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2. [15 marks} Here we work with the game from Q4 Assignment 9. At each time step: an individual on it stays oanith probability 1,15. 1" 4- - - -1 _____ 1" moves to Ywith probability 2/5 an individual on Y ' stays on Y with probability 1 {5. moves to X with probability BIS 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 Jrn and yn be the expected number of individuals on nodes X and if at time n. Find a set of linear recursive equations for x and y at time n+ 1 in terms of their values at time :2. 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 xu = l and y.) = D. Solve the system of [a]. Write separate equations for x\" and y\". Report them in as simple a form as you can. [c] [3 marks] Check your formulae from [b] in the following manner. First use your equations to calculate its and y5. Display these expressions in their simplest form and evaluate them. Then put your recursive equations on a spread sheet and use it to tabulate x\" and 3:1.1 for sns. Show that you get the same answer with each method. Include a copy of the table obtained from the spreadsheet listing all values from n = CI to n = 5. [d] [4 marks} Suppose the population is at its steady state. Let or = average age of the counters on X ,3 = average age of the counters on Y Use a recursive argument to calculate a and ,3. To be more precise about the calculation of age. note that a newborn individual on X [new born means that it sprouted from a seed planted in the previous time-step] has age If]. If it survives to the next time step {and the probability ofthat is US + 2,8 = 3/5} it will be ofage 1. Ifit survives another time step, it will be age 2. etc. [For a question that is similar to this, see Qll]I on the April 2 019 exam]