Exponential growth continues forever, which no real population does. Resources impose a ceiling.
Step 1: Let's Learn
Read it, or press Listen and follow the words.
The logistic equation
dP/dt = kP(1 − P ÷ L), where L is the carrying capacity. The bracket is what slows growth near the ceiling.
Early on it looks exponential
When P is much smaller than L, the bracket is nearly 1 and the equation behaves like plain exponential growth.
Near the ceiling
As P approaches L the bracket approaches zero, so growth slows to nothing. The solution levels off at L.
Fastest growth is halfway
The growth rate peaks at P = L ÷ 2. That is the inflection point of the solution curve.
Two equilibria
P = 0 and P = L both make dP/dt zero. The carrying capacity is stable; zero is not.
Growth with a ceiling
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 unlimited exponential growth.
The S-shaped curve
The solution rises slowly, then steeply, then levels off at M. The inflection point — the moment of fastest growth — occurs at exactly half the carrying capacity, which is a memorable and examined fact.
Two equilibrium solutions
P = 0 and P = M make the derivative zero, so both are constant solutions. M is stable, attracting nearby solutions; 0 is unstable. Identifying equilibria from the equation alone is a standard task.
The long-run behaviour
For any positive starting population, P approaches M as t grows. Stating that limit without solving the differential equation is possible directly from the equation's structure.
Step 2: Try It Yourself
Tap and try it out.
- The equationdy/dx = a·y
- Through(0, 1)
Every short line shows the slope the equation demands at that point. The curve simply follows them, and the marked point is the initial condition that picks it out of the family.
Step 3: Watch an Example
One step at a time.
Watch Kofi Find the Fastest Growth
A population satisfies dP/dt = 0.1P(1 − P ÷ 1000).
- Step 1
The carrying capacity is read straight off: L = 1000.
Step 4: Your Turn
Practice makes it stick.
The Capacity
Problem 1 of 2
dP/dt = 0.1P(1 − P ÷ 1000). What is the carrying capacity?
The Fastest Point
Problem 2 of 2
Same equation. At what population is growth fastest?
Growth With a Ceiling
1 of 8
Carrying capacity 800. At what population is growth fastest?
2 of 8
Carrying capacity 2000. Fastest growth at what population?
3 of 8
dP/dt = 0.2P(1 − P ÷ 500). Carrying capacity?
4 of 8
As P approaches L, what does dP/dt approach?
5 of 8
How many equilibrium solutions does a logistic equation have?
6 of 8
When P is far below L, does it behave like exponential growth? 1 yes, 0 no.
7 of 8
Sort each statement by which model it describes.
Tap something to move it.
- Empty
- Empty
8 of 8
Carrying capacity 1200. Fastest growth at what population?
Step 5: Quick Check
Show what you know.
Question 1 of 2
Carrying capacity 3000. At what population is growth fastest?
Question 2 of 2
What does the bracket in the logistic equation do?
What You Learned
- The logistic equation adds a carrying capacity to exponential growth.
- Growth is fastest at half the capacity, which is the inflection point.
- The solution levels off at the capacity.