Growth slows as the population fills the space available
The bracket is the brake. When the population N is far below the carrying capacity K it is close to 1, so growth is nearly exponential at rate r. As N approaches K the bracket shrinks toward zero and growth stalls. Plotted over time this traces an S-shaped curve that flattens at K. Solved out, N(t) = K / (1 + A e−rt), where A = (K − N₀) / N₀.
Growth rate against how full the habitat is
Population growth is fastest not when N is smallest but at the halfway point, N = K/2. This is why maximum sustainable yield in fisheries and forestry is taken there.
| Population (N / K) | Brake (1 − N/K) | Growth rate |
|---|---|---|
| 10% | 0.90 | Low — few breeders |
| 25% | 0.75 | Rising |
| 50% | 0.50 | Maximum (rK/4) |
| 75% | 0.25 | Falling |
| 100% | 0.00 | Zero — at K |
The peak growth rate is r K/4, reached at N = K/2. A pond with K = 10,000 fish and r = 0.15 grows fastest at 5,000 fish, adding about 375 fish a year.
Reading the model
- r is the intrinsic growth rate — the per-capita rate when resources are unlimited. Estimate it from two counts: r = (ln N₂ − ln N₁) / (t₂ − t₁).
- K is not fixed. Drought, seasons and habitat change all shift it, so real populations oscillate around K rather than sitting exactly on it.
- Exponential vs logistic. Drop the bracket and you get dN/dt = rN, unbounded exponential growth. The (1 − N/K) term is what makes the logistic model realistic.
Common questions
What is carrying capacity (K)?
K is the largest population an environment can support indefinitely, set by food, water, space, predators and disease. In the logistic model it is the ceiling the population levels off at. The symbol comes from the German Kapazitaetsgrenze, meaning capacity limit.
What is the logistic growth equation?
dN/dt = r times N times (1 minus N over K). Growth speeds up while the population is small and slows as it nears K, tracing an S-shaped curve. When N equals K the growth rate is zero.
Why does a population grow fastest at half the carrying capacity?
Growth is the product of two things: how many individuals can breed (N) and how much room is left (1 minus N over K). One rises as the other falls, and their product peaks exactly at N equal to K over 2. That is why the fastest growth rate equals r times K divided by 4.
