Lab 1 (Logistic Models) written by Prof. Jason Osborne 1 Assigned: Thursday, Oct. 3rd Due: Tuesday, Oct. 15th at the beginning of class. Directions: Read over Lab 1a and Lab 1b. Choose one of these two problems that you find the most interesting and write a lab report detailing your understanding about it (see 3 for general lab writing details). 1 Lab 1a Problem Statement: In this lab, we consider logistic models of population growth that have been modified to include terms that account for “harvesting”. In particular, you should imagine a fish population subject to various degreees and types of fishing. The differential equation models are given below. Using the value you find from the (On Title Page) paragraph of Section 3, you should identify the parameter values k,N,a 1 and a 2 in this choice/row of the Table below. The values should be used by your team throughout the rest of the lab. In your report, you should include a discussion of the meaning of each variable and parameter and an explanation of why the models are written the way they are. We have discussed three general approaches that can be employed to study DE: Numerical techniques yield graphs of approximate solutions, geometric/qualitative techniques provide predications of the long- term behavior of the solution and, in special cases, analytic techniques provide explicit formulas for the solution. In your report, you should employ these techniques as you find appropriate to help convey your understanding of these problems. Choice k N a 1 a 2 1 0.25 4 0.16 0.25 2 0.50 2 0.21 0.25 3 0.20 5 0.21 0.25 4 0.20 5 0.16 0.25 5 0.25 4 0.09 0.25 6 0.20 5 0.09 0.25 7 0.50 2 0.16 0.25 8 0.20 5 0.24 0.25 9 0.25 4 0.21 0.25 10 0.50 2 0.09 0.25 1. (Logistic growth with constant harvesting) The equation dp dt = kp · 1 - p N - a (1) represents a logistic model of a population growth with constant harvesting at a rate a. For a = a 1 , what will happen to the fish population for various initial conditions? Note: Equation (1) is autonomous, so you can take advantage of the special techniques that are available for autonomous equation. Any plots that you might include using these techniques should follow the style layed out in the (On Figues and Plots) paragraph of Section 3. 2. (Logistic growth with periodic harvesting) The equation dp dt = kp · 1 - p N - a(1 + sin(bt)) (2) is a non-autonomous (time dependent) equation that considers periodic harvesting. What do the parameters a and b represent? Let b = 1. If a = a 1 , what will happen to the fish population for various initial conditions? 3. Consider the same equation as in Number 2 above, but let a = a 2 . What will happen to the fish population for various initial conditions with this value of a? 1 This lab consists of two choices which are essentially Lab 1.3 and Lab 1.5 of Blanchard, Devaney, and Hall(BDH, Differential Equations, 4th ed. pg. 144). 1
Transcript
Lab 1 (Logistic Models)written by Prof. Jason Osborne1
Assigned: Thursday, Oct. 3rdDue: Tuesday, Oct. 15th at the beginning of class.
Directions: Read over Lab 1a and Lab 1b. Choose one of these two problems that you find the most interestingand write a lab report detailing your understanding about it (see 3 for general lab writing details).
1 Lab 1a Problem Statement:
In this lab, we consider logistic models of population growth that have been modified to include terms thataccount for “harvesting”. In particular, you should imagine a fish population subject to various degreees andtypes of fishing. The differential equation models are given below. Using the value you find from the(On Title Page) paragraph of Section 3, you should identify the parameter values k,N, a1 anda2 in this choice/row of the Table below. The values should be used by your team throughout the restof the lab. In your report, you should include a discussion of the meaning of each variable and parameterand an explanation of why the models are written the way they are.
We have discussed three general approaches that can be employed to study DE: Numerical techniquesyield graphs of approximate solutions, geometric/qualitative techniques provide predications of the long-term behavior of the solution and, in special cases, analytic techniques provide explicit formulas for thesolution. In your report, you should employ these techniques as you find appropriate to help convey yourunderstanding of these problems.
1. (Logistic growth with constant harvesting) The equation
dp
dt= kp ·
(1 − p
N
)− a (1)
represents a logistic model of a population growth with constant harvesting at a rate a. For a = a1,what will happen to the fish population for various initial conditions?
Note: Equation (1) is autonomous, so you can take advantage of the special techniquesthat are available for autonomous equation. Any plots that you might include using thesetechniques should follow the style layed out in the (On Figues and Plots) paragraph of Section3.
