Population model

Quote:

Originally Posted by tsal15

The question is stated exactly as follows:

"Suppose $A$ has a strictly dominant eigenvalue $\lambda_1$ . Age class evolution is given by: $x^{(k)} = Ax^{(k-1)}$ . Initial Population is $x^{(0)}$ .

Initially, let $x^{(0)}$ be the linear combination $a_1x_1 + a_2x_2 + ... + a_nx_n$ , of A's linearly independent eigenvectors, with each constant $a_i \neq 0$ . Find two expressions for the population distribution $x^{(1)}$ after one generation; one using products involving matrix $A$ , the other in terms of $A$ 's eigenvalues."

Now, the question is a little blurry, and so I've come to an answer of which i'm not sure if is correct...

My Answer:

Expression #1 => $x^{(1)}$ = $Ax^{(0)}$
Expression #2 => $x^{(1)}$ = $\lambda x^{(0)}$

Please indicate for me if I'm wrong...and if so could you please direct me on how i should go about this question.

Many thanks :)

Or should I expand the multiplication of the above?