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A qualitative look at a fish population Example. A population of fish is being monitored in a lake. The fish population at any time t we'll call p(t), where p is measured in thousands of kilograms and t is measured in weeks. Then suppose p(t) satisfies the DE (fishpop) p' = p -0.2p² - 0.7, where p' = dp/dt. You'll see in a later lesson how such an equation can be determined. For now, you'll do some qualitative analysis of the equation and see what you can predict about the growth or decline of the population using only the DE itself, without knowing the solutions. Fill in the details of the following steps: 1. First, notice that the population doesn't change if p' = 0. The values of p at which this happens are called the equilibrium points. Find these equilibrium points to 1 decimal place accuracy. (Do this by setting the right-hand side of equation (fishpop) equal to zero and then solving the resulting quadratic equation for p.) These equilibrium values are constant solutions of the DE, that is, they are constant functions of time which are solutions. (Can you verify this?) 2. Now you'll learn how to draw a phase line plot of the solutions. A phase line plot is a 1- dimensional plot of a p-axis that shows where the solutions are increasing and decreasing. Use the p-axis shown below, and mark the equilibrium positions on it; you should have found these to be: p₁ = 0.8 and P₂ = 4.2. 0 Pi Р In between the equilibrium values, the population p is increasing if p' is positive, or decreasing if p' is negative. For the values of p for which p'>0, draw small arrows on the phase line pointing to the right, and for p' <0, point the arrows to the left. Some of the arrows have been drawn for you; fill in the rest. (Hint: To figure out where p' is positive or negative, either plug some p values into the right-hand side of (fishpop), or sketch a graph of p' vs. p).

A qualitative look at a fish population Example. A population of fish is being monitored in a lake. The fish population at any time t we'll call p(t), where p is measured in thousands of kilograms and t is measured in weeks. Then suppose p(t) satisfies the DE (fishpop) p' = p -0.2p² - 0.7, where p' = dp/dt. You'll see in a later lesson how such an equation can be determined. For now, you'll do some qualitative analysis of the equation and see what you can predict about the growth or decline of the population using only the DE itself, without knowing the solutions. Fill in the details of the following steps: 1. First, notice that the population doesn't change if p' = 0. The values of p at which this happens are called the equilibrium points. Find these equilibrium points to 1 decimal place accuracy. (Do this by setting the right-hand side of equation (fishpop) equal to zero and then solving the resulting quadratic equation for p.) These equilibrium values are constant solutions of the DE, that is, they are constant functions of time which are solutions. (Can you verify this?) 2. Now you'll learn how to draw a phase line plot of the solutions. A phase line plot is a 1- dimensional plot of a p-axis that shows where the solutions are increasing and decreasing. Use the p-axis shown below, and mark the equilibrium positions on it; you should have found these to be: p₁ = 0.8 and P₂ = 4.2. 0 Pi Р In between the equilibrium values, the population p is increasing if p' is positive, or decreasing if p' is negative. For the values of p for which p'>0, draw small arrows on the phase line pointing to the right, and for p' <0, point the arrows to the left. Some of the arrows have been drawn for you; fill in the rest. (Hint: To figure out where p' is positive or negative, either plug some p values into the right-hand side of (fishpop), or sketch a graph of p' vs. p).

Functions and Change: A Modeling Approach to College Algebra (MindTap Course List)
Functions and Change: A Modeling Approach to College Algebra (MindTap Course List)
6th Edition
ISBN:9781337111348
Author:Bruce Crauder, Benny Evans, Alan Noell
Publisher:Bruce Crauder, Benny Evans, Alan Noell
Chapter5: A Survey Of Other Common Functions
Section5.3: Modeling Data With Power Functions
Problem 3TU
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A qualitative look at a fish population Example. A population of fish is being monitored in a lake. The fish population at any time t we'll call p(t), where p is measured in thousands of kilograms and t is measured in weeks. Then suppose p(t) satisfies the DE (fishpop) p' = p 0.2p²0.7, where p' = dp/dt. You'll see in a later lesson how such an equation can be determined. For now, you'll do some qualitative analysis of the equation and see what you can predict about the growth or decline of the population using only the DE itself, without knowing the solutions. Fill in the details of the following steps: 1. First, notice that the population doesn't change if p' = 0. The values of p at which this happens are called the equilibrium points. Find these equilibrium points to 1 decimal place accuracy. (Do this by setting the right-hand side of equation (fishpop) equal to zero and then solving the resulting quadratic equation for p.) These equilibrium values are constant solutions of the DE, that is, they are constant functions of time which are solutions. (Can you verify this?) 2. Now you'll learn how to draw a phase line plot of the solutions. A phase line plot is a 1- dimensional plot of a p-axis that shows where the solutions are increasing and decreasing. Use the p-axis shown below, and mark the equilibrium positions on it; you should have found these to be: P₁ = 0.8 and P₂ = 4.2. 0 P₁= P₂= In between the equilibrium values, the population p is increasing if p' is positive, or decreasing if p' is negative. For the values of p for which p'>0, draw small arrows on the phase line pointing to the right, and for p' <0, point the arrows to the left. Some of the arrows have been drawn for you; fill in the rest. (Hint: To figure out where p' is positive or negative, either plug some p values into the right-hand side of (fishpop), or sketch a graph of p' vs. p).
