LESSON 9 A quantitative look at the fish population Now you'll study the same fish population as in Lesson 8, but this time you'll answer some quantitative questions. Recall that the fish population, p(t), satisfies the DE p' = p - 0.2p² - 0.7, (fishpop) where p is measured in thousands of kilograms and t is in weeks. The above DE has two equilibrium populations, p₁=0.8 and p₂ = 4.2, both of which are constant solutions of the equation. Since they're solutions, no other solution can cross through them; this can be seen by applying the Existence and Uniqueness Theorem (see your text) to the problem. Thus, the other solutions either approach them asymptotically as t-, or move away from them. Your phase line plot from the last lesson should show this. Task 1. Use the Euler approximation method with a reasonable step size to answer the following questions. 1. If p(0) = 6 (i.e., 6,000 kg), what is the population after 10 weeks? 2. How close is this to the equilibrium point it's approaching? 3. At about what time does the population stop dropping by 100 kg per week? 4. When does the population come within 50 kg of equilibrium?
LESSON 9 A quantitative look at the fish population Now you'll study the same fish population as in Lesson 8, but this time you'll answer some quantitative questions. Recall that the fish population, p(t), satisfies the DE p' = p - 0.2p² - 0.7, (fishpop) where p is measured in thousands of kilograms and t is in weeks. The above DE has two equilibrium populations, p₁=0.8 and p₂ = 4.2, both of which are constant solutions of the equation. Since they're solutions, no other solution can cross through them; this can be seen by applying the Existence and Uniqueness Theorem (see your text) to the problem. Thus, the other solutions either approach them asymptotically as t-, or move away from them. Your phase line plot from the last lesson should show this. Task 1. Use the Euler approximation method with a reasonable step size to answer the following questions. 1. If p(0) = 6 (i.e., 6,000 kg), what is the population after 10 weeks? 2. How close is this to the equilibrium point it's approaching? 3. At about what time does the population stop dropping by 100 kg per week? 4. When does the population come within 50 kg of equilibrium?
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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![**Lesson 9: A Quantitative Look at the Fish Population**
In this lesson, you will explore quantitative aspects of the fish population studied in Lesson 8.
**Differential Equation (DE):**
The fish population, \( p(t) \), satisfies the differential equation:
\[ p' = p - 0.2p^2 - 0.7 \]
where \( p \) is measured in thousands of kilograms, and \( t \) is in weeks.
**Equilibrium Populations:**
The DE has two equilibrium populations, \( p_1 = 0.8 \) and \( p_2 = 4.2 \). These are constant solutions of the equation. Due to the Existence and Uniqueness Theorem, no other solutions can cross these equilibrium points. Solutions will either approach them asymptotically as \( t \to \infty \) or move away. Your phase line plot from the last lesson should demonstrate this.
**Task 1: Euler Approximation Method**
Use the Euler approximation method with a reasonable step size to answer the following questions:
1. If \( p(0) = 6 \) (i.e., 6,000 kg), what is the population after 10 weeks?
2. How close is this to the equilibrium point it’s approaching?
3. At what time does the population stop dropping by 100 kg per week?
4. When does the population come within 50 kg of equilibrium?
5. If \( p(0) = 0.5 \), when does the population die out?
6. If \( p(0) = 1.2 \), when does the population come within 100 kg of equilibrium?](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F15c022aa-6711-4b29-ab7f-4e5d70ca82f7%2Fbb5f6f3d-c2b4-4f79-a954-e0060f1311bb%2Fvifrch7_processed.png&w=3840&q=75)
Transcribed Image Text:**Lesson 9: A Quantitative Look at the Fish Population**
In this lesson, you will explore quantitative aspects of the fish population studied in Lesson 8.
**Differential Equation (DE):**
The fish population, \( p(t) \), satisfies the differential equation:
\[ p' = p - 0.2p^2 - 0.7 \]
where \( p \) is measured in thousands of kilograms, and \( t \) is in weeks.
**Equilibrium Populations:**
The DE has two equilibrium populations, \( p_1 = 0.8 \) and \( p_2 = 4.2 \). These are constant solutions of the equation. Due to the Existence and Uniqueness Theorem, no other solutions can cross these equilibrium points. Solutions will either approach them asymptotically as \( t \to \infty \) or move away. Your phase line plot from the last lesson should demonstrate this.
**Task 1: Euler Approximation Method**
Use the Euler approximation method with a reasonable step size to answer the following questions:
1. If \( p(0) = 6 \) (i.e., 6,000 kg), what is the population after 10 weeks?
2. How close is this to the equilibrium point it’s approaching?
3. At what time does the population stop dropping by 100 kg per week?
4. When does the population come within 50 kg of equilibrium?
5. If \( p(0) = 0.5 \), when does the population die out?
6. If \( p(0) = 1.2 \), when does the population come within 100 kg of equilibrium?
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Transcribed Image Text:**Task 3.** The differential equation (fishpop) can be solved analytically. Do it. Then find the solution which satisfies \( p(0) = 6 \) and plot it. What happens to the population as time goes on?
*Explanation*:
- **Differential Equation (fishpop)**: Solve the given differential equation analytically.
- **Initial Condition**: The solution must satisfy \( p(0) = 6 \).
- **Plot**: Create a plot of the solution to visualize changes over time.
- **Population Analysis**: Analyze the trend of the population as time progresses.
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