Population/Reproduction and extinction/Exercise

An animal population consists of babies (first year), freshers (second year), Halbstarke (third year), mature ones (fourth year) and veterans (fifth year), these animals can not become older. The total stock of these animals in a given year ${}j$ is given by a ${}5$-tuple ${}B_{j}=(b_{1,j},b_{2,j},b_{3,j},b_{4,j},b_{5,j})$.

During a year ${}7/8$ of the babies become freshers, ${}9/10$ of the freshers become Halbstarke, ${}5/6$ of the Halbstarken become mature ones and ${}2/3$ of the mature ones reach the fifth year.

Babies and freshes can not reproduce yet, then they reach the sexual maturity and ${}10$ Halbstarke generate ${}5$ new pets and ${}10$ of the mature ones generate ${}8$ new babies, and the babies are born one year later.

a) Determine the linear map (i.e. the matrix), which expresses the total stock ${}B_{j+1}$ with respect to the stock ${}B_{j}$.

b) What will happen to the stock ${}(200,150,100,100,50)$ in the next year?

c) What will happen to the stock ${}(0,0,100,0,0)$ in five years?

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