Question

#1, 2, 3

о Problem 1 Denote the owl and wood rat populations at time k by x = where k RA is the time in months, O, is the number of owls in the region studies, and Rt is the number of rats (measured in thousands). Suppose Ox+1 0.50 +0.4R Rx+1 -0.1040x + 1.1R In the first equation, the term 0.50x says that with no wood rats for food, only 50% of the owls will survive each month, and the term 0.4Rx says that owl population will rise if rats are plentiful. In the second equation, the term -0.1040, measures the deaths of rats due to predation by owls, and the term 1.1 R$ says that rat population will grow by 10% each month with no owls as predators. (1) The population dynamics is described by the difference equation Xx+1 = Axx, k = 0,1,2,... What is the matrix A? (2) The eigenvalues of A are = 1.01 and 12 - 0.58 with corresponding eigenvectors 10 V1 = and V2 = 13 = [13] [i], respectively. If the eigenvector decomposition of Xo is xo = +qva, find e, and co, if Xo = [ 25 (3) Write down the eigenvector decomposition of X* (i.e. expression of Xi in terms of 20 the eigenvalues and eigenvectors) for Xo = If k is sufficiently large, what is the approximate ratio of the populations of owls and rats? 15

Answer 1

(1) хи с [ On Ru Outl 0.5 On tory Rk -0.104 Ok tlol RX RIAI ARK K= 0, 1, 2, - If akt 0.41 Then A = 1.1 -0.104 The eigen ralues [os of A is n = 0.015, 12 =0.58. with corresponding eigen sectors - [it] - [1] [...] Vi Now, xoa and GN & G2 5 106 756 [s] - Lives 13G of G 10 G & S G 20 13 G 7 @ 19. G = 2, G ad , > There fore, so ilov, * 2.02