###### Exercise 5

Suppose we have the female population of a species is divided into juveniles, yearlings, and adults and that each year

90% of the juveniles live to be yearlings.

80% of the yearlings live to be adults.

60% of the adults survive to the next year.

50% of the adults give birth to a juvenile.

Set up a system of the form \(\xvec_{k+1}=A\xvec_k\) that describes this situation.

Find the eigenvalues of the matrix \(A\text{.}\)

What prediction can you make about these populations after a very long time?

If the birth rate goes up to 80%, what prediction can you make about these populations after a very long time? For every 100 adults, how many juveniles, and yearlings are there?