Population Growth
Project a population forward with the exponential or logistic growth model.
Exponential growth has no upper bound — it accelerates without limit.
Exponential growth follows N = N₀ · e^(r·t). With N₀ = 100, r = 0.5, and t = 3, you get N = 100 · e^1.5 ≈ 448. The logistic model adds a carrying capacity K, so growth slows as the population fills the environment and levels off near K instead of rising without limit.
Two ways a population grows
The exponential model assumes unlimited resources: each individual reproduces at the same per-capita rate, so the population keeps accelerating. The growth rate r is the intrinsic per-capita rate of increase per time unit, and t is elapsed time in the same units. The logistic model is more realistic — it adds a carrying capacity K, the largest population the environment can sustain. As N approaches K, growth slows toward zero and the curve flattens out.
Exponential growth. Logistic: N = K / (1 + ((K − N₀)/N₀) · e^(−r·t))
Worked example
A culture starts with N₀ = 100 cells and grows exponentially at r = 0.5 per hour. How many are there after t = 3 hours?
- 1 Write the exponential formula. N = N₀ · e^(r·t), where N₀ is the starting population, r the per-capita growth rate, and t the elapsed time.
- 2 Compute the exponent r · t. r · t = 0.5 · 3 = 1.5. This is the total growth experienced over the period.
- 3 Raise e to that power. e^1.5 ≈ 4.4817, so the population multiplies by about 4.48.
- 4 Multiply by the starting population. N = 100 · e^1.5 ≈ 100 · 4.4817 ≈ 448. So the culture grows from 100 to about 448 cells in 3 hours.
Exponential vs logistic growth
Both start the same; they diverge as the population grows.
| Feature | Exponential | Logistic |
|---|---|---|
| Equation | N = N₀ · e^(r·t) | N = K / (1 + ((K − N₀)/N₀) · e^(−r·t)) |
| Curve shape | J-curve | S-curve (sigmoid) |
| Upper limit | None — unbounded | Levels off at carrying capacity K |
| Resources assumed | Unlimited | Limited (competition increases with N) |
| Best for | Short bursts, early growth | Long-term, resource-limited populations |
How to read the result
Exponential growth is unrealistic over the long run. No real population can keep multiplying forever — food, space, and other resources eventually run short. The J-curve is a good approximation only early on, when the population is far below what its environment can support.
Logistic growth fixes this by adding the carrying capacity K. Near K, the per-capita growth slows to zero and the S-curve flattens, so the population settles rather than overshooting indefinitely. If N₀ starts above K, the logistic model declines toward K instead.
What r means. The growth rate r is the intrinsic per-capita rate of increase per time unit: r = 0.5 means the population would grow by roughly 50% per unit if unconstrained. A larger r makes both curves climb faster; in the logistic model it only changes how quickly the population approaches K, not the final level.