# example of fundamental theorem of demography

Assume a population with a (age, sex, etc.) that is described by a vector $x(t)$, where ${x}_{1}(t),\mathrm{\dots},{x}_{n}(t)$ represents the number of individuals in the population who possess the characteristic at a level $1,\mathrm{\dots},n$, at time $t$.

For example, consider age-groups, and assume ${x}_{0}(t)$ is the number of individuals in the population that are aged 0 to 1 year, ${x}_{1}(t)$ 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 chain^{}.

The fundamental theorem of demography then states that if the matrix $A(t)$ satisfies the required properties, then the distribution^{} vector $x(t)$ converges to the eigenvector associated to the dominant eigenvalue, *regardless* of the behavior of the total population $\parallel x(t)\parallel $.

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 |