Question 2.7 (5 marks) The species X from two questions ago want to communicate with you and arrange themselves in the shape of the area described by the following set M: M = {(32321) e R2 : a? +92 5 36} n {(56:21) E R2 : (ac 2.5)2 + (y 2.5)2 2 4} n {(93:20) E R2 : (a:+2)2+(y2)2 2 1} 0 {(93:31) E R2 : 3:2 + 9(y + 2.5)2 2 9}. Describe the individual sets and then present a neat sketch of the area determined by M. Question 2.5 (8 marks) Your spaceship made it through the wormhole and now you've landed on a new planet. You observe that this planet is inhabited by two species, called X and Y, both represented by huge amorphic masses. When you arrive (day 0) you observe that there are 370 cubic metres of X and k0 cubic metres of Y. The next day, you observe that the original 930 cubic metres of X turned into gaze cubic metres of X and %x0 cubic metres of Y; moreover, the original k0 cubic metres of Y turned into 3kg cubic metres of Y and glee cubic metres of X. The day after, the same happens with the 3:1 and 161 cubic metres of X and Y you assume that this is a general reproduction pattern! You keep on measuring the sizes and write act- and ki for the number of X and Y on day i. a) Determine (1:1 and k1 in terms of 51:0 and kg. b) Find a matrix A such that the reproduction pattern is described by (:i+1) = A (:3) forz' 2 0. 2+1 1' c) Soon the X and Y have become aware of your arrival and they suddenly change their behavior. You observe that their new reproduction pattern is now described by (1%\") 3(191) where B (1/3 5/3 . You want to see what happens in the long run. Do the following: (i) Compute the eigenvalues of B. (ii) For each eigenvalue of B nd one eigenvector. (iii) Find a diagonal matrix D and an invertible matrix P such that B = PDP'I