# Population model

• Jan 30th 2009, 05:16 AM
tsal15
Population model
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...

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

Many thanks :)
• Feb 3rd 2009, 03:25 AM
tsal15
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...

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