How to use this calculator
- Enter or choose initial population N₀.
- Enter or choose growth rate per period.
- Enter or choose number of periods.
- Enter or choose model.
- The population after t periods updates as you type. Use Copy result or Share to keep it.
Method
Continuous: N = N₀ × e^(r t); per period: N = N₀ × (1 + r)^t; doubling time = ln 2 ÷ k, where k = r or ln(1 + r)
- N₀
- initial population
- r
- growth rate per period (as a decimal)
- t
- number of periods
- k
- equivalent continuous growth rate
Worked example
- r t = 0.02 × 10 = 0.2.
- N = 1,000 × e^0.2 = 1,000 × 1.22140 = 1,221.40.
- Doubling time = ln 2 ÷ 0.02 = 0.693147 ÷ 0.02 = 34.66 periods.
A population of 1,000 growing continuously at 2% per period reaches about 1,221 after 10 periods and doubles every 34.7 periods.
What the result means
Exponential growth describes the early phase of a population with plenty of resources (bacteria in fresh medium, an invasive species). Over longer spans resources run short and growth slows, which the logistic model captures.
Assumptions and limits
- A constant growth rate (births minus deaths, plus net migration) over the whole period.
- Real populations are limited by resources, so long projections overstate growth. Use the logistic model when a carrying capacity matters.
Questions
What is the rule of 70?
Doubling time ≈ 70 ÷ growth rate in %. It comes from ln 2 ≈ 0.693; at 2% it gives 35 periods, close to the exact 34.66.
Continuous or per period?
Use per period when growth happens in discrete steps (annual breeding seasons); continuous for overlapping generations such as bacteria or human populations.