project

The predator-prey model

MTH 347, Project.

DUE Thursday, December 5, 2019.

Instructions. You can work alone or in groups of 2 or 3 and hand in one report per group.

All members of the group are to be involved in the computer part as well as in the write-up.

You should respond to the questions posed below and write your report in the form of a paper

with an introduction and conclusion. Your report should include at least two pictures.

Preliminary step. Open MATLAB and type pplane8 in the command line. If this program

is on your computer (and in the MATLAB path) then the pplane window will open. If not,

then you need to download the program by visiting

http://math.rice.edu/~dfield/

Go to this site in a browser and select ‘pplane8.m’ and save the file as pplane8.m on your

computer. As long as this file is in the MATLAB path, you will be able to enter the word

pplane8 in the command line and the program will run. It produces menu driven windows that

are somewhat self-explanatory.

When you run pplane8, notice that one of the menus of the setup screen is labeled gallery.

You will want to select ’predator prey’ from that menu for the problem below. Then you enter

various parameter values and hit ’proceed’ to go to the next step. That opens a display screen

with its own menus. Under the heading solutions you will want to try ’plot several solutions’

and ’find an equilibrium point’.

The problem. Investigate the non-linear, predator-prey