Question

OK Problem 1 Denote the owl and wood rat populations at time k by xk = where k R is the time in months, Ox is the number of owls in the region studies, and Rx is the number of rats (measured in thousands). Suppose Ox+1 0.50 +0.4Rk Rx+1 -0.1040+1.1 RK 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 -0.1040x measures the deaths of rats due to predation by owls, and the term 1.1 Re says that rat population will grow by 10% each month with no owls as predators. Vi = - [i], respectively. If the eigenvector decomposition of xo is (1) The population dynamics is described by the difference equation Xk+1 = Axk, k 0,1,2,.... What is the matrix A? (2) The eigenvalues of A are 1 1.01 and 2 0.58 with corresponding eigenvectors and v2 = 13 20 Xo = V1 + c2V2, find c and c if Xo (3) Write down the eigenvector decomposition of xi (i.e. expression of xx 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?

Answer 1

3 The owl and wood rat population at timme k by Ok. [ ar Where K is the time in months Rk Ok is the no. q outs in the region studies Rk is the no. of rats (measured in thousands). Suppose , Oker = 0.5 Ok + 0.486 Rkel = -0.104, Okt 1.1 kk In matrix form 0.5 0.4 Xkel Xk -0.104 다. lil Okel Ox When Xkant and X 2 RKTI Rk ANSWERS : Đ The population dynamics described by the difference equanian Xkel AXA K , 0, 1,4 The matrix 0.5 0.4 Az 2 -0,104 (2 The eigenvalues of a can be found by the characteristics equation [A-911-0

The eigenvalues of A 2, z 101 and and 2 2 0.58 with corresponding eigennectors lo -13 and & 6-[: respectively. If the eigenrector decomposition صلى Xo i3 Xo=4 VT GV - [is] Thin To find : 4 & G we have, Xo a Greek ما => ليا ( * )= a[i)e also [:) ( 1o 4 e 56 ك۱ 134 TG logf5G 220 ? 24+ 4 z 4 -ý >) 13G + C₂ z 17 134 + 4 zis Solving & & 2 ศพ ศ. 24+4+4 139 1/4 =15 Hl4 2711 7 (Gal 24 + 204 2(1) 1 4 2 4 ما این کار از » G z 4-2 (622 Hency 1421 & /y = 2 & Xoa +2k

Eigenrecter decomposition og in terms of the regenvalues & eigenvectors) for Xo = xx (it expression q Xx -[is] Since, Yo - G V TE V2 we have, X1 A Xo = G AV AGAR Bowego Gair tern [ --Avsar] X - AX, 2 GAAN & GA AVL 9(x, prit GAPV In general XI - AXKA G(a,iky Y 16(82)kų 노 Xkel I AXk a(2)kl ve 2(22) kelk 2 here, 9.211 & 4 z 2 Hence, Xx = (21) * v, + (%20% v Yk 2 (1.01)* [!!] + (OS) [:] But, we are interested i of k i Sufficiently large, nie. Ko AS Kto, 0.58)— -O (3 (0.53)<] to for large K, Xk ~ (101k (lopk []

and , Xkti ñ (1.01) ~ 1.01 XK The last approximation says that eventually both the arts grow by population of rats and the population of a factor of almost 101 per month, it. growth rate Ich The ratio lo to of the enties T in Xx remain the same for So, for every lo owls there are 13 thors and rats. rie The approximal ratio of the population of onls an rats 10 ; 13,000 (Aris)