Advanced Engineering Mathematics
10th Edition
ISBN: 9780470458365
Author: Erwin Kreyszig
Publisher: Wiley, John & Sons, Incorporated
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I'm currently encountering challenges in solving this problem using only matrix notation, and I'm seeking your guidance. The problem necessitates a solution exclusively in matrix notation, without any other methods. Could you kindly provide me with a detailed, step-by-step explanation using matrix notation to help me understand and solve the problem until we reach the final solution?
this has to be done using the matrix format please
![# Population Dynamics of Owls and Wood Rats
## Mathematical Representation of Populations
We denote the owl and wood rat populations at time \( k \) by \( x_k = \begin{bmatrix} O_k \\ R_k \end{bmatrix} \), where \( k \) is the time in months. Here, \( O_k \) is the number of owls in the region studied, and \( R_k \) is the number of rats (measured in thousands). Suppose the evolution of populations follows the equations:
\[
O_{k+1} = (0.5) \cdot O_k + (0.4) \cdot R_k
\]\
\[
R_{k+1} = (-0.125) \cdot O_k + (1.1) \cdot R_k
\]
### Explanation of Equations
- **First Equation**: \(O_{k+1} = (0.5) \cdot O_k + (0.4) \cdot R_k \)
- The term \((0.5)O_k\) indicates that, without any wood rats for food, only half of the owl population will survive each month.
- The term \((0.4)R_k \) implies that if there are plenty of wood rats, the owl population will increase due to ample food.
- **Second Equation**: \(R_{k+1} = (-0.125) \cdot O_k + (1.1) \cdot R_k\)
- The term \((1.1)R_k\) suggests that in the absence of owls as predators, the rat population will grow by 10% per month.
- The term \((-0.125)O_k\) measures the deaths of rats due to owl predation.
## Example: Initial Populations
To determine the evolution of this system, consider the following starting populations:
\[
\begin{bmatrix} O_1 \\ R_1 \end{bmatrix} = \begin{bmatrix} 100 \\ 500 \end{bmatrix}
\]
Given these initial values, one can use the provided equations to compute the populations of owls and rats for subsequent months.
This model provides insights into the dynamic interaction between predator (owls) and prey (wood rats) populations over time, illustrating concepts](https://content.bartleby.com/qna-images/question/0113a2ae-503a-4a14-a5d4-d8fff765856f/c65ca3bc-124b-419b-bd03-54b65ab9ef2a/med00g_processed.jpeg)
Transcribed Image Text:# Population Dynamics of Owls and Wood Rats
## Mathematical Representation of Populations
We denote the owl and wood rat populations at time \( k \) by \( x_k = \begin{bmatrix} O_k \\ R_k \end{bmatrix} \), where \( k \) is the time in months. Here, \( O_k \) is the number of owls in the region studied, and \( R_k \) is the number of rats (measured in thousands). Suppose the evolution of populations follows the equations:
\[
O_{k+1} = (0.5) \cdot O_k + (0.4) \cdot R_k
\]\
\[
R_{k+1} = (-0.125) \cdot O_k + (1.1) \cdot R_k
\]
### Explanation of Equations
- **First Equation**: \(O_{k+1} = (0.5) \cdot O_k + (0.4) \cdot R_k \)
- The term \((0.5)O_k\) indicates that, without any wood rats for food, only half of the owl population will survive each month.
- The term \((0.4)R_k \) implies that if there are plenty of wood rats, the owl population will increase due to ample food.
- **Second Equation**: \(R_{k+1} = (-0.125) \cdot O_k + (1.1) \cdot R_k\)
- The term \((1.1)R_k\) suggests that in the absence of owls as predators, the rat population will grow by 10% per month.
- The term \((-0.125)O_k\) measures the deaths of rats due to owl predation.
## Example: Initial Populations
To determine the evolution of this system, consider the following starting populations:
\[
\begin{bmatrix} O_1 \\ R_1 \end{bmatrix} = \begin{bmatrix} 100 \\ 500 \end{bmatrix}
\]
Given these initial values, one can use the provided equations to compute the populations of owls and rats for subsequent months.
This model provides insights into the dynamic interaction between predator (owls) and prey (wood rats) populations over time, illustrating concepts
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