exponentialgrowthcalculator.com

Population Growth Calculator

Enter a starting population, a birth rate, and a death rate to see the net growth rate and projected population update as you type.

Population growth calculator

The population at time zero
%
Percent added per period
%
Percent lost per period
Any unit, years, cycles, censuses
Projected Population P(t) 51,507.95 after 15 periods at 1.70% net rate
Net Growth Rate +1.70% birth rate − death rate

Period-by-Period Population

Period tPopulation P(t)Change
0 40,000.00 N/A
1 40,680.00 680.00
2 41,371.56 691.56
3 42,074.88 703.32
4 42,790.15 715.27
5 43,517.58 727.43
6 44,257.38 739.80
7 45,009.76 752.38
8 45,774.92 765.17
9 46,553.10 778.17
10 47,344.50 791.40
11 48,149.35 804.86
12 48,967.89 818.54
13 49,800.35 832.45
14 50,646.95 846.61
15 51,507.95 861.00

What the table shows: at the default settings, 40,000 people growing at a net rate of 1.70% (2.8% births minus 1.1% deaths) for 15 periods, the population rises by 680.00 in period 1, then by 861.00 in period 15 as the larger base compounds the same percentage into a bigger absolute gain. The population finishes at 51,507.95 after 15 periods, an increase of 11,507.95 over the starting figure.

Concept

What Drives Population Growth?

Population growth is exponential when births and deaths occur in a steady proportion to the population that already exists, producing a constant net growth rate that compounds every period.

A population does not grow by a fixed head count each period. It grows by a fixed percentage of whatever its current size happens to be. A town adding 2.8% of its residents through births while losing 1.1% to deaths each year adds more people in absolute terms once it is larger than it did when it was smaller, even though both rates stay exactly the same. That is the same compounding behavior seen everywhere else on this site, just measured in people instead of dollars or bacteria.

The two rates rarely matter individually once combined. What determines the shape of the curve is the net growth rate, birth rate minus death rate, a single number that plays the same role as "r" in the general exponential growth formula. A 2.8% birth rate paired with a 1.1% death rate collapses to a 1.7% net rate, and from that point forward the population behaves exactly like any other quantity growing at 1.7% per period.

This is also why two very different-looking populations can produce identical projections. A population with 4% births and 2.3% deaths and a population with 9% births and 7.3% deaths both carry a 1.7% net rate, and the formula cannot tell them apart. It only ever sees the difference between the two rates, never their individual size.

The net rate concept is exactly what the site's general exponential growth calculator already computes from a single rate input; this page simply adds the extra step of deriving that rate from birth and death figures first.

Equation

The Population Growth Formula

The population growth formula is P(t) = P0(1 + (birth − death))t for discrete periods, and P(t) = P0ert for continuous demographic growth. Both describe the same rising curve; they differ only in whether growth is applied in steps or smoothly.

Discrete, applied per period
P(t) = P0(1 + (birth − death))t

Use this when births and deaths are tallied in identifiable steps: an annual census, a yearly vital-statistics report, a generation count.

Continuous, applied every instant
P(t) = P0ert

Use this when births and deaths happen continuously rather than in batches, with r as the instantaneous net growth rate.

Variables in the population growth equation
SymbolNameWhat it representsExample
P(t)Projected populationThe population size after t periods.51,507.95 people
P0Initial populationThe population at t = 0, before this projection begins.40,000 people
birthBirth rateNew individuals added per period, as a percentage of the population.2.8%
deathDeath rateIndividuals lost per period, as a percentage of the population.1.1%
rNet growth ratebirth − death, the single rate that drives the formula.0.017 (that is 1.7%)
tElapsed timeHow many periods have passed.15 periods

What the table shows: birth and death are only ever entered separately for convenience. The formula itself uses their difference, r, and never needs the two original figures again once that subtraction is done. P(t) is always the computed result, never entered directly.

Method

How to Project Population Growth

To project population growth, subtract the death rate from the birth rate to get a net rate, add 1, raise that sum to the power of the elapsed periods, then multiply by the starting population.

  1. Record the starting population

    Identify P₀, the population currently on record before any projection is applied.

  2. Convert the birth rate to a decimal

    Divide the percentage rate by 100. A birth rate of 2.4% per year becomes 0.024.

  3. Convert the death rate to a decimal

    Do the same for the death rate over the same period length. A death rate of 1.6% per year becomes 0.016.

  4. Subtract to get the net growth rate

    Net rate = birth − death. Here, 0.024 − 0.016 = 0.008, or 0.8% per year.

  5. Apply the formula

    Compute (1 + net rate)^t, then multiply by P₀. With a net rate of 0.8% across 8 years, 1.008^8 = 1.065821.

Worked example, coastal town population

A coastal town has 8,500 residents. Its birth rate runs 2.4% per year and its death rate runs 1.6% per year. The population growth formula gives its projected population 8 years from now.

