(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

In the Leslie population model, suppose matrixAhas a strictly dominant eigenvalue [tex]\lambda_1[/tex]. Age class evolution is given by: [tex]x^{(k)} = Ax^{(k-1)}[/tex]; initial population is x^{(0)}.

(i) Initially, let [tex]x^{(0)}[/tex] be the linear combination [tex]a_1x_1 + a_2x_2 + ... + a_nx_n[/tex], ofA's linearly independent eigenvectors, with each constant [tex]a_i \neq 0[/tex]. Find two expressions for the population distribution [tex]x^{(1)}[/tex]after one generation; one using products involving matrixA, the other in terms ofA's eigenvalues. Then, repeat this process to find the population distribution in the k-th generation.

(ii) Repeat (i) to find the population distribution [tex]x^{(k)}[/tex]

(iii) Use (ii) to find the consequence of [tex]\lambda_1[/tex] , being the strictly dominant

eigenvalue. For suitably large values of k , what can be said of the

relative sizes of the terms in the expression found in ii. Hint:Consider the limit [tex]lim_k\rightarrow\infty (\frac{x^{(k)}}{\lambda_1 ^{k}})[/tex].

(iv) Explain why the population approaches a stable population division into its classes as k gets large, with stable population growth rate of [tex]\lambda_1[/tex].

(v) If A has nostrictly dominanteigenvalue, how will the conclusions differ?

2. Relevant equations

diagonalization of matrices

3. The attempt at a solution

(i) [tex]x^{(1)} = L^{-1} D L x^{(0)}[/tex]

[tex]x^{(1)} = \lambda_n a_nx_n[/tex]

(ii) [tex]x^{(k)} = \lambda_n ^{(k)} a_nx_n[/tex]

(iii) >>> I'm stuck on this question

(iv) as the [tex]x^k[/tex] the population age classes becomes stable, due to k getting bigger, [tex]\lambda_1[/tex] will become the more significant eigen value, therefore population growth rate approaches [tex] \lambda_1[/tex], which leads to a stable population division which is proportional to its eigenvector.

(v) >>> I'm stuck on this question

I'm not confident about any of my answers, I'd appreciate if you would also comment on my answers as well as giving tips on the questions i'm not sure of.Thank you ever so much

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# Eigenvalues with a leslie population model

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