Exponential and logistic population models, doubling time and carrying capacity, checked against OpenStax Biology 2e and Calculus Volume 2
Updated October 2026
Exponential growth assumes unlimited resources: the population grows in proportion to its size, dN/dt = rN, which gives N = N₀e^(rt). The growth rate r is births minus deaths per individual per time unit. Doubling time is ln 2 ÷ r, about 0.693 ÷ r.
Logistic growth adds a carrying capacity K, the largest population the environment can support: dN/dt = rN(K − N)/K. When N is small the factor (K − N)/K is near 1 and growth looks exponential; as N approaches K it falls toward 0. The solution is N = K N₀e^(rt) ÷ (K − N₀ + N₀e^(rt)). Growth is fastest at N = K/2, where dN/dt = rK/4.
| Hours | Bacteria (from 1,000) |
|---|---|
| 0 | 1,000 |
| 1 | 2,000 |
| 2 | 4,000 |
| 3 | 8,000 |
| 6 | 64,000 |
| 12 | 4,096,000 |
| 24 | 16,777,216,000 |
Biology 2e notes that after a day of hourly doublings a flask of 1,000 bacteria would hold more than 16 billion, which is why unlimited exponential growth never lasts in nature.
OpenStax Calculus Volume 2 models a deer population starting at 900,000 with r = 0.2311 per year and a carrying capacity of 1,072,764 (39,732 square miles × 27 deer per square mile). Unchecked, the herd would double about every three years; the logistic model keeps it below K:
| Year | Exponential (doubling every 3 years) | Logistic (K = 1,072,764) |
|---|---|---|
| 0 | 900,000 | 900,000 |
| 1 | 1,133,929 | 930,935 |
| 2 | 1,428,661 | 957,043 |
| 3 | 1,800,000 | 978,830 |
| 5 | 2,857,322 | 1,011,614 |
| 10 | 9,071,432 | 1,052,725 |
| 20 | 91,434,301 | 1,070,743 |
After 3 years the logistic model gives about 978,830 deer, as in the textbook, against 1.8 million for pure doubling.
For a growth rate of p% per time unit, the exact doubling time is ln 2 ÷ ln(1 + p/100); dividing 70 by p is a quick approximation that works well for small rates.
| Growth per time unit | Exact doubling time | Rule of 70 |
|---|---|---|
| 0.5% | 139.0 | 140.0 |
| 1% | 69.7 | 70.0 |
| 2% | 35.0 | 35.0 |
| 3% | 23.4 | 23.3 |
| 5% | 14.2 | 14.0 |
| 7% | 10.2 | 10.0 |
It uses a constant r and K. Age structure, predator-prey cycles, harvesting and random variation need other models; for money growing at compound interest, use the compound interest calculator.