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- 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- LA Tn 0.8 02 0.2] [z Yn| = (0.1 0.7 0.2) |Yn_1] - 02 Zn 0.1 01 0.6] |2n1 0T (a) Find the eigenvalues and eigenvectors of the matrix. uppose we start (n = 0) wit ogs 1n total, on X, b) Supp 0) with 100 frogs in total, 60 on X, 40 on Y, and none on Z. Find an expression for the numbers in X, and Z at time n. Write separate equations for xn, yn and zn. (c) What happen to these numbers as n approaches infinity? Show that they converge to a limiting distribution