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Formula
Exponential: N = N₀ × e^(rt). Discrete: N = N₀ × (1 + r)^t. Logistic: N = K ÷ (1 + ((K − N₀) ÷ N₀) × e^(−rt))
- N
- population after time t
- N₀
- initial population
- r
- growth rate per time period as a decimal (birth rate − death rate); negative for a decline
- t
- number of time periods (years, hours, generations), in the same unit as r
- K
- carrying capacity, the largest population the environment can sustain
- e
- Euler's number, about 2.71828
What population growth measures
Population growth describes how the number of individuals in a group changes over time, whether that group is bacteria in a flask, deer in a forest or people in a country. Ecologists model it with a per-capita growth rate, r, which is the birth rate minus the death rate. When r is positive the population grows, and when it’s negative the population shrinks.
Because each new individual can also reproduce, populations grow by a percentage of their current size rather than by a fixed amount. That makes the growth exponential, at least until food, space or disease start to hold it back.
How to use the population growth calculator
Pick what you want to find:
- Final population – enter the initial population N₀, the growth rate r as a percentage and the time t.
- Growth rate – enter N₀, the final population N and the time.
- Time – enter N₀, N and the rate. For a decline, use a smaller N and a negative rate.
- Logistic – enter N₀, the carrying capacity K, the intrinsic rate r and the time.
In the first three modes, choose the exponential model for continuous growth or the discrete model for growth that compounds once per period. Keep the units consistent: if r is per year, t is in years. The results include the doubling or halving time, the total change, the equivalent rate in the other model and a table of the population over time.
Population growth formulas
Exponential growth
N = N₀ × e^(rt)
Say a town of 2,000 grows continuously at 6% a year for 5 years. N = 2,000 × e^(0.06 × 5) = 2,000 × e^0.3 = 2,000 × 1.34986 = 2,699.72.
To find time instead, rearrange: t = ln(N ÷ N₀) ÷ r. The same town takes ln 3 ÷ 0.06 = 18.31 years to triple.
Discrete (geometric) growth
N = N₀ × (1 + r)^t
At 6% a year compounded once a year, the town reaches 2,000 × 1.06^5 = 2,676.45, slightly less than with continuous growth. This model suits animals that breed once a season and figures quoted as simple annual percentages.
Logistic growth with carrying capacity
Real environments can only support so many individuals. The logistic model adds a carrying capacity K, and the growth rate becomes dN/dt = rN(1 − N/K). When N is small, growth is nearly exponential. As N approaches K, the term (1 − N/K) shrinks toward zero and growth levels off. The solution is:
N = K ÷ (1 + ((K − N₀) ÷ N₀) × e^(−rt))
For 50 rabbits introduced to an island that can support 800, growing at r = 0.5 per year: after 4 years, N = 800 ÷ (1 + 15 × e^−2) = 800 ÷ 3.03 = 264.02. Growth is fastest when the population reaches K ÷ 2 = 400, which happens at t = ln((K − N₀) ÷ N₀) ÷ r = ln 15 ÷ 0.5 = 5.42 years.
Doubling time by growth rate
Computed with ln 2 ÷ r (continuous) and ln 2 ÷ ln(1 + r) (discrete).
| Growth rate per period | Continuous | Discrete | Rule of 70 |
|---|---|---|---|
| 1% | 69.31 | 69.66 | 70 |
| 2% | 34.66 | 35 | 35 |
| 3% | 23.1 | 23.45 | 23.3 |
| 5% | 13.86 | 14.21 | 14 |
| 8% | 8.66 | 9.01 | 8.8 |
| 10% | 6.93 | 7.27 | 7 |
| 25% | 2.77 | 3.11 | 2.8 |
The rule of 70 is a mental-math shortcut. It’s close for small rates and drifts as rates grow.
How to calculate population growth rate in geography
Geography and APES courses often work from crude rates per 1,000 people. The natural increase rate is (birth rate − death rate) ÷ 10, in percent. A country with 22 births and 8 deaths per 1,000 has a natural increase of 1.4% a year. Add net migration per 1,000 the same way to get the overall growth rate, then enter that rate in the calculator to project the population or its doubling time.
Real-life uses
- Microbiology – estimating how long a bacterial culture takes to reach a target density.
