2. [15 marks) Here we work with the game from Q4 Assignment 9. At each time step: an individual on X stays onX with probability 1/5, 1 5 {. . . _ 1 1 moves to Y with probability 2/5 ----- 25. an individual on Y T stays on Y with probability 1/5, moves to X with probability 3 f 5 In addition an individual, upon arrival on Y, on Y plants a single offspring seed on X that becomes a new-born individual on X at the next time step. Note that an individual who moves from Y to Y in a time step is considered a new arrival and gets the offspring again. [a] (4 marks} Let x\" and y" be the expected number of individuals on nodes X and Y at time :1. Find a set of linear recursive equations for x and y at time n+1 in terms of their values at time n. Find the matrix of this system and calculate its eigenvalues and eigenvectors. Conclude that this population has a steady state and nd the proportions of the population on X and Y in this steady state. [b] [4 marks) Suppose we begin with x0 = 100,000 and ya = 0. Solve the system of (a). Write separate equations for x\" and 3!\". Report them in as simple a form as you can