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Thread: Population model

  1. #1
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    Population model

    The question is stated exactly as follows:

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

    Initially, let $\displaystyle x^{(0)}$ be the linear combination $\displaystyle a_1x_1 + a_2x_2 + ... + a_nx_n$, of A's linearly independent eigenvectors, with each constant $\displaystyle a_i \neq 0$. Find two expressions for the population distribution $\displaystyle x^{(1)}$ after one generation; one using products involving matrix $\displaystyle A$, the other in terms of $\displaystyle A$'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 => $\displaystyle x^{(1)}$ = $\displaystyle Ax^{(0)}$
    Expression #2 => $\displaystyle x^{(1)}$ = $\displaystyle \lambda x^{(0)}$

    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
    Last edited by tsal15; Feb 3rd 2009 at 02:36 AM.
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  2. #2
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    Melbourne
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    Quote Originally Posted by tsal15 View Post
    The question is stated exactly as follows:

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

    Initially, let $\displaystyle x^{(0)}$ be the linear combination $\displaystyle a_1x_1 + a_2x_2 + ... + a_nx_n$, of A's linearly independent eigenvectors, with each constant $\displaystyle a_i \neq 0$. Find two expressions for the population distribution $\displaystyle x^{(1)}$ after one generation; one using products involving matrix $\displaystyle A$, the other in terms of $\displaystyle A$'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 => $\displaystyle x^{(1)}$ = $\displaystyle Ax^{(0)}$
    Expression #2 => $\displaystyle x^{(1)}$ = $\displaystyle \lambda x^{(0)}$

    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?
    Last edited by tsal15; Feb 3rd 2009 at 02:36 AM.
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