- Wildlife management – projecting a herd or fish stock toward its carrying capacity and setting harvest levels. Harvesting near K ÷ 2 takes advantage of the fastest growth.
- Demography and planning – estimating how many people a town will have in 10 or 20 years.
- Conservation – working out how quickly a declining species will halve.
The same exponential maths describes radioactive decay, covered by the half-life calculator, and investment growth, covered by the CAGR calculator.
Common mistakes
- Mixing units. A rate per year with a time in months gives a result twelve times too large or small.
- Using a percentage as a decimal. Enter 3 for 3% in the calculator. In the formula itself, r is 0.03.
- Confusing total growth with the growth rate. A 25% rise over three years isn’t 25% a year. The percentage change calculator gives total change, while this tool gives the per-period rate.
- Applying exponential growth for too long. No population grows exponentially forever. Over long periods, use the logistic mode with a realistic K.
Glossary
- Intrinsic growth rate (r) – the per-capita growth rate under ideal conditions, birth rate minus death rate.
- Carrying capacity (K) – the largest population an environment can support over time.
- Doubling time – the time a growing population takes to double at a constant rate.
- J-shaped and S-shaped curves – the typical shapes of exponential and logistic growth.
Worked examples
A bacterial culture (exponential)
500 cells growing continuously at 4% per hour for 10 hours: N = 500 × e^0.4.
Inputs (Final population): Growth model = Exponential: N = N₀ · e^(rt); Initial population (N₀) = 500; Growth rate (r) per time period = 4 %; Time (t) = 10 periods
Model: exponential (continuous), r = 4% = 0.04 per period N = 500 × e^(0.04 × 10) = 500 × 1.491825 = 745.91 Doubling time = ln 2 ÷ 0.04 = 17.3287
Answer: Exponential (continuous) growth from 500 at 4% per period for 10 periods gives 745.91.
A deer herd that breeds once a year (discrete)
1,200 deer growing 3.5% per breeding season for 8 seasons: N = 1,200 × 1.035^8.
Inputs (Final population): Growth model = Discrete: N = N₀ · (1 + r)^t; Initial population (N₀) = 1200; Growth rate (r) per time period = 3.5 %; Time (t) = 8 periods
Model: discrete (per period), r = 3.5% = 0.035 per period N = 1,200 × (1 + 0.035)^8 = 1,200 × 1.316809 = 1,580.17 Doubling time = ln 2 ÷ |ln(1 + 0.035)| = 20.1488
Answer: Discrete (per period) growth from 1,200 at 3.5% per period for 8 periods gives 1,580.17.
Growth rate of a town
A population that quadrupled from 200 to 800 in 12 years: r = ln 4 ÷ 12.
Inputs (Growth rate): Growth model = Exponential: N = N₀ · e^(rt); Initial population (N₀) = 200; Final population (N) = 800; Time (t) = 12 periods
Model: exponential (continuous), r = 11.5525% = 0.115525 per period r = ln(800 ÷ 200) ÷ 12 = 1.386294 ÷ 12 = 0.115525 Doubling time = ln 2 ÷ 0.115525 = 6
Answer: Exponential (continuous) growth from 200 at 11.5525% per period for 12 periods gives 800.
Time to double at 7% per period
How many periods a population growing 7% per period needs to double: ln 2 ÷ ln 1.07.
Inputs (Time): Growth model = Discrete: N = N₀ · (1 + r)^t; Initial population (N₀) = 1000; Final population (N) = 2000; Growth rate (r) per time period = 7 %
Model: discrete (per period), r = 7% = 0.07 per period t = ln(2,000 ÷ 1,000) ÷ ln(1 + 0.07) = 0.693147 ÷ 0.067659 = 10.2448 Doubling time = ln 2 ÷ |ln(1 + 0.07)| = 10.2448
Answer: Discrete (per period) growth from 1,000 at 7% per period for 10.2448 periods gives 2,000.
Fish stocked in a pond (logistic)
200 fish in a pond that can support 5,000, growing at an intrinsic 40% per year: growth slows as the pond fills.
