Miniproject 5

Initial due date: Sunday, April 2 at 11:59pm ET

Overview

This miniproject goes deeper into an idea we have encountered in class: Using differential equations to model population growth, featuring the logistic model.

Prerequisites: You'll need to be familiar with examples from class on setting up differential equations to model real-world systems; and how to use technology tools to generate symbolic and graphical solutions to DE's.

Background

The relevant background knowledge for this miniproject, including a definition of the logistic differential equation which is at the heart of this assignment, is given in your textbook, Section 2.7.1. Please read that entire section; it includes a very detailed example that might be helpful below.

Assignment

1. Suppose a population \(P(t)\) exhibits logistic growth, using the general logistic model:
   \[\frac{dP}{dt} = 0.02P\left(1-\frac{P}{500}\right), P(0) = 100\]

   (a) For which values of \(P\) is \(P\) an increasing function? Explain your reasoning using both English and math. (Note: This is not asking for the values of \(t\) for which \(P\) is increasing.)
   (b) Use a computer tool to produce a direction field for this differential equation. You can either generate it in your notebook, or generate it elsewhere and import a screenshot. But make sure that the viewing window on it is such that it highlights all the useful behaviors of the direction field.
   © What are the equilibrium solutions to this differential equation? Explain your reasoning using both English and math.
   (d) At what precise value of \(P\) (not \(t\)!) is the population increasing at its fastest rate? Explain your reasoning using both English and math.
   (e) If we multiply out the right side of the DE, we get
   \[\frac{dP}{dt} = 0.02P - 0.00004P^2\]
   Using this simplified form of the DE, solve the initial value problem using Wolfram|Alpha. Here is an example of how to use Wolfram|Alpha to solve an IVP. It will help if you take the result that Wolfram|Alpha gives you and use Wolfram|Alpha again to simplify it; you can do this by opening up a new Wolfram|Alpha tab, typing "simplify", and then entering the result you got initially. The simplified version will not contain any very large or extremely small numbers. Use this solution to find the time value where the population reaches 95% of its carrying capacity. (Both of these can be done using a computer, but be sure to include the code or a screenshot of what you enter.)

2. Consider a fish population that grows according to the model
   \[\frac{dP}{dt} = 0.05P - 0.000005P^2\]
   where \(t\) is measured in years and \(P\) is measured in thousands of fish.
   (a) Determine the population of fish at time \(t\) if initially \(P(0) = 1000\). You may use a computer to do any math you need, but be sure to include your code.
   (b) What is the carrying capacity of this population? Explain your reasoning using both English and math.
   © Suppose that the fish population is growing according to the model above, but it doesn't take into account fish being removed from the lake in which they live. Suppose that fish are harvested at a flat rate of 20,000 fish per year. Modify the differential equation to account for this. State the new differential equation clearly and explain why the new model correctly accounts for the harvesting of the fish.
   (d) Plot a direction field for the updated differential equation from the previous part along with 3-4 particular solutions. Are there any equilibrium solutions for the population? Explain your reasoning using both English and math.
   (e) Finally, suppose that the fish population is growing according to the original logistic model we started with, but instead of harvesting 20,000 fish per year, the number of fish harvested per year is 100 per year in the first year, 200 per year in the second, 300 pear year in the third, and so on. Modify the differential equation to account for this update. State the new differential equation clearly and explain why the new model correctly accounts for the harvesting of the fish. Then, plot a direction field for the updated differential equation from the previous part along with 3-4 particular solutions and discuss how this direction field is different from the one you made in part (d).

Notes on this Miniproject

• All mathematical computations, including finding algebraic solutions to DE's, can be done using a computer. But if you do, you need to "show your code". This means that you must clearly state what you entered into the computer to get your answer, so that the reader can reproduce your results. If you are using SymPy, this is no extra work because the code is part of the submission. If you use a different tool, for example if you use one of the direction field websites to create a direction field, then you will need to embed an image of your results (see below) and a clear statement of the code you entered to get it. If you use Wolfram|Alpha for anything, give the URL for the page that has your computation on it.
• If you use the Bluffton direction field website (https://homepages.bluffton.edu/~nesterd/apps/slopefields.html) then show your code as follows. First, once your direction field has been made, download the image using the download button in the upper right:

And then upload the image into your Jupyter notebook.

And then, grab a link to the code used to create the direction field and include that link in your submission. You can get this link by going to the Share button, clicking it, then looking for Short link, then clicking the Copy button next to it:

Do not click on the text "Short Link" because this just reloads the page. Click on "Copy". Then paste the link into your submission.

Submission and Grading

Formatting and special items for grading

Please review the section on Miniprojects in the document Standards For Student Work in MTH 302 before attempting to write up your submission. Note that all Miniprojects:

• Must be typewritten. If any portion of the submission has handwritten work or drawings, it will be marked Incomplete and returned without further comment.
• Must represent a good-faith effort at a complete, correct, clearly communicated, and professionally presented solution. Omissions, partial work, work that is poorly organized or sloppily presented, or work that has numerous errors will be marked Incomplete and returned without further comment.
• Must include clear verbal explanations of your work when indicated, not just math or code. You can tell when verbal explanations are required because the problems say something like "Explain your reasoning".

Your work here is being evaluated partially on whether your math and code are correct; but just as much on whether your reasoning is correct and clearly expressed. Make sure to pay close attention to both.

Miniproject 5 must be done in a Jupyter notebook using SymPy or another computer tool to carry out all mathematical calculations. A sample notebook, demonstrating the solution to a Calculus problem, can be found here. Study this first before writing up your work.

And please review the requirements above for including your code.

How to submit

You will submit your work on Blackboard in the Miniproject 5 folder under Assignments > Miniprojects. But you will not upload a PDF for Miniprojects. Instead you will share a link that allows me (Talbert) to comment on your work. As explained in one of the Jupyter and Colab tutorials, the process goes like this:

1. In the notebook, click "Share" in the upper right.
2. Do not share with me by entering my email. Instead, go to General Access, and in the pulldown menu select "Anyone with the link", then set the permissions to "Commenter".
3. Then click "Copy Link".
4. On Blackboard, go to the Assignments area, then Miniprojects. Select Miniproject 5.
5. Under Assignment Submission, where it says Text Submission, click "Write Submission".
6. Paste the link to your notebook in the text area that appears.
7. Then click "Submit" to submit your work.

I will then evaluate your work using the link. Specific comments will be left on the notebook itself. General comments will be left on Blackboard.