Population Growth Formula
Calculate exponential population growth using the continuous growth model.
Used in ecology, demographics, and microbiology.
Need to calculate, not just reference? Use the interactive version.
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The Formula
N(t) = N₀ × e^(rt)
The exponential growth model predicts population size over time when resources are unlimited. It assumes a constant growth rate with no environmental constraints.
Variables
| Symbol | Meaning |
|---|---|
| N(t) | Population at time t |
| N₀ | Initial population size |
| r | Growth rate (per unit time) |
| t | Time elapsed |
| e | Euler's number (approximately 2.71828) |
Example 1
A population of 1,000 rabbits grows at 5% per year. Find the population after 10 years.
N₀ = 1,000, r = 0.05, t = 10
N(10) = 1,000 × e^(0.05 × 10) = 1,000 × e^0.5
N(10) = 1,000 × 1.6487 ≈ 1,649 rabbits
Example 2
A bacterial culture of 500 cells grows at a rate of 0.3 per hour. Find the count after 8 hours.
N₀ = 500, r = 0.3, t = 8
N(8) = 500 × e^(0.3 × 8) = 500 × e^2.4
N(8) = 500 × 11.023 ≈ 5,512 cells
When to Use It
Use the population growth formula when:
- Modeling population growth in ecology or demographics
- Predicting bacterial or viral spread in early stages
- Estimating future population size with a known growth rate
- Resources are abundant and environmental limits have not been reached
Key Notes
- Exponential growth only applies when resources are unlimited — in reality, every real population eventually follows logistic growth and levels off at the carrying capacity K; the exponential formula is most useful for early-stage or short-term predictions
- Doubling time: t_d = ln(2) / r ≈ 0.693 / r — a population growing at r = 0.1/year doubles in ≈ 6.9 years; this is the biological equivalent of the financial "Rule of 72" and is a quick sanity check on growth rate estimates
- Negative r models population decline: when mortality exceeds birth rate, N(t) = N₀ × e^(rt) with r < 0 gives exponential decay; the same formula applies to endangered species, radioactive decay, and cooling rates
- Continuous (e^rt) vs discrete ((1+r)ᵗ) growth: for large r or short intervals they diverge noticeably; the continuous model is used in ecology and microbiology, while the discrete form is common in finance (annual compounding)