Question

Deep in the forests, dusky-footed wood rats provide up to 80% of the diet for the spotted own, the main predator of the wood rat. Denote the owl and wood rat population at time k by P2 = and assume that 7 Pk+1 = 0.5 0.4) Pk -0.104 1.1 a. The eigenvalues of A are 11 = 1.02 and 12 = 0.58 where A1 is the dominant eigenvalue. Find corresponding eigenvectors v1 and 02. 5 Suppose Po Q and write the vector po as a linear combination of the eigenvectors V1, V2. Write Pk as a function of k, C1, C2, 11, 12, Vi, and v2. d. Find a simple big k approximation of Px as we did in class. e. Use the big k approximation to calculate the eventual growth rate of the combined population of owls and rats. f. Use the big k approximation to calculate the eventual ratio of owls to rats.

Answer 1

Giren that, Per = [as a la intore Puz [ ] 0.5 0.4 Ket A= 1 1.1 0.104 (as The eigen ratues of A are d=102, d =0.58. for An = 1.02. we have to find eigen rectors, (A - A, I) x = 0, (al-[0] -0.184 - 0.52x torunyo g -0.52 .:] [:]

Let x=0 6.52 e or4 13 10 g C = X E L 13 130 for A2 = 0.58 (A - 227) X Or08 = 0, 0.4 60 0.104 0152 9 ) Orofa of oil o.uy

Kel 2. » ya +0.08xC 0.4 allo c [*] V hence, the eigen rectors are (3] [s. (62 Po = [12] set Po FG Niagra 10G 456 13G + ₂ st 9 LOG + Sty=5, 139 4 b = 12. Gal , C2 = =1 a Tranny PS 1. [**] +60 31 T =g