Inputs (Logistic (carrying capacity)): Initial population (N₀) = 200; Carrying capacity (K) = 5000; Intrinsic growth rate (r) per period = 40 %; Time (t) = 6 periods
Model: logistic, K = 5,000, r = 40% = 0.4 per period A = (K − N₀) ÷ N₀ = (5,000 − 200) ÷ 200 = 24 N = K ÷ (1 + A × e^(−rt)) = 5,000 ÷ (1 + 24 × e^(−2.4)) = 1,573.7 dN/dt = rN(1 − N ÷ K) = 0.4 × 1,573.7 × (1 − 0.314739) = 431.36
Answer: Logistic growth from 200 toward a carrying capacity of 5,000 at r = 40% gives 1,573.7 after 6 periods (31.47% of K).
A declining population (edge case)
A population shrinking continuously at 5% per year falls to a quarter (5,000 → 1,250) in ln 4 ÷ 0.05 years.
Inputs (Time): Growth model = Exponential: N = N₀ · e^(rt); Initial population (N₀) = 5000; Final population (N) = 1250; Growth rate (r) per time period = -5 %
Model: exponential (continuous), r = −5% = −0.05 per period t = ln(1,250 ÷ 5,000) ÷ −0.05 = −1.386294 ÷ −0.05 = 27.7259 Halving time = ln 2 ÷ 0.05 = 13.8629
Answer: Exponential (continuous) growth from 5,000 at −5% per period for 27.7259 periods gives 1,250.
Assumptions and limitations
- Exponential mode assumes continuous growth at a constant per-capita rate r with unlimited resources: N = N₀e^(rt). Discrete mode compounds once per period: N = N₀(1 + r)^t.
- Logistic mode uses the Verhulst solution N = K ÷ (1 + ((K − N₀) ÷ N₀)e^(−rt)), in which growth slows as N approaches the carrying capacity K. It solves for N only, with r greater than 0.
- Doubling time is ln 2 ÷ r for continuous growth and ln 2 ÷ ln(1 + r) for discrete growth. For a negative rate the same expressions give the halving time.
- Migration, age structure, random events and changes in r or K over time are not modeled. r is a net rate, so fold births, deaths and net migration into it before entering it.
- Populations are shown to 2 decimal places, rates to 4. A projected value below 1 means the population has effectively died out.
Frequently asked questions
How do you calculate population growth rate?
For continuous growth, r = ln(N ÷ N₀) ÷ t. A population that grows from 4,000 to 5,000 in 3 years has r = ln 1.25 ÷ 3 = 0.0744, or 7.44% a year. For growth that compounds once a year, use r = (N ÷ N₀)^(1/t) − 1, which gives 7.72%.
How do you calculate population growth percentage?
Divide the change by the starting population and multiply by 100. Going from 4,000 to 5,000 is a 1,000 increase, so 1,000 ÷ 4,000 × 100 = 25% total growth. That's different from the yearly growth rate, which spreads the change over time.
How do birth and death rates give the growth rate?
The per-capita growth rate r is the birth rate minus the death rate. With 18 births and 11 deaths per 1,000 people a year, r = 0.018 − 0.011 = 0.007, or 0.7% a year. Add the net migration rate as well if people move in or out.
What is the difference between exponential and logistic growth?
Exponential growth assumes unlimited resources, so the population keeps accelerating in a J-shaped curve. Logistic growth adds a carrying capacity K. Growth is fastest at K ÷ 2 and then slows, giving an S-shaped curve that levels off at K.
What is the population growth formula with carrying capacity?
It's the logistic equation dN/dt = rN(1 − N/K). Its solution is N = K ÷ (1 + ((K − N₀) ÷ N₀)e^(−rt)), which this calculator uses in logistic mode.
What is the rule of 70 for doubling time?
Dividing 70 by the growth rate in percent gives a quick estimate of doubling time. At 2% a year, 70 ÷ 2 = 35 years. It works because ln 2 ≈ 0.693, and it's accurate for small rates under continuous growth.
Should I use the exponential or the discrete model?
Use the exponential model for organisms that reproduce continuously, such as bacteria, or for human populations measured with a continuous rate. Use the discrete model for species with one breeding season a year, or when a rate is quoted as a simple yearly percentage.
Sources
- Biology 2e, 45.3 Environmental Limits to Population Growth — OpenStax, Rice University. Checked October 10, 2026.
- Calculus Volume 2, 4.4 The Logistic Equation — OpenStax, Rice University. Checked October 10, 2026.