0 0.6 A population with four age classes has a Leslie matrix L = 0 0 1 2 0 DON 0 5 0.7 0 0 0 0.4 0 If the initial population vector is x = 10 10 10 10 compute x₁, x₂, and X3.
0 0.6 A population with four age classes has a Leslie matrix L = 0 0 1 2 0 DON 0 5 0.7 0 0 0 0.4 0 If the initial population vector is x = 10 10 10 10 compute x₁, x₂, and X3.
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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![**Leslie Matrix Population Model**
A population with four age classes can be modeled using a Leslie matrix \( L \). The Leslie matrix is given by:
\[
L = \begin{bmatrix}
0 & 1 & 2 & 5 \\
0.6 & 0 & 0 & 0 \\
0 & 0.7 & 0 & 0 \\
0 & 0 & 0.4 & 0
\end{bmatrix}
\]
The initial population vector \( \mathbf{x}_0 \) is:
\[
\mathbf{x}_0 = \begin{bmatrix}
10 \\
10 \\
10 \\
10
\end{bmatrix}
\]
Using the Leslie matrix, compute the population vectors \( \mathbf{x}_1 \), \( \mathbf{x}_2 \), and \( \mathbf{x}_3 \).
**Computed Population Vectors:**
\[
\mathbf{x}_1 = \begin{bmatrix}
80 \\
6 \\
7 \\
4
\end{bmatrix}
\]
\[
\mathbf{x}_2 = \begin{bmatrix}
66 \\
80 \\
1.5 \\
1.6
\end{bmatrix}
\]
\[
\mathbf{x}_3 = \begin{bmatrix}
\boxed{} \\
\boxed{} \\
\boxed{} \\
\boxed{}
\end{bmatrix}
\]
**Explanation of Computation:**
- The Leslie matrix is used to project the growth of populations across different age classes.
- Each element in the resulting vector \( \mathbf{x}_n \) is calculated using the multiplication of the Leslie matrix with the previous population vector \( \mathbf{x}_{n-1} \).
Understanding the matrix and computing these vectors is crucial in the study of population dynamics, allowing researchers to predict changes in population structure over time.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fa1e36031-6f93-4211-8c57-892d2fd1b551%2Fc3a219c1-c4a3-45d3-b9ca-9544f0cce601%2Fn8zqjin_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Leslie Matrix Population Model**
A population with four age classes can be modeled using a Leslie matrix \( L \). The Leslie matrix is given by:
\[
L = \begin{bmatrix}
0 & 1 & 2 & 5 \\
0.6 & 0 & 0 & 0 \\
0 & 0.7 & 0 & 0 \\
0 & 0 & 0.4 & 0
\end{bmatrix}
\]
The initial population vector \( \mathbf{x}_0 \) is:
\[
\mathbf{x}_0 = \begin{bmatrix}
10 \\
10 \\
10 \\
10
\end{bmatrix}
\]
Using the Leslie matrix, compute the population vectors \( \mathbf{x}_1 \), \( \mathbf{x}_2 \), and \( \mathbf{x}_3 \).
**Computed Population Vectors:**
\[
\mathbf{x}_1 = \begin{bmatrix}
80 \\
6 \\
7 \\
4
\end{bmatrix}
\]
\[
\mathbf{x}_2 = \begin{bmatrix}
66 \\
80 \\
1.5 \\
1.6
\end{bmatrix}
\]
\[
\mathbf{x}_3 = \begin{bmatrix}
\boxed{} \\
\boxed{} \\
\boxed{} \\
\boxed{}
\end{bmatrix}
\]
**Explanation of Computation:**
- The Leslie matrix is used to project the growth of populations across different age classes.
- Each element in the resulting vector \( \mathbf{x}_n \) is calculated using the multiplication of the Leslie matrix with the previous population vector \( \mathbf{x}_{n-1} \).
Understanding the matrix and computing these vectors is crucial in the study of population dynamics, allowing researchers to predict changes in population structure over time.
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