Stable Age Distribution Calculator

Enter Population Inputs

Use comma separated or line separated values. Survival values must be one less than fecundity values because the last age class has no next class.

Age Class Labels

Example: Age 0-14, Age 15-29, Age 30-44

Fecundity Rates

Enter one value per age class. Values must be zero or greater.

Survival Probabilities

Enter n-1 values between 0 and 1.

Initial Population Vector

Used for projections and as the starting vector for iteration.

Projection Periods

Number of future periods to project.

Normalization Total

Stable shares are also scaled to this total.

Max Iterations

Higher values allow tighter convergence checks.

Tolerance

Smaller values demand stricter convergence.

Example Data Table

This sample uses the default values shown in the calculator. Stable shares are illustrative outputs from the same input structure.

Age Class	Fecundity	Survival to Next	Example Stable Share (%)
Age 0-14	0.0000	0.9200	36.8332
Age 15-29	0.8000	0.8800	26.6192
Age 30-44	1.2000	0.8100	18.4011
Age 45-59	0.3000	0.7000	11.7084
Age 60+	0.0000	—	6.4382

Formula Used

Population projection:
n(t + 1) = L · n(t)

Leslie matrix structure:
| f1  f2  f3  ...  fn |
| s1   0   0  ...   0 |
|  0  s2   0  ...   0 |
|  .   .   .   .    . |
|  0   0   0 ... s(n-1) 0 |

Stable age distribution:
L · w = λ · w

Normalized stable share for class i:
p_i = w_i / Σw_i

Intrinsic growth rate:
r = ln(λ)

Here, f values are fecundity rates, s values are survival probabilities, λ is the dominant growth factor, and w is the dominant eigenvector.

The calculator uses power iteration to approximate the dominant eigenvector and then converts it into proportions and percentages for interpretation.

How to Use This Calculator

Enter age class labels for each row of your population model.
Provide one fecundity value for every age class.
Provide survival probabilities for transitions into the next class.
Enter the initial population vector used for projections.
Choose projection periods, normalization total, iteration limit, and tolerance.
Click the calculate button to see the stable age shares above the form.
Review the graph, matrix, and projection totals.
Use the CSV or PDF buttons to export the results.

Frequently Asked Questions

1) What is a stable age distribution?

It is the long run proportion of a population found in each age class when fertility and survival rates stay constant over time.

2) What does the dominant growth factor mean?

The dominant growth factor, λ, shows whether the modeled population grows, shrinks, or stays steady from one projection period to the next.

3) Why must survival values be one less than age classes?

A standard Leslie matrix only tracks transitions into the next class. The final class has no next class, so one fewer survival value is required.

4) Why does the calculator use power iteration?

Power iteration is an efficient way to estimate the dominant eigenvector and eigenvalue for nonnegative population matrices without external math libraries.

5) What does the normalization total change?

It does not change the proportions. It only rescales the stable distribution to a convenient total such as 100, 1,000, or any positive value.

6) Can I use this for teaching and research examples?

Yes. It is useful for classroom demonstrations, demographic exercises, ecology models, and sensitivity testing with structured populations.

7) What if the result does not converge fully?

Increase the maximum iterations, tighten or loosen the tolerance carefully, and recheck the input rates for unrealistic values or data entry mistakes.

8) Does the initial population change the stable distribution?

It affects short run projections, but not the long run stable distribution when the matrix is valid and the dominant eigenvalue is unique.