Problem 12 In a forest, the owls dines mainly on squirrels. Denoting respectively Ok and Sk the population of owls and squirrels at a given time (say months). We have the populations for the (k + 1) th month are Ok+1 = 0.40k + 0.3Sk Sk+1 = -pok + 1.25k for p a parameter. (1) Denoting Ek = [* ]. Check that Ek+1 = TA(TK) for the matrix A = 10 [-10p 12]. (2) Express xk in terms of k, A and To- (3) Compute XA(x) = det(A - x12). (4) Assume p (predator rate) is 0.325. (a) Show that the eigenvalues of A (i.e. the zeroes of XA) are s and 105. (b) Find a basis of the vector subspace 55 Ker(TA - 100 IdR?) = {[: ] ER', T(A-$5. [2) 160 12) ([22]) = [81} =TA([x; ])- 100 [ 32] (c) Find a basis of the vector subspace Ker(TA - 105 100 IdR2) = ([ ] ER', (A-1512) ([2]) = [81} =TA([x2])-100 [x2] (d) (without long calculation) Show that Ker(TA - 3IdR2 ) = {0}. (e) Show that there is a basis (that we will denote Bo.325) of R2 made up of a vector of Ker(TA - 100 IdR2 ) and a vector of Ker(TA - 100 IdR2). (f) Show that if we place in the forest twice as many owls as squirrels, the owls and the squirrels eventually perish. hint: we know the action of TA on ] (g) Show that for any other ratio between the initial populations, both populations grow. hint: notice that there is a basis of R made-up of a vector of Ker(TA - 106 Id 55 Id2) and Ker(TA - 106 ]d - 105 Id2) and we know the action of TA on those vectors. (5) Assume p is 0.5. (a) Show that both owls and squirrels will eventually perish (regardless of the initial ratio of the populations) hint: findthe new eigenvalues and eigenvectors (6) What happens when p =