where the populations x(t) and y(t) are measured in thousands and t in years. Use a numerical solver to analyze the populations over a long period of time for each of the following cases:
where the populations x(t) and y(t) are measured in thousands and t in years. Use a numerical solver to analyze the populations over a long period of time for each of the following cases:
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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![### 12. Consider the competition model defined by
\[ \frac{dx}{dt} = x(2 - 0.4x - 0.3y) \]
\[ \frac{dy}{dt} = y(1 - 0.1y - 0.3x) \]
where the populations \( x(t) \) and \( y(t) \) are measured in thousands and \( t \) in years. Use a numerical solver to analyze the populations over a long period of time for each of the following cases:
(a) \( x(0) = 1.5 \), \( y(0) = 3.5 \)
(b) \( x(0) = 1 \), \( y(0) = 1 \)
(c) \( x(0) = 2 \), \( y(0) = 7 \)
(d) \( x(0) = 4.5 \), \( y(0) = 0.5 \)
### Analysis
In this exercise, we will use the competition model described by the given differential equations to study the dynamic behavior of two competing populations \( x(t) \) and \( y(t) \). The populations are measured in thousands, and \( t \) represents time in years.
The given model accounts for the interactions between two species where the growth rates depend not only on their respective current populations but also on the interaction between the two (competition).
### Instructions
1. **Model Understanding**: Understand the interaction terms and how they affect the growth rates.
2. **Numerical Solver Implementation**: Use a numerical solver like Euler’s method, Runge-Kutta method, or any built-in solver in software like MATLAB, Python with libraries, or any computational tool.
3. **Long-term Behavior Analysis**: Simulate the populations for a long period of time for each of the given initial conditions and analyze how the populations evolve.
### Cases for Analysis
1. **Case (a)**: Initial populations \( x(0) = 1.5 \), \( y(0) = 3.5 \)
2. **Case (b)**: Initial populations \( x(0) = 1 \), \( y(0) = 1 \)
3. **Case (c)**: Initial populations \( x(0) = 2 \), \( y(0) = 7 \](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F4fcf4b0b-d82c-43e8-86bb-da9864dd2a82%2F94c0239c-43f9-4008-8af0-103e03fad7cf%2F81299na_processed.png&w=3840&q=75)
Transcribed Image Text:### 12. Consider the competition model defined by
\[ \frac{dx}{dt} = x(2 - 0.4x - 0.3y) \]
\[ \frac{dy}{dt} = y(1 - 0.1y - 0.3x) \]
where the populations \( x(t) \) and \( y(t) \) are measured in thousands and \( t \) in years. Use a numerical solver to analyze the populations over a long period of time for each of the following cases:
(a) \( x(0) = 1.5 \), \( y(0) = 3.5 \)
(b) \( x(0) = 1 \), \( y(0) = 1 \)
(c) \( x(0) = 2 \), \( y(0) = 7 \)
(d) \( x(0) = 4.5 \), \( y(0) = 0.5 \)
### Analysis
In this exercise, we will use the competition model described by the given differential equations to study the dynamic behavior of two competing populations \( x(t) \) and \( y(t) \). The populations are measured in thousands, and \( t \) represents time in years.
The given model accounts for the interactions between two species where the growth rates depend not only on their respective current populations but also on the interaction between the two (competition).
### Instructions
1. **Model Understanding**: Understand the interaction terms and how they affect the growth rates.
2. **Numerical Solver Implementation**: Use a numerical solver like Euler’s method, Runge-Kutta method, or any built-in solver in software like MATLAB, Python with libraries, or any computational tool.
3. **Long-term Behavior Analysis**: Simulate the populations for a long period of time for each of the given initial conditions and analyze how the populations evolve.
### Cases for Analysis
1. **Case (a)**: Initial populations \( x(0) = 1.5 \), \( y(0) = 3.5 \)
2. **Case (b)**: Initial populations \( x(0) = 1 \), \( y(0) = 1 \)
3. **Case (c)**: Initial populations \( x(0) = 2 \), \( y(0) = 7 \
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