Project a declining population or reverse-solve the initial count, periodic decrease rate, or time, with annualization, half-life, milestone timing, and scenario projections.

Apply a constant decrease to the population remaining each period.

Constant percentage scenario, not a forecast. Fractional periods interpolate the compound formula; counts are rounded for display only.

Optional settings

Population Decrease Formula

The population decrease calculator estimates how many individuals remain after a population declines by the same percentage each selected rate period. This is a compound decrease scenario: the percentage applies to the remaining population. The formulas below use annual rates and years; for monthly rates, use elapsed months as the exponent.

FP = IP × (1 - DR / 100)TS
Variable Meaning
FP Final population after the decrease period
IP Initial population at the starting point
DR Annual decrease rate entered as a percent, such as 2 for 2%
TS Time span in years

In practical terms, the formula multiplies the initial population by the yearly retention factor. If the decrease rate is 2%, then the population keeps 98% of its size each year.

Equivalent Forms

If you know any three values, the missing value can be solved with these rearrangements:

IP = (FP) / ((1 - DR / 100)TS)
DR = 100 × (1 - ((FP) / (IP))1 / TS)
TS = (ln(FP / IP)) / (ln(1 - DR / 100))

These forms estimate a starting population, infer a periodic decline rate, or determine time to a target. Select the rate period and time unit consistently; the calculator converts years to 12 months when needed. Fractional periods interpolate the compound formula. Constant-rate scenarios are not demographic forecasts.

How to Use the Calculator

  1. Select the value to find: final population, initial population, periodic decrease rate, or time.
  2. Select an annual or monthly rate period and years or months for time.
  3. Enter the visible known populations, periodic decrease percentage and time span.
  4. Optionally enter a lower planning threshold, then select Calculate. The selected result has no input field; leave only optional inputs blank.

Enter a decrease rate from 0% up to, but below, 100%. Zero means a constant population and cannot determine time to a lower target or threshold. The calculator excludes 100%, zero populations and nonpositive known time spans because its inverse logarithms require a positive retention factor and population.

Example

A town begins with 100,000 residents and declines by 2% per year for 5 years.

FP = 100000 × (1 - 2 / 100)⁵ ≈ 90392.08

Rounded to whole people, the final population is 90,392. That means the town loses about 9,608 residents over the 5-year period.

You can also measure the total number lost and the percent remaining:

L = IP - FP
R = (FP) / (IP) × 100

Here, L is the total population lost and R is the percentage of the original population still remaining after the decline period.

Total Percentage Decrease Over the Full Time Span

The annual decrease rate and the total decrease across the full period are not the same thing. To find the overall percentage drop from start to finish, use:

TD = (1 - (1 - DR / 100)TS) × 100

In the example above, the annual decrease rate is 2%, but the total 5-year decrease is about 9.61%, not exactly 10%. This happens because each year’s decrease is applied to a smaller remaining population.

When This Calculator Is Appropriate

  • City or regional population projections
  • School enrollment decline estimates
  • Wildlife population reduction modeling
  • Subscriber, customer, or membership attrition
  • Any scenario with a constant percentage decrease over equal time intervals

Important Assumptions

  • The decrease percentage stays constant for each selected annual or monthly rate period.
  • The model assumes percentage-based decline, not a fixed number lost each year.
  • The formula uses elapsed rate periods: years for an annual rate or months for a monthly rate. Fractional-period values use exponential interpolation.
  • Population values must be positive. Counts are model estimates: display rounding does not change the unrounded values used for rates, half-life or threshold timing.

If your problem involves losing the same number of individuals each year rather than the same percentage, a linear decrease model would be more appropriate than this calculator.

Common Questions

Is this a linear or exponential model?
This calculator uses an exponential decay model because the decrease is compounded over time.

Why is the final population not equal to the initial population minus rate times years?
Because the rate is applied to the remaining population each year, not to the original population only once.

Can I use decimals for the rate?
Yes. For example, 1.5 means 1.5% per selected rate period: year or month.