P0
8,500 people
birth − death
2.4% − 1.6% = 0.8% → 0.007999999999999998
t
8 years
P(t) = P0(1 + r)t P(8) = 8,500 × (1 + 0.008)8 P(8) = 8,500 × (1.008)8 P(8) = 8,500 × 1.065821 P(8) = 9,059.48 people

The town gains about 559.48 residents over the 8 years, noticeably more than the 544-person gain a flat "0.8% times 8 years" estimate would predict, because each year's growth compounds on an already-larger base.

Limits

Doubling Time and the Limits of This Model

Doubling time applies to population growth the same way it applies to any exponential quantity: at a fixed net rate, a population doubles after a predictable number of periods, and that number shrinks as the net rate rises. At this page's default 1.7% net rate, the population doubles roughly every 41 periods. The dedicated periods-to-double tool works this out directly from any net growth rate without needing to separate it back into births and deaths.

Every projection above assumes the net growth rate stays fixed no matter how large the population becomes, the same assumption the flagship exponential growth calculator on this site's homepage makes for any quantity it models. That assumption holds reasonably well over short stretches, but no real population keeps a constant birth-minus-death rate indefinitely. Food supply, living space, water, and disease pressure all tend to push birth rates down and death rates up as a population grows denser, which is exactly the feedback this discrete model has no way to represent.

Left unchecked, this model implies growth without limit, an ever-larger population producing an ever-larger absolute increase forever. Real populations instead tend to slow as they approach the carrying capacity of their environment, the maximum size that available resources can sustain. The S-curve population model builds that correction in directly, bending the curve toward a ceiling instead of letting it climb without end, and is the natural next step once a projection is being carried out far enough into the future for resource limits to matter.

Questions

Frequently Asked Questions

What does "net growth rate" mean in population growth?

The net growth rate is the birth rate minus the death rate, and it is the single percentage that actually determines how a population changes each period. A population with a 2.8% birth rate and a 1.1% death rate has a net growth rate of 1.7%. Every other detail of the two rates cancels out once they are combined this way.

This net figure is exactly the "r" that goes into the exponential growth formula, so the calculator above only ever needs birth and death rates as a convenience for entering it.

What is the population growth formula?

The population growth formula is P(t) = P₀(1 + (birth − death))t for periods counted in whole steps, and P(t) = P₀ert for continuous growth. P₀ is the starting population, birth and death are the periodic rates as decimals, r is the continuous net growth rate, and t is elapsed time.

Both forms are the same exponential growth equation used across this site. Population growth has no special mathematics of its own, only a demographic interpretation of the same r.

How do birth and death rates combine into one growth figure?

Birth and death rates combine by simple subtraction, net rate = birth rate − death rate, because both are measured as a percentage of the same existing population over the same period. A 3% birth rate and a 3% death rate produce a net rate of 0%, meaning the population stays flat even though people are being born and dying the entire time.

The two rates must cover the same period length before subtracting them; an annual birth rate cannot be combined with a monthly death rate without converting one of them first.

What does a negative net growth rate mean?

A negative net growth rate means the death rate exceeds the birth rate, so the population shrinks by a fixed proportion every period instead of growing. The formula does not change. P(t) = P₀(1 + r)t still applies, but with r below zero, the growth factor drops below 1 and each period's result is smaller than the last.

A population declining this way traces the same falling, ever-flattening curve as any exponential decay process, just applied to people instead of a physical quantity.

How does doubling time apply to population growth?

Doubling time applies to population growth exactly as it does to any exponential process: it is the number of periods needed for the population to reach twice its current size at a fixed net growth rate. At the calculator's default 1.7% net rate, that works out to roughly 41 periods.

The dedicated doubling time calculator solves this directly from any net growth rate, without needing to separate it back into birth and death components.

Why is this exponential model unrealistic over very long periods?

This exponential model is unrealistic over very long periods because it assumes birth and death rates stay fixed no matter how large the population gets, ignoring the fact that food, space, and resources are finite. A real population's growth rate typically slows as it approaches the environment's carrying capacity, rather than compounding forever at the same percentage.

The logistic growth calculator adds exactly that correction, flattening the curve as the population nears a defined ceiling instead of letting it grow without bound.

What time period should the birth and death rates be measured in?

The birth and death rates should be measured over the same period as the time value entered for t, whether that is a year, a decade, or a census cycle. Mixing an annual birth rate with a value of t counted in decades will overstate or understate the projection by an order of magnitude.

Government and demographic sources typically publish birth and death rates per year per 1,000 people; divide that figure by 10 to convert it to the percentage form this calculator expects.

Can this calculator model non-human populations too?

Yes, this calculator models any population where individuals are added and removed as a percentage of the current total, including wildlife herds, insect colonies, and microbial cultures. The formula does not know or care what is being counted.

For a population that reproduces by splitting rather than by separate births and deaths, the bacterial growth calculator is built around that specific doubling-based process instead.