Thayer Watkins
Silicon Valley
& Tornado Alley

The Matrix Version of
the Cohort Survival Method
of Population Projection

The Cohort Survival method for projecting the future population of a country uses the present age-specific population data to forecast the future population based upon survival ratios and fertility ratios to forecast the births. The discription of the method will be for the female population because in population growth it is the female population that is the limiting factor. The male population will be derived from later as an adjunct to the female population.

Let Fi(t) be the number of females in age group i in year t. A typical implementation of cohort survival projection using five-year age groups and the method is easier to explain if a specific age group, say five-years is presumed.

If Si is the proportion of females of age-group i who survive five years then

Fi(t+5) = Si-1Fi-1(t)

for any age-group except the youngest age-group, the one for girls of age 0 to 4, or the oldest age-group.

The youngest age group is found by projecting births. This is done using fertility ratios. Let fi be the proportion of females in age-group i who have a female baby in a year. Usually these fertility ratios would be for having a baby of either sex and then the births are multiplied by the ratio of female births to total births, approximately 49%. The approach used here is to present the model in its simplest terms. This approach also allows the analysis for a time when parents may be able to choose the sex of their offspring.

For the age-groups beyond the fertile years the value of fi is zero. The number of female babies born in a year is then


This is for one year. For a five-year projection that number must be multiplied by approximately 5.0. The number that should be used is actually less than 5.0 because not all survive. The number born the first year have to survive four years, the ones born the second year have to survive three years and so on. The symbol K will be used to denote this factor, the sum of these survival ratios.

The projection for the oldest age group must be handled somewhat differently than the other age-groups. The number in that oldest age group can either be survivors from the next younger age group or survivors from that same oldest age group; i.e., if the oldest age group is n then

Fn(t+5) = Sn-1Fn-1(t) + SnFn(t)

A Matrix Formulation of the Model

Let the age-group populations be represented as a vector

|   F1(t)   |
|   F2(t)   |
|   F3(t)   |
| .........   |
| Fn-1(t)  |
|   Fn(t)   |


This vector of populations for year t is converted into the vector of population for year t+5 by multiply by the following matrix:

| 0    0   0   Kf4  Kf5 ... 0 |
| S1   0   0   0   0 0 |
| 0   S2  0   0   0 0 |
|   ..  .. .. .. .. .. |
|   ..  .. .. .. .. .. |
|   0   0   0 0 Sn-1 Sn |


If this matrix is represented as S and the population vector by F then the cohort survival projection model is represented in matrix form as

F(t+5) = S*F(t)

The population projection ten years ahead is expressed as

F(t+10) = S*F(t+5) = S*(S*F(t+5)) = S2*F(t)

Or in general

F(t+5m) = Sm*F(t)

The projection of the male population can now be represented in matrix form as

M(t+5) = S'*M(t) + S"*F(t)

where M is the vector of male population age-groups and S' is the corresponding matrix involving the male survival rates but zeroes in the first row for the fertility rates and S" is a matrix that has the fertility rates for bearing male children weighted by some factor K" and having zeroes everywhere else.

The projection ten years ahead is given by

M(t+10) = S'*M(t+5) + S"*F(t+5)
= S'[S'*M(t) + S"*F(t)] + S"[S*F(t)]
= S'2*M(t) + [S'*S"+S"S]*F(t)

The general solution for F(t+5m) was found above so the general solution for M(t+5m) is

M(t+5m) = S'mM(t) + [S'*S"+S"*S]*Sm-2F(t)

Equilibrium Age Distribution
of the Population

An equilibrium age distribution would be one such that the proportions in the age groups stays constant. This means that the populations in the various age groups all increase by the same factor;

F(t+5) = λF(t)

This would mean that

S*F = λF

A vector F such that this is true is called an eigenvector of the matrix S and the value of λ is called an eigenvalue of S. The eigenvalue λ gives the equilibrium growth rate of the population.

The equation for the eigenvalue and eigenvector can be put into the form

(S - λI)F = 0

where I is the identity matrix and 0 denotes a column vector of zeroes. If the matrix (S-λI) has an inverse then the only solution is F=0, which is called the trivial solution. The only way that there can be a nontrivial solution is if the matrix (S-λI) does not have an inverse. That will be the case if the determinant of (S-λI) is equal to zero. If S is an n×n matrix then det(S-λI) is an n degree polynomial equation. This can be solved for λ. The value of λ is equal to (1+g)5 where g is the annual growth rate of the population.

The eigenvector F is found by putting value of λ into the eigenvector equation SF=λF and solving for all the components of F in terms of one component. The eigenvector F can then be put into the form in which the sum of the components is equal to unity. The components of F are then the equilibrium proportions of the population. Note that the growth rate and equilibrium distribution of the female population depend only upon the matrix S, nothing else.

The rule of thumb for long term population stability in the developed world is that the total total fertility rate of females, the average number of children born per female, must be 2.1. The more precise statement would be that the number of female children born per woman must be 1.05. The 0.05 is to allow for the mortality between birth and the end of the fertile age years and other factors which interfere with a woman producing one daughter. In countries with lower survival ratios the increment above 1.0 had to be greater. The precise conditions for a stable population can now be stated in terms of the above model. An equilibrium population means λ must equal 1.0. This means that det(S-I) = 0, or in expanded form, that the determinant of the following matrix be zero:

| -1    0   0   Kf4  Kf5 ... 0 |
| S1   -1   0   0   0 0 |
| 0   S2  -1   0   0 0 |
|   ..  .. .. .. .. .. |
|   ..  .. .. .. .. .. |
|   0   0   0 0 Sn-1 Sn-1 |


This can be further simplified by eliminating those age groups above fertility. Let q be the highest age group with positive female fertility. (For five-year age groups q is about 10.) Then the condition for population stability is that the determinant of the matrix below be zero.

| -1    0   0   Kf4      Kf5 ...    Kfq |
| S1   -1   0   0 0    0 |
| 0   S2  -1   0 0    0 |
|   ..  .. .. .. .. .. |
|   ..  .. .. .. .. .. |
|   0   0   0 0 Sq-1 -1 |


The determinant can be conveniently expanded by the first column or less conveniently by the first row.

The Male Population

Now consider an equilibrium growth rate and distribution for the male population; i.e., M and μ such that

μM = S'*M + S"F
and thus
M = (μI - S')-1S"*F

Since F(t) for equilibrium growth has to be λtF it has to hold that μ=λ. Therefore

M = (λI - S')-1S"*F

Since the growth rate of the male population is equal to λ it means that the growth rate of the population depends only upon the female baby birth rates and the female survival factors. The equilibrium distribution M does depend upon the male survival rates and the male baby birth rates but the equilibrium growth of the male population is independent of those parameters (so long as there are some male births). Many species survive quite well while operating with breeding populations involving one male to twenty females.

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