Lab Goals and Instructions
Please read this section carefully. Not following the instructions, including the writing guidelines, can have a very negative impact on your grade!
This lab is due in lecture on Friday, December 5. YOU MUST ALSO SUBMIT YOUR PROJECT TO AMESS. Late labs are not accepted. You are encouraged to work in groups of no more than 3 people ( you may work with anyone currently in 2360, and you turn in only copy of the project), though you may work alone.
NOTE: to upload to AMESS your file must be in .pdf format. The computer labs in the engineering building have a version of Microsoft Word that will allow you to save in .pdf format. If you want the option to save in .pdf format on your personal version of Word 2007 then you will have to download the additional add in from the Microsoft website.
On your title page, clearly mark the following information for all group members:
If you do not include this information, as much as 5% will be deducted from your final score. Format is worth 20% of your grade. Please refer to the following writing guidelines for the expository sections of this report:
Writing Guidelines for Labs
You are required to know and follow the Writing Guidelines for all three labs.
Take a look at this sample project and examples of "HOW TO" and "HOW NOT TO" present your results.
Getting help: If you are having trouble using Matlab, some resources can be found on the Differential Equations Lab website. In addition, TA's will be holding regular office hours in the Applied Math Undergraduate Lab (ECCR 143) during the week that the lab is due.
Refer to InteractionModels.html and read the course textbook, pages 98-104 for an introduction to the predator-prey model.
This lab demonstrates the modeling predator-prey populations by first order systems. There are three models: (1) the predator-prey system, (2) the logistic predator-prey system, and (3) the predator-prey system with migration
Let x be the population of foxes (in hundreds) and let y be the population of rabbits (in hundreds) on an island. Note that x and y are functions of t, time in days. In this scenario, the foxes are the only predators of the rabbits and the rabbits are the only prey for the foxes. Some examples of systems of differential equations that describe the changes in the population of these two species are:
Predator Prey Model
y' = c y - d x y
(1) is the simplest of the predator-prey models.
Logistic Predator Prey Model
y' = c y (1 - k y) - d x y
(2) is a modified version of (1). In this scenario we assume that, in the absence of predators, the prey population obeys a logistic rather than an exponential growth model. The logistic model is based on two additional assumptions:
Predator Prey Model with Migration
y' = c y - d x y + k sin(w t)
(3) is yet another modification of (1). In this scenario, we add the assumption of periodic emigration and immigration of prey. Suppose that, by some means, the prey can leave the island but the predators chose to stay. We use a sine function as the forcing term that describes the migration. The amplitude of the migration term is given by k and the period is determined by w.
Questions and Issues to Address
Your lab report should answer the following questions and address the following issues. All plots and calculations mentioned in these questions should be included in the report.
For the following questions, let a = 1.4, b = 1, c = 2.3 and d = 1.5, unless otherwise stated.
This Help File gives some tips on using MVT, and som MatLab code for plotting vector fields, phase portraits, and such.
This lab was created by Keith Wojciechowski in March 2001 and revised in April 2002. Revised again by DBdR, April 2004, again by Andrew Barker, November 2005, and again by Dan Kaslovsky, November 2008.
Borrelli, Robert and Coleman, Courtney, Differential Equations: A Modeling Perspective, pp. 285-293, first edition, John Wiley & Sons, 1998
Diacu, Florin, An Introduction to Differential Equations, pp. 189-197, 225-228, first edition, W.H. Freeman and Company, 2000
Boyce, William and DiPrima, Richard, Elementary Differential Equations and Boundary Value Problems, pp. 493-499, fourth edition, John Wiley & Sons, 1986
Blanchard, Devaney and Hall, Differential Equations, pp. 136-143, 466, preliminary edition, Brooks/Cole, 1996
Farlow, Hall, McDill, West, Differential Equations and Linear Algebra, pp. 98-104, Prentice Hall, 2002