example of fundamental theorem of demography

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

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.

单词	ExampleOfFundamentalTheoremOfDemography
释义	example of fundamental theorem of demography Assume a population with a (age, sex, etc.) that is described by a vector $x(t)$ , where $x_{1}(t),\\ldots,x_{n}(t)$ represents the number of individuals in the population who possess the characteristic at a level $1,\\ldots,n$ , 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 $\\\|x(t)\\\|$ . 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.
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