This questions refers to another question so i have provided the game it is refering to below the questions. Please answer 2a,b,c,d

4. [10 marks-2 marks for each part] At the right is pictured a 2-node dynamical system. At each time step: an individual on X 0.2 0.2 stays on X with probability 0.2, moves to Y with probability 0.4, X 0.4, Y an individual on Y 0.6 stays on Y with probability 0.2, moves to X with probability 0.6 In addition an individual on Y Plants a single offspring seed on X that becomes a new-born individual on X at the next time step.2. (15 marks) Here we work with the game from Q4 Assignment 9. At each time step: an individual on X stays on X with probability 1/5, 1/5 !. . ... 1/5 moves to Y with probability 2/5 X 2/5 Y an individual on Y stays on Y with probability 1/5, 3/5 moves to X with probability 3/5 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 *n and In be the expected number of individuals on nodes X and Y 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 n. Find the matrix of this system and calculate its eigenvalues and eigenvectors. Conclude that this population has a steady state and find the proportions of the population on X and Y in this steady state. (b) (4 marks) Suppose we begin with x, = 100,000 and yo = 0. 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 x's and ys. 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 y, for Osns5. 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 = 0 to n = 5. (d) (4 marks) Suppose the population is at its steady state. Let or = average age of the counters on X B = average age of the counters on Y Use a recursive argument to calculate a and B. 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 0. If it survives to the next time step (and the probability of that is 1/5 + 2/5 = 3/5) it will be of age 1. If it survives another time step, it will be age 2, etc. [For a question that is similar to this, see Q10 on the April 2019 exam.]