Question

2. (Adapted from problem 9.9 in the text) A population of insects is divided into two stages. Females in the first stage produce 3 female offspring each, and 25 percent survive to the second stage. The females in the second stage produce 7 female offspring and then die. When a new female is born (from a female in either stage), it becomes first stage female. a (a) Write down a matrix model for these populations that determines how the pop- ulations in each stage change from find a matrix A such that a(t+1) = A (t) for any t > 0, where (t) is the column vector of populations in each of the two stages at time t. (b) Find all eigenvalues and eigenvectors for A. one time step to the next. More specifically: (c) What is the long-run growth rate for this population? I.e., after a large number of time steps, by what factor do the populations approximately change? (Hint: consider one of the eigenvalues). (d) Determine the steady-state population vectors, i.e. those that don't change from one time step to the next. (e) What is the long-run ratio of the stage one females to stage two females? (Hint: consider an eigenvector).

Answer 1

2) Let po population firat atac fema Ce a at tima t (7 Sccond blと) 3a 7 b(+ altナ) b(t+ 2 s a (t) alt) 3 a (t) -) bttti) .2S ) ut (t 3 A = - 2 S 13-7 JA->エ) =0 =) 7 b) Eigenvelues : 2, S 2-32 4 ^2-12a-7=o 7 3. S 7 14 7 A 2 V 2 2 3.SJ .2 S Eigcnvalue and Eigenvectors 2

C2 c) C terrm Long at C2 by factor Population uu incruare 3. S 2 c) 2 A =) J1-A72(t) o (14) -2 - 7 Q= (7 - .2 S C2 e) tSeconic Ration One 14 Atage 14