APPM 2360 Lab 3 Predator-Prey Systems

Lab Goals and Instructions

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:

• Names
• Student ID numbers
• Recitation section number, TA, and professor

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:

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.

The Models

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

 x' = - a x + b x y y' = c y - d x y (1)

(1) is the simplest of the predator-prey models.

Logistic Predator Prey Model

 x' = - a x + b x y y' = c y (1 - k y) - d x y (2)

(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:

• If the population is small, the rate of growth of the population is proportional to its size.
• If the population is too large to be supported by its environment and resources, the population will decrease.
Notice that without interaction with the foxes, the rabbit population is governed by y' = c y (1 - k y).  Letting  y' = f(y), we have f(y) = c y (1 - k y), which corresponds to the above assumptions.

Predator Prey Model with Migration

 x' = - a x + b x y y' = c y - d x y + k sin(w t) (3)

(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.

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.

Model (1)

1. Briefly discuss all the terms in system (1).  For example, what does the coefficient to the x y term in x' represent?
2. Find all of the equilibria for (1).
3. Volterra's Law of Averages states:  In system (1), the average predator and prey populations over the period of a cycle are, respectively, c/d and a/b.  In other words, the average population of each species over any of the cycles is a constant.  Compare these constants to the equilibria you have found above.  Explain.
4. Plot the flow (vector field) of (1).  You may find pplane to be very helpful for such plots; you can use either the online version or download the matlab versions(dfield and pplane, you must save both of these and use pplane to make it work) Also see Worksheet/HW 8 on the APPM 2460 website.
• For this question, create two plots, one for -6< x < 6, -6 < y < 6 and one for 0 < x < 6, 0 < y < 6.
• Use the first plot to inspect the equilibria (solution curves are not needed here).  Classify the equilibrium point (0, 0).  What is occurring at the other equilibrium point?
• Plot some solution curves ("orbits" or "trajectories" in the phase plane - see page 101 in the text for help with terminology) over the second vector field plot.
5. Let x(0) = 0.5 and y(0) = 1.0.  Use an ODE integrator to plot the curves x(t) and y(t), called component curves (in MVT they are "time series")(the ODE integrator in MVT is easy to use, or try ode45 in Matlab(learn from Worksheet/HW 8 on the APPM 2460 website) ) .
6. Examine the effect of the parameters on this system:  First change the value of a from 1.4 to 2.8.  Plot the solution curves using the same initial values as given above, x(0) = 0.5 and y(0) = 1.0.  Plot the solutions in the phase plane and describe the effect of this change and why this makes sense.
7. Reset a = 1.4 and change d from 1.5 to 5.0.  Repeat (6) with this new parameter value.
8. Refer to Section 7.2 of Farlow, Hall, McDill, and West for help in performing the following analysis.  In particular, the table on page 437 will be useful.
• Find the Jacobian matrix for the system.  Do this for each equilibrium point.
• Find the eigenvalues of the Jacobian matrix for each equilibrium point.
• Use the eigenvalues to classify the system (see p. 437 of the textbook).

Model (2)

1. Consider system (2) with k = 1/2.  Plot the flow (vector field) of (2); again, you only need to create a plot in the first quadrant. What are the equilibria for this system?
2. Let x(0) = 0.5 and y(0) = 1.0.  Use an ODE integrator to plot the component curves over a time period of 0 < t < 25.  Overlay these plots and label the curves appropriately (once again, you may want to use different colors for each curve).
1. Repeat for x(0) = 1.5 and y(0) = 0.5.
2. Repeat again for x(0) = 1.5 and y(0) = 1.5.
3. Compare the component curves to the solution curves that you produced above in part 1 (for model (2)).  How do these plots relate to one another?
4. Comment on each of these component curves; are they periodic?  Do they behave asymptotically?  What is the behavior of the populations in each case?

Model (3)

1. Consider system (3), set k = 0.2 and w = 5/(2*Pi).  What can you say about the long-term behavior of solutions?  Interpret your observations in terms of the behavior of the populations.  To answer this question, do the following:
1. Plot the component curves of (3) starting with the initial condition (5/3, 10/9).
2. Plot the solutions in the phase plane with an ODE integrator and compare them to the component curves.
3. Make sure you follow solutions long enough to be confident you are seeing the "long-term" behavior.
2. Do you see any problems with this model when the initial value of y is 0?
3. Compare the results of the three models.  Which model do you think is most realistic and why?  What are the shortcomings of these models, and what could one do to improve them?

Mathematical Visualization Toolkit.  This link is where you can find some of the necessary mathematical tools needed for this lab.

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.

References:
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