(Tucker) A growth model for elephants (E) and mice (M) predicts population changes from decade to decade. E3E+M M+2E+ 4M a. Determine the eigenvalues and associated eigenvectors for this system. E b. Suppose that the initial population is p = [] = [5]. Write p as a linear combination of the eigenvectors. c. Use the information in part b to estimate the elephant and mice populations in eight decades.
(Tucker) A growth model for elephants (E) and mice (M) predicts population changes from decade to decade. E3E+M M+2E+ 4M a. Determine the eigenvalues and associated eigenvectors for this system. E b. Suppose that the initial population is p = [] = [5]. Write p as a linear combination of the eigenvectors. c. Use the information in part b to estimate the elephant and mice populations in eight decades.
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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![**(Tucker) A Growth Model for Elephants and Mice: Predicting Population Changes**
This educational resource utilizes a growth model to predict population changes for elephants (E) and mice (M) from decade to decade, using the following system of linear equations:
\[ E \leftarrow 3E + M \]
\[ M \leftarrow 2E + 4M \]
**Tasks:**
a. **Determine the Eigenvalues and Associated Eigenvectors for this System.**
Use the given system to find the eigenvalues (λ) and the corresponding eigenvectors (v). This involves solving the characteristic equation to find λ, then solving for v in the equation \((A - λI)v = 0\), where A is the coefficient matrix.
b. **Express the Initial Population as a Linear Combination of Eigenvectors.**
Suppose the initial population vector is:
\[
\mathbf{p} = \begin{bmatrix} E \\ M \end{bmatrix} = \begin{bmatrix} 5 \\ 5 \end{bmatrix}
\]
Write the given population vector \(\mathbf{p}\) as a combination of the system's eigenvectors. This generally involves solving for coefficients in the equation \(\mathbf{p} = c_1 \mathbf{v}_1 + c_2 \mathbf{v}_2\), where \(\mathbf{v}_1\) and \(\mathbf{v}_2\) are eigenvectors of the system.
c. **Estimate the Population in Eight Decades.**
Utilize the information obtained from part b to estimate the populations of elephants and mice after eight decades. This involves applying the eigenvalues raised to the power of the number of decades to the initial condition expressed in terms of eigenvectors.
**How to Approach the Problems:**
1. **Finding Eigenvalues and Eigenvectors:**
- Construct the coefficient matrix \(A\) from the given system of equations.
- Calculate \(A - λI\) and solve \(\det(A - λI) = 0\) for eigenvalues \(λ\).
- For each eigenvalue, solve \((A - λI)v = 0\) to find the corresponding eigenvectors \(v\).
2. **Writing the Initial Population as a Linear Combination of Eigenvectors:**
- Decompose the initial population vector into the eigenvector basis by](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fe6e77107-ef1d-4e39-badf-4decafa47f1b%2Ffc50bb74-18ca-4bb8-95a8-6122b8724919%2F0r8hi16_processed.png&w=3840&q=75)
Transcribed Image Text:**(Tucker) A Growth Model for Elephants and Mice: Predicting Population Changes**
This educational resource utilizes a growth model to predict population changes for elephants (E) and mice (M) from decade to decade, using the following system of linear equations:
\[ E \leftarrow 3E + M \]
\[ M \leftarrow 2E + 4M \]
**Tasks:**
a. **Determine the Eigenvalues and Associated Eigenvectors for this System.**
Use the given system to find the eigenvalues (λ) and the corresponding eigenvectors (v). This involves solving the characteristic equation to find λ, then solving for v in the equation \((A - λI)v = 0\), where A is the coefficient matrix.
b. **Express the Initial Population as a Linear Combination of Eigenvectors.**
Suppose the initial population vector is:
\[
\mathbf{p} = \begin{bmatrix} E \\ M \end{bmatrix} = \begin{bmatrix} 5 \\ 5 \end{bmatrix}
\]
Write the given population vector \(\mathbf{p}\) as a combination of the system's eigenvectors. This generally involves solving for coefficients in the equation \(\mathbf{p} = c_1 \mathbf{v}_1 + c_2 \mathbf{v}_2\), where \(\mathbf{v}_1\) and \(\mathbf{v}_2\) are eigenvectors of the system.
c. **Estimate the Population in Eight Decades.**
Utilize the information obtained from part b to estimate the populations of elephants and mice after eight decades. This involves applying the eigenvalues raised to the power of the number of decades to the initial condition expressed in terms of eigenvectors.
**How to Approach the Problems:**
1. **Finding Eigenvalues and Eigenvectors:**
- Construct the coefficient matrix \(A\) from the given system of equations.
- Calculate \(A - λI\) and solve \(\det(A - λI) = 0\) for eigenvalues \(λ\).
- For each eigenvalue, solve \((A - λI)v = 0\) to find the corresponding eigenvectors \(v\).
2. **Writing the Initial Population as a Linear Combination of Eigenvectors:**
- Decompose the initial population vector into the eigenvector basis by
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