###### Exercise10

Consider a small rodent that lives for three years. Once again, we can separate a population of females into juveniles, yearlings, and adults. Suppose that, each year,

• Half of the juveniles live to be yearlings.

• One quarter of the yearlings live to be adults.

• Adult females produce eight female offspring.

• None of the adults survive to the next year.

1. Writing the populations of juveniles, yearlings, and adults in year $$k$$ using the vector $$\xvec_k=\threevec{J_k}{Y_k}{A_k}\text{,}$$ find the matrix $$A$$ such that $$\xvec_{k+1} = A\xvec_k\text{.}$$

2. Show that $$A^3=I\text{.}$$

3. What are the eigenvalues of $$A^3\text{?}$$ What does this say about the eigenvalues of $$A\text{?}$$

4. Verify your observation by finding the eigenvalues of $$A\text{.}$$

5. What can you say about the trajectories of this dynamical system?

6. What does this mean about the population of rodents?

7. Find a population vector $$\xvec_0$$ that is unchanged from year to year.

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