Exponential growth is unbounded, which no real population manages. Food, space and disease eventually intervene.
Step 1: Let's Learn
Read it, or press Listen and follow the words.
The equation
dP/dt = kP(1 − P/M). The extra factor shrinks toward zero as P approaches the carrying capacity M.
Early on
When P is small the bracket is close to 1, so the growth is nearly exponential. The limit is not yet felt.
Later
As P approaches M the bracket approaches zero and growth stalls. The curve flattens against the capacity.
The fastest moment
Growth is fastest at exactly half the carrying capacity. That is the inflection point of the S-shaped curve.
Two equilibria
P = 0 is unstable and P = M is stable. Any positive starting population ends at M.
Growth that runs into a limit
dP/dt = kP(1 − P/M) grows nearly exponentially when P is small and slows as P approaches the carrying capacity M. It is the standard correction to unbounded growth.
Two equilibria
P = 0 and P = M are both constant solutions. M is stable and attracts every positive starting population; 0 is unstable. That is readable from the equation without solving it.
Fastest growth at half capacity
The solution curve has its inflection point at P = M/2, where growth is quickest. Before that growth accelerates and after it decelerates, producing the characteristic S shape.
It separates, with partial fractions
The equation is separable, and the integral requires splitting the rational expression into partial fractions. It is a good demonstration that integration technique is a prerequisite here.
Step 2: Try It Yourself
Tap and try it out.
- Point(2, 1.39)
- Slope of the tangent1
Step 3: Watch an Example
One step at a time.
Watch Leo Find the Fastest Growth
Leo has dP/dt = 0.4P(1 − P/800) and wants the population growing fastest.
- Step 1
He reads the carrying capacity from the equation as M = 800.
Step 4: Your Turn
Practice makes it stick.
The Capacity
Problem 1 of 2
dP/dt = 0.2P(1 − P/1500). What is the carrying capacity?
The Peak
Problem 2 of 2
A carrying capacity of 600. At what population is growth fastest?
Growth With a Ceiling
1 of 8
A carrying capacity of 1000. At what population is growth fastest?
2 of 8
dP/dt = 0.3P(1 − P/240). What is the carrying capacity?
3 of 8
A population exactly at the carrying capacity. What is dP/dt?
4 of 8
A population of 0. What is dP/dt?
5 of 8
A logistic model with capacity 900. What is the long-run population from any positive start?
6 of 8
A population well above the carrying capacity. Does it rise or fall? Enter 1 for rise, 2 for fall.
7 of 8
Order the stages of a logistic population starting small.
- 1Fastest growth at half the carrying capacity
- 2Growth slowing as the ceiling nears
- 3Levelling off at the carrying capacity
- 4Nearly exponential growth while the population is small
8 of 8
A carrying capacity of 50. At what population is growth fastest?
Step 5: Quick Check
Show what you know.
Question 1 of 2
A carrying capacity of 4000. At what population is growth fastest?
Question 2 of 2
Why does logistic growth level off?
What You Learned
- 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.