example of fundamental theorem of demography

Assume a population with a (age, sex, etc.) that is described by a vector x(t), where x1(t),,xn(t) represents the number of individuals in the population who possess the characteristic at a level 1,,n, at time t.

For example, consider age-groups, and assume x0(t) is the number of individuals in the population that are aged 0 to 1 year, x1(t) is the number of individuals aged 1 to 2 years, etc.

Suppose that the transition from one class to another is described by a matrix A(t). In the case of age-groups, this matrix will for example describe mortality in a given age-group. This matrix, in the case of non deterministic modelling, will define a Markov chainMathworldPlanetmath.

The fundamental theorem of demography then states that if the matrix A(t) satisfies the required properties, then the distributionPlanetmathPlanetmath vector x(t) converges to the eigenvector associated to the dominant eigenvalue, regardless of the behavior of the total population x(t).

Hence, in the case of age-groups, the proportion of individuals aged, say, 1 to 2 years, tends to a fixed value, even if the total population increases.

Title example of fundamental theorem of demography
Canonical name ExampleOfFundamentalTheoremOfDemography
Date of creation 2013-03-22 14:15:59
Last modified on 2013-03-22 14:15:59
Owner mathcam (2727)
Last modified by mathcam (2727)
Numerical id 6
Author mathcam (2727)
Entry type Example
Classification msc 92D25
Classification msc 37A30