Population/Reproduction and extinction/Exercise

An animal population consists of babies (first year), freshers (second year), rockers (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 rockers, ${}5/6$ of the rockers become mature ones, and ${}2/3$ of the mature ones reach the fifth year.

Babies and freshers can not reproduce yet, then they reach sexual maturity, and ${}10$ rockers 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) that 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?

Create a solution