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
icon
Related questions
Question
**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?
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?
Expert Solution
trending now

Trending now

This is a popular solution!

steps

Step by step

Solved in 3 steps

Blurred answer
Follow-up Questions
Read through expert solutions to related follow-up questions below.
Follow-up Question
**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.
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.
Solution
Bartleby Expert
SEE SOLUTION
Similar questions
Recommended textbooks for you
Advanced Engineering Mathematics
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Mathematics For Machine Technology
Mathematics For Machine Technology
Advanced Math
ISBN:
9781337798310
Author:
Peterson, John.
Publisher:
Cengage Learning,
Basic Technical Mathematics
Basic Technical Mathematics
Advanced Math
ISBN:
9780134437705
Author:
Washington
Publisher:
PEARSON
Topology
Topology
Advanced Math
ISBN:
9780134689517
Author:
Munkres, James R.
Publisher:
Pearson,