CHAPTER 9

1

Use pencil and paper.

2

2a

Use pencil and paper.

2b

Eigenvalues[{{c,2c},{c,0}}]

2c

Find the optimal clutch size under Godfray's model:

s=S Exp[-h(n-1)];
a=D[n s,n];
Solve[Numerator[Together[a]]==0,n]

Define c as a function of n; plot 2c, the dominant eigenvalue, against n, and observe that, approximately, 2c < 1 when n > 4; and then find the critical value of n above which 2c < 1 exactly.

c=(1-.5^n)S/(n s)/.h->1/n;

Plot[2c,{n,1,10}]

FindRoot[2c==1,{n,4}]

3, 4, 5

Use pencil and paper.

6

6a

Evaluate f[n] for different values of n:

A={{1/2,1/2},{1,0}};
f[n_]:=MatrixPower[N[A],n]//MatrixForm

Find the eigenvalues and left eigenvectors of the matrix and hence identify the dominant left eigenvector:

Eigensystem[Transpose[A]]

6b

Repeat this exercise for the matrix in Equation 20.

7, 8

Use pencil and paper.