1. The population of chimpanzees in a particular country consists of juveniles and adults. Juvenile chimpanzees do not reproduce.

In a study, the numbers of juvenile and adult chimpanzees were estimated at the start of each year. A model for the population satisfies the matrix system $$\binom { J _ { n + 1 } } { A _ { n + 1 } } = \left( \begin{array} { c c } a & 0.15
0.08 & 0.82 \end{array} \right) \binom { J _ { n } } { A _ { n } } \quad n = 0,1,2 , \ldots$$ where \(a\) is a constant, and \(J _ { n }\) and \(A _ { n }\) are the respective numbers of juvenile and adult chimpanzees \(n\) years after the start of the study.

1. Interpret the meaning of the constant \(a\) in the context of the model. At the start of the study, the total number of chimpanzees in the country was estimated to be 64000 According to the model, after one year the number of juvenile chimpanzees is 15360 and the number of adult chimpanzees is 43008
2. 1. Find, in terms of \(a\) $$\left( \begin{array} { c c } a & 0.15
      0.08 & 0.82 \end{array} \right) ^ { - 1 }$$
   2. Hence, or otherwise, find the value of \(a\).
   3. Calculate the change in the number of juvenile chimpanzees in the first year of the study, according to this model. Given that the number of juvenile chimpanzees is known to be in decline in the country,
3. comment on the short-term suitability of this model. A study of the population revealed that adult chimpanzees stop reproducing at the age of 40 years.
4. Refine the matrix system for the model to reflect this information, giving a reason for your answer.
   (There is no need to estimate any unknown values for the refined model, but any known values should be made clear.)