Question

-O In old-growth forests of Douglas fir, the spotted owl dines mainly on flying squirrels. Suppose the predator-prey matrix for the two populations is A= 0.5 0.7 and assume that an 1.4 initial vector Xo has an eigenvector decomposition Xo = C1V1 + ... +n Vn such that C1 >0. Show that if the predation parameter p is 0.275, both populations grow. Estimate the long-term growth rate and the eventual ratio of owls to flying squirrels. one If p = 0.275, the eigenvalues of A are 1.05,0.85 . Both populations grow because (Use a comma to separate answers as needed.) of these eigenvalues is/are greater than 1. The long-term growth rate of both populations is about 5 %. Eventually, the two populations will be in the simplified ratio of approximately spotted owl(s) to every thousand flying squirrels. (Type whole numbers.)

Please answer the last quetion starting with "eventually"

Answer 1

1 0.5 0.77 A= -0.275 1.4 This has eigenvalues 1.05 and 0.85 14 Eigenvectors are y = 11 General solution is x(t) = 44+ [14 (1.05)' +2 11 1 Eventually, the population tends towards 3() = a[11]1.057 as him(0.85¥ =0 So the required ratio is 14c,(1.05)*_14 114,(1.05) 11 That is, the ratio will be 14 owls for every 11 squirrels