Question

Question 13 (10 marks) Scientists observed populations of sea turtles on two islands: Turtle Island (TI) and Lizard Island (LI). The first year of observation, they noted that 700 turtles lived on Turtle Island and 200 turtles lived on Lizard Island. Let x;, zyi denote the numbers of turtles (in hundreds) on Turtle Island and Lizard Island after i years. Write this information as - F the vector po = |Y0 The scientists returned the following year and made the following observations Births on TI =03( Yo) Births on LI = 0 Deaths on LI = 03y0 Deaths on TI = .03x0 Migration from TI to LI = .01(xo + y0) Migration from LI to TI= 0 (a) Using the table above, write p1 = | y1 (b) Assuming that the changes in population are the same from year to year, find a matrix A Xi+1 such that = A Yi+1 (c) Show that the matrix A has eigenvalues X = .97 and A2 = 1 associated with eigenvectors and v2 = V1 = (d) Find matrices P and D such that A = PDP-1. (e) Use this diagonalization to express in terms of Yo (f) What happens as n -> oc?

Answer 1

Potulation atr Cussent potudation Bisthe-Deaths-Migeation But Migeakion tn ene yeas + on Tistle tsland, potulation afte year One en TI - Deaths On TI -Migaon fom +Mgration foa TI to LI LI Jo TI XT O.03 (x0+y)- 0.03 X-0.01 (Xo+Y.) O.0290 en Lizasd aland potulation after gear one -Migeation fron ,yBisths on Li-Deaths on LI TI to LI LI to TI+ Migeation fon 9,-9% O- o.03y- O+0.01 +

X,0.9900.02yo + 9, =0.01X0+ 0.98 0 C 0.99 O.02 0.98 N-NI-CHN 0.99 7 O02 ue have 1 2 D.98 2 6.97 2.03 deom yeae to Same ale (the Changas in the population then iai O.99X+ 0.02; FO.OKi+ 0.989 diti O.02 O.99 O.98 O.O1 O.99 O.02 A= O.98 O.O1

Exgon Values of A Sadis fes l0.99- det (A-dI)O 6.02 O.98-d O. O 0.9702-1.97 O.0002 O +0.97 3 O TIOTT+ 0.97=0 d--P -) = o Cd-1) CA-0.91) O I0.9 / Hene the Eigen Values ale Eigen Vecto's ted,0.97 AV,-0.97 V 0.99 O.0272 O.97 メ2 O.98 X2 O.O 02,0.97 0.99x,+ O.O Hene the Eigen Vectos s

AV2= V2 0.99 O.02 X2 o.98 O.01 O.99X1+0 02 X2=X 2 X2 X hence the Eien Vect E1 2 D - D-F 2 P= D= 0.97 Veridey that A= PD P One ton To.97 2 A= -aX A -p ニA 9n-2 A

O.97) 4 Co.91) A (o.91) :-(o1) CO.97) 3 Co97 1 A A' 7 2 (0.97)+6 Co-97)" () as 0.97) O hance 33 Steady state Pofulation Vect is as