2. (Logistic growth with periodic harvesting) The equation
dp
dt= kp ·
(1 − p
N
)− a(1 + sin(bt)) (2)
is a non-autonomous (time dependent) equation that considers periodic harvesting. What do theparameters a and b represent? Let b = 1. If a = a1, what will happen to the fish population for variousinitial conditions?
3. Consider the same equation as in Number 2 above, but let a = a2. What will happen to the fishpopulation for various initial conditions with this value of a?
1This lab consists of two choices which are essentially Lab 1.3 and Lab 1.5 of Blanchard, Devaney, and Hall(BDH, DifferentialEquations, 4th ed. pg. 144).
1
In your report you should address these three questions, one at a time, in the form of an essay. Begin questions1 and 2 with a description of the meaning of each of the variables and parameters and an explanation of whythe differential equation is the way it is. You should include pictures and graphs of data and of solutionsof your models as appropriate. Refer to Section 3 for additional details on the format of this essay. Note,one carefully chosen picture with a well written caption is extremely helpful and worthwhile, but a thousandpicture isn’t worth much of anything.
The person on your team with first letter of Last (Given) name closest to A should record the last digitof their U#. Consider a 0 digit to represent the number 10. Using this value, identify the correspondingrow from the Table. This row is your data to use in answering questions 1-3 above.
2 Lab 1b Problem Statement:
(continued on next page)
2
3
3 Lab 1a and Lab 1b Report Writing Details:
• (On General Report Grading Rubric) Your lab reports will be graded largely on presentationand readability as well as correctness of solution. Your reports must be typed2 and must bereadable as technical documents should be read, namely, left to right and top to bottom.It will affect your grade significantly if your report requires the reader to “connect the dots” to followyour arguements.
• (On Lab Report Length and Style Issues) There is a limit of four (4) pages for this report. This4 page limit includes figures/tables of data/plots. That is, you have 4 pages (10pt font, 1 inch margins,single spacing, front page only) to convey your message to a reader. Material on pages > 5 will not beconsidered and discarded.
– (On Title Page) On the title page (which does not count toward the 4 page limit) you shouldinclude the names (Last (Family) Name, First (Given) Name and U# of each of your teammates.If you choose Lab 1a, the person on your team with first letter of Last (Given) name closest to A
should record the last digit of their U#. Consider a 0 digit to represent the number 10. Includethis number on your title page as it corresponds to your data set choice (in Lab 1a).
– (On Figures and Plots) Figures and Plots (like Direction Fields or Phase Lines or Solutionsy(t)) must have captions that clearly explain what the figure depicts. The caption should bereadable independent of the text of your report. That is, the readers should be able to read theFigure caption and understand what it is showing without having to refer to the text of the maindocument. Think of the caption as a mini-summary.
– (On Tables) You will likely have data to show in your report. The formatting should be doneas in the Table from this document. See also Tables 1.3 and 1.4 (BDH, Differential Equation 4thed, pg. 55) for how to format Euler’s method (MATLAB, DEtools or other) numerical output.
– (Equations) All significant equations should be on a seperate line and centered and num-bered like the equation
dy
dt= a(t)y + b(t) (3)
– (On References) You may use any resources at your disposal. MATLAB or Wolfram alpha orMathematica or Excel or whatnot. If you use Wolframe alpha be sure to say so by referencingthe question you asked and the output you received.
• (On Teamwork) You will/must work on the solutions and the official lab report in teamsof, ideally, 2 or, acceptably, 3. If you feel that a member of your team is not keeping up with theirresponsibilities, then you should ask them to join another team or start their own team. If you arehaving trouble finding a team, try to meet someone in your discussion group or in lecture next to you. Irealize that many engineering students are used to working in group and probably already know manyother engineering people, but I would encourage you to choose groups with a diversity of backgrounds.For example, an engineering group of 2 with 1 math major would be a well balanced group. How about1 math person, 1 physics person, and 1 chemistry person? Each subject has a different but equallyinteresting way of approaching problems, so it can only help to work with different people.
2Using Microsoft Word (where equations or, preferably and ideally, in LATEX (which was used to create this document). Forthe purposes of this lab report, LATEX has a learning curve, but once you get it you will be able to typeset nearly anythingpretty easily. See me (and/or the .tex sample on our website) if you would like to use LATEX.
