Population Growth Calculator
Model continuous population growth using P(t) = P₀·e^(rt). Enter starting population, growth rate and years to see final population, doubling time and a growth chart.
How to use this tool
- Enter initial population, annual growth rate and years in the fields above.
- Results update instantly as you type — or click Calculate.
- Read your final population and the full breakdown beneath it.
Continuous population growth is modelled by P(t) = P₀ · e^(rt), where P₀ is the starting population, r the continuous growth rate, and t time in years. The doubling time is ln(2)/r.
Formula
P(t) = P0 × ert
Doubling time = ln(2) / r
Where P0 = initial population, r = continuous annual growth rate (decimal), t = years, e = Euler's number (≈ 2.71828).
How it works
This calculator uses the continuous exponential growth model P(t) = P0ert, which assumes population grows proportionally to its current size at every instant. The doubling time is derived analytically as ln(2) / r. Continuous growth is a standard first approximation in demography; real populations are limited by carrying capacity, resource availability, and demographic variability, so long-range projections should be interpreted cautiously.
Worked example
- Initial population P₀ = 1,000,000, annual growth rate r = 2% = 0.02, t = 50 years.
- Apply the formula: P(50) = 1,000,000 × e^(0.02 × 50) = 1,000,000 × e^1.
- e^1 = 2.71828..., so P(50) = 1,000,000 × 2.71828 = 2,718,282 (rounded to integer).
- Doubling time = ln(2) / 0.02 = 0.6931 / 0.02 = 34.657 years.
Final population: 2,718,282; Doubling time: 34.657 years.
Common mistakes to avoid
- Entering the growth rate as a whole number percentage (e.g. 2.5 for 2.5%) instead of a decimal (0.025) — the continuous formula P0*e^(rt) uses r as a decimal, so 2.5 models 250% annual growth.
- Applying a constant rate over decades without accounting for demographic transition — real population growth rates change significantly as economies develop.
- Confusing this continuous model with discrete annual compounding (P0*(1+r)^t) — the two give slightly different results because e^r does not equal (1+r) except for very small r.
Key terms
- Continuous growth rate (r)
- The instantaneous per-capita growth rate, applied in the exponent. Differs slightly from the discrete percentage rate used in period-by-period models.
- e (Euler's number)
- The mathematical constant ≈ 2.71828, the base of the natural logarithm and the foundation of continuous exponential functions.
- Doubling time
- The number of years for the population to double, equal to ln(2) / r ≈ 0.6931 / r. Commonly approximated by the Rule of 70 (70 / r%).
- Growth factor
- The ratio of final population to initial population, e^(rt). A growth factor of 2 means the population doubled.
- Carrying capacity
- The maximum population an environment can sustain, which this simple exponential model does not account for. Logistic growth models incorporate this limit.
Frequently asked questions
- What is the difference between continuous and annual compounding?
- Continuous uses e^(rt); annual uses (1+r)^t. For small r they are close, but continuous growth is slightly faster and is the standard in population biology.
- Can growth rates be negative?
- Yes — a negative rate models population decline or any shrinking quantity.