A population with four age classes has a Leslie matrix L- 0 0.4 0 0 0 0 0.6 0 0.7 0 0 0 If the initial population vector is Xo 10 10 10 compute x₁, x₂, and x
A population with four age classes has a Leslie matrix L- 0 0.4 0 0 0 0 0.6 0 0.7 0 0 0 If the initial population vector is Xo 10 10 10 compute x₁, x₂, and x
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
![In this exercise, we explore population dynamics using a Leslie matrix, which models the population of a species divided into age classes. The matrix, \( L \), is given as:
\[
L = \begin{bmatrix}
0 & 1 & 2 & 5 \\
0.7 & 0 & 0 & 0 \\
0 & 0.4 & 0 & 0 \\
0 & 0 & 0.6 & 0
\end{bmatrix}
\]
The initial population vector \( \mathbf{x_0} \) is given by:
\[
\mathbf{x_0} = \begin{bmatrix}
10 \\
10 \\
10 \\
10
\end{bmatrix}
\]
The task is to compute the population vectors \( \mathbf{x_1} \), \( \mathbf{x_2} \), and \( \mathbf{x_3} \).
### Steps to Solve:
1. **Calculate \( \mathbf{x_1} \):**
Multiply the matrix \( L \) with the initial vector \( \mathbf{x_0} \).
\[
\mathbf{x_1} = L \cdot \mathbf{x_0}
\]
\[
\mathbf{x_1} = \begin{bmatrix}
80 \\
7 \\
4 \\
6
\end{bmatrix}
\]
2. **Calculate \( \mathbf{x_2} \):**
Multiply the matrix \( L \) with the vector \( \mathbf{x_1} \).
\[
\mathbf{x_2} = L \cdot \mathbf{x_1}
\]
\[
\mathbf{x_2} = \begin{bmatrix}
80 \\
1.4 \\
3 \\
2.4
\end{bmatrix}
\]
3. **Calculate \( \mathbf{x_3} \):**
Multiply the matrix \( L \) with the vector \( \mathbf{x_2} \).
\[
\mathbf{x_3} = L \cdot \mathbf{x_2}
\]
Provide the resulting vector based on the calculations.
This Leslie matrix describes transitions of the population through different age classes, using birth rates and](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fa1e36031-6f93-4211-8c57-892d2fd1b551%2F0331bc90-a211-4e17-9bca-f9e02ed6894e%2Fk9njgxv_processed.jpeg&w=3840&q=75)
Transcribed Image Text:In this exercise, we explore population dynamics using a Leslie matrix, which models the population of a species divided into age classes. The matrix, \( L \), is given as:
\[
L = \begin{bmatrix}
0 & 1 & 2 & 5 \\
0.7 & 0 & 0 & 0 \\
0 & 0.4 & 0 & 0 \\
0 & 0 & 0.6 & 0
\end{bmatrix}
\]
The initial population vector \( \mathbf{x_0} \) is given by:
\[
\mathbf{x_0} = \begin{bmatrix}
10 \\
10 \\
10 \\
10
\end{bmatrix}
\]
The task is to compute the population vectors \( \mathbf{x_1} \), \( \mathbf{x_2} \), and \( \mathbf{x_3} \).
### Steps to Solve:
1. **Calculate \( \mathbf{x_1} \):**
Multiply the matrix \( L \) with the initial vector \( \mathbf{x_0} \).
\[
\mathbf{x_1} = L \cdot \mathbf{x_0}
\]
\[
\mathbf{x_1} = \begin{bmatrix}
80 \\
7 \\
4 \\
6
\end{bmatrix}
\]
2. **Calculate \( \mathbf{x_2} \):**
Multiply the matrix \( L \) with the vector \( \mathbf{x_1} \).
\[
\mathbf{x_2} = L \cdot \mathbf{x_1}
\]
\[
\mathbf{x_2} = \begin{bmatrix}
80 \\
1.4 \\
3 \\
2.4
\end{bmatrix}
\]
3. **Calculate \( \mathbf{x_3} \):**
Multiply the matrix \( L \) with the vector \( \mathbf{x_2} \).
\[
\mathbf{x_3} = L \cdot \mathbf{x_2}
\]
Provide the resulting vector based on the calculations.
This Leslie matrix describes transitions of the population through different age classes, using birth rates and
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