Stretch Yourself
Mixed problems. Work out what kind of question it is before you start.
4 problemsOrder the stages of a logistic population starting small.
Answer 1. Nearly exponential growth while the population is small 2. Fastest growth at half the carrying capacity 3. Growth slowing as the ceiling nears 4. Levelling off at the carrying capacity
Why The peak growth is in the middle, not at the start.
A carrying capacity of 50. At what population is growth fastest?
Answer 25
Why Half of 50.
The Capacity: dP/dt = 0.2P(1 − P/1500). What is the carrying capacity?
Answer 1500
Why 1500.
The Peak: A carrying capacity of 600. At what population is growth fastest?
Answer 300
Why 300.