5. [6 marks] There are three ponds, each occupied by a large number of frogs. Every day, a certain fraction of the frogs from each pond mi- 0.8 grate to one of the two neighbouring ponds according to the di- agram at the right. Thus the number of frogs in each pond changes from day to day according to the following dynamical sys- AN Iy 0.8 0.2 0.2 Tn1 Yn| = 0.1 0.7 0.2 Yn1 Zn 0.1 0.1 0.6 Py f 0.1 0.7 0.6 (a) Find the eigenvalues and eigenvectors of the matrix. (b) Suppose we start (n = 0) with 100 frogs in total, 60 on X, 40 on Y. and none on Z. Find an expression for the numbers in X, Y and Z at time n. Write separate equations for x,, y, and z,. () What happen to these numbers as n approaches infinity? Show that they converge to a limiting distribution