Chapter 7.5, Problem 36E

In 1990, the population of the African country Benin was about 4.6 million people. Its composition by age was as follows:

We represent these data in a state vector whose components are the populations in the various age brackets, in millions:
$\stackrel{\to }{x}\left(0\right)=4.6\left[\begin{array}{c}0.466\\ 0.257\\ 0.147\\ 0.084\\ 0.038\\ 0.008\end{array}\right]\approx \left[\begin{array}{c}2.14\\ 1.18\\ 0.68\\ 0.39\\ 0.17\\ 0.04\end{array}\right]$
We measure time in increments of 15 years, with $t=0$ in 1990. For example, $\stackrel{\to }{x}\left(3\right)$ gives the age composition in the year $2035\left(1990+3\cdot 15\right)$ . If current age-dependent birth and death rates are extrapolated, we have the following model:
$\stackrel{\to }{x}\left(t+1\right)=\left[\begin{array}{cccccc}1.1& 1.6& 0.6& 0& 0& 0\\ 0.82& 0& 0& 0& 0& 0\\ 0& 0.89& 0& 0& 0& 0\\ 0& 0& 0.81& 0& 0& 0\\ 0& 0& 0& 0.53& 0& 0\\ 0& 0& 0& 0& 0.29& 0\end{array}\right]\stackrel{\to }{x}\left(t\right)=A\stackrel{\to }{x}\left(t\right)$
a. Explain the significance of all the entries in the matrix A in terms of population dynamics.
b. Find the eigenvalue of A with the largest modulus and an associated eigenvector (use technology).What is the significance of these quantities in terms of population dynamics? (For a summary on matrix techniques used in the study of age-structured populations, see Dmitrii O. Logofet, Matrices andGraphs: Stability Problems in Mathematical Ecology, Chapters 2 and 3, CRC Press, 1993.)