Model how a population changes over time with the exponential (continuous), discrete (geometric) or logistic equation. Enter a starting size, growth rate and carrying capacity to get the population at any time, doubling time, inflection point and a year-by-year table and chart.
Last updated:
Growth model
% / unit
per-capita rate per time unit; negative = decline
find when the population reaches this size
Enter your values to see the result.
Advertisement
Advertisement
How to use the Population Growth Calculator
Pick a model: exponential for unlimited growth, logistic when resources set a carrying capacity, or discrete for populations that breed once per period.
Enter N₀ and the growth rate r as a percentage per time unit (use a negative rate for a declining population).
Set the carrying capacity K (logistic) and the time span, and choose the time unit.
Optionally add a target population to find out when it will be reached.
Press Calculate and read the result, the step-by-step working, the chart and the table.
The three growth models compared
Model
Equation
Rate form
Shape
Use it for
Exponential (continuous)
N(t) = N₀·e^(rt)
dN/dt = rN
J-curve
Bacteria in fresh medium, early invasions, any population with unlimited resources
Discrete (geometric)
N(t) = N₀·λ^t
N(t+1) = λ·N(t)
J-curve in steps
Species that breed once a season, annual census data, simple % per year projections
Logistic
N(t) = K / (1 + A·e^(−rt))
dN/dt = rN(1 − N/K)
S-curve
Populations limited by food, space or other resources
Here A = (K − N₀) / N₀ and λ = e^r. With r given as a percentage, divide by 100 first: a rate of 40% per year means r = 0.4.
Doubling times for common growth rates
Continuous rate r
Doubling time (ln 2 / r)
Discrete rate (λ − 1)
Doubling time (ln 2 / ln λ)
1%
69.3 periods
1%
69.7 periods
2%
34.7
2%
35.0
5%
13.9
5%
14.2
10%
6.93
10%
7.27
25%
2.77
25%
3.11
69.3%
1.00
100%
1.00
The rule of 70 (doubling time ≈ 70 / rate in %) is a quick mental estimate of the same numbers.
Reading the logistic curve
Lag and acceleration: while N is far below K the curve is nearly exponential.
Inflection at K/2: the population adds individuals fastest here, at rK/4 per time unit. Wildlife and fisheries managers use this idea (maximum sustainable yield).
Deceleration: above K/2 each step adds fewer individuals as resources run short, and N levels off near K.
Above K: a population that overshoots, or is stocked above K, shrinks back toward it.
Frequently asked questions
What is the exponential population growth formula?
N(t) = N₀ · e^(rt), where N₀ is the starting population, r is the per-capita growth rate per unit of time (as a decimal) and t is time. It assumes unlimited resources, so the population keeps growing faster as it gets larger. The doubling time is ln 2 / r, about 0.693 / r.
What is the logistic growth equation?
N(t) = K / (1 + A·e^(−rt)) with A = (K − N₀) / N₀. Its rate form is dN/dt = rN(1 − N/K): growth looks exponential while N is small, is fastest at N = K/2, and slows to zero as the population approaches the carrying capacity K.
What is carrying capacity?
Carrying capacity (K) is the largest population an environment can sustain over time with its available food, space, water and other resources. In the logistic model a population below K grows toward it and a population above K declines toward it.
What is the difference between r and λ?
r is the continuous (instantaneous) per-capita growth rate and λ is the finite rate of increase per period for populations that reproduce in discrete pulses. They are linked by λ = e^r. A discrete growth rate of 10% per year (λ = 1.10) equals a continuous r of ln 1.10 ≈ 9.53% per year.
When is the logistic population growing fastest?
At the inflection point, when N = K/2. The maximum growth rate there is rK/4 individuals per time unit. The calculator also gives the time to reach K/2, t = ln(A) / r, when the starting population is below half of K.
Can I model a declining population?
Yes. Enter a negative growth rate. The exponential and discrete models then show a half-life instead of a doubling time. In the logistic model, a starting population above K also declines toward K even with a positive rate.
Are these models accurate for real populations?
They are simplified models used in biology and ecology classes. Real populations are affected by age structure, migration, environmental change, predators and random events, so treat the results as projections under the stated assumptions rather than forecasts.
Are all the tools on FreeToolCabin tested for quality and accuracy?
Yes! Every tool on our site undergoes automated Q/A testing in real browser environments to ensure high precision, correct output, and zero errors. If you happen to spot a bug or have a suggestion, please contact us at [email protected] and our team will review it right away.