Transcribed Image Text:A qualitative look at a fish population Example. A population of fish is being monitored in a lake. The fish population at any time t we'll call p(t), where p is measured in thousands of kilograms and t is measured in weeks. Then suppose p(t) satisfies the DE (fishpop) p' = p 0.2p²0.7, where p' = dp/dt. You'll see in a later lesson how such an equation can be determined. For now, you'll do some qualitative analysis of the equation and see what you can predict about the growth or decline of the population using only the DE itself, without knowing the solutions. Fill in the details of the following steps: 1. First, notice that the population doesn't change if p' = 0. The values of p at which this happens are called the equilibrium points. Find these equilibrium points to 1 decimal place accuracy. (Do this by setting the right-hand side of equation (fishpop) equal to zero and then solving the resulting quadratic equation for p.) These equilibrium values are constant solutions of the DE, that is, they are constant functions of time which are solutions. (Can you verify this?) 2. Now you'll learn how to draw a phase line plot of the solutions. A phase line plot is a 1- dimensional plot of a p-axis that shows where the solutions are increasing and decreasing. Use the p-axis shown below, and mark the equilibrium positions on it; you should have found these to be: P₁ = 0.8 and P₂ = 4.2. 0 P₁= P₂= In between the equilibrium values, the population p is increasing if p' is positive, or decreasing if p' is negative. For the values of p for which p'>0, draw small arrows on the phase line pointing to the right, and for p' <0, point the arrows to the left. Some of the arrows have been drawn for you; fill in the rest. (Hint: To figure out where p' is positive or negative, either plug some p values into the right-hand side of (fishpop), or sketch a graph of p' vs. p).
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In between the equilibrium values, the population p is increasing if p' is positive, or decreasing if p' is negative. For the values of p for which p'>0, draw small arrows on the phase line pointing to the right, and for p' <0, point the arrows to the left. Some of the arrows have been drawn for you; fill in the rest. (Hint: To figure out where p' is positive or negative, either plug some p values into the right-hand side of (fishpop), or sketch a graph of p' vs. p). 8-1 You now have a phase line plot of the solutions, which you can use to get an overview of the behavior. Use it to answer the following questions. Task 1. Suppose that the lake is initially stocked with 1200 kg of fish. Translate this statement into an initial condition for the DE. Over the next few weeks, what will the population do -- grow? die out? remain the same? Task 2. Same question as in Task 1, but suppose the initial population is 5000 kg of fish. Task 3. Draw a phase line plot for the DE dy dt = 4.ly - y² - 4.2.
Transcribed Image Text:In between the equilibrium values, the population p is increasing if p' is positive, or decreasing if p' is negative. For the values of p for which p'>0, draw small arrows on the phase line pointing to the right, and for p' <0, point the arrows to the left. Some of the arrows have been drawn for you; fill in the rest. (Hint: To figure out where p' is positive or negative, either plug some p values into the right-hand side of (fishpop), or sketch a graph of p' vs. p). 8-1 You now have a phase line plot of the solutions, which you can use to get an overview of the behavior. Use it to answer the following questions. Task 1. Suppose that the lake is initially stocked with 1200 kg of fish. Translate this statement into an initial condition for the DE. Over the next few weeks, what will the population do -- grow? die out? remain the same? Task 2. Same question as in Task 1, but suppose the initial population is 5000 kg of fish. Task 3. Draw a phase line plot for the DE dy dt = 4.ly - y² - 4.2.
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