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Population growth and transitions

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Juveniles Adults
Subpopulation size (aged under 15 years) (aged 15 years and over)
(in millions) 1.89 5.31
Birth rate 0 0.0332
Death rate 0.0086 0.0295

The diagram below shows a network model of the changes in the subpopulation sizes from year to year. Here Jn and An denote the sizes (in millions) of the subpopulations of juveniles and adults, respectively, n years after the year 1880. It has been assumed that, in any year, 1/15 of the juveniles become adults.

(a) Explain why the number labeling the pipe from J to An+1 is 0.0661 and the number labeling the pipe from Jn to Jn+1 is 0.9253.
(h) The network model above can be written as a matrix equation of the form
where M is a 2 x 2 matrix. Write down the matrix M.

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The solution shows how to convert the network diagram to a matrix-based system.

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