Answered: ro 1 2 0.5 L - 0. 51 0.7 0 0.3 0 10 10…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Question

A population with four age classes has a Leslie matrix L. If the initial population vector x0, compute x1, x2, and x3

Expert Solution

Step 1

Given: Leslie matrix, $L = [\begin{matrix} 0 & 1 & 2 & 5 \\ 0.5 & 0 & 0 & 0 \\ 0 & 0.7 & 0 & 0 \\ 0 & 0 & 0.3 & 0 \end{matrix}]$ and Initial population vector, $x_{0} = [\begin{matrix} 10 \\ 10 \\ 10 \\ 10 \end{matrix}]$
To find: The value of vectors $x_{1}, x_{2}, and x_{3} .$

Step 2

We know that the general equation for the Leslie matrix is given by :
$x_{n + 1} = L x_{n} At n = 0, x_{1} = L x_{0} = {[\begin{matrix} 0 & 1 & 2 & 5 \\ 0.5 & 0 & 0 & 0 \\ 0 & 0.7 & 0 & 0 \\ 0 & 0 & 0.3 & 0 \end{matrix}]}_{4 \times 4} {[\begin{matrix} 10 \\ 10 \\ 10 \\ 10 \end{matrix}]}_{4 \times 1} = {[\begin{matrix} 0 + 10 + 20 + 50 \\ 5 + 0 + 0 + 0 \\ 0 + 7 + 0 + 0 \\ 0 + 0 + 3 + 0 \end{matrix}]}_{4 \times 1} = {[\begin{matrix} 80 \\ 5 \\ 7 \\ 3 \end{matrix}]}_{4 \times 1}$

Step 3

Similarly,
$At n = 1, x_{2} = L x_{1} = {[\begin{matrix} 0 & 1 & 2 & 5 \\ 0.5 & 0 & 0 & 0 \\ 0 & 0.7 & 0 & 0 \\ 0 & 0 & 0.3 & 0 \end{matrix}]}_{4 \times 4} {[\begin{matrix} 80 \\ 5 \\ 7 \\ 3 \end{matrix}]}_{4 \times 1} = {[\begin{matrix} 0 + 5 + 14 + 15 \\ 40 + 0 + 0 + 0 \\ 0 + 3.5 + 0 + 0 \\ 0 + 0 + 2.1 + 0 \end{matrix}]}_{4 \times 1} = {[\begin{matrix} 34 \\ 40 \\ 3.5 \\ 2.1 \end{matrix}]}_{4 \times 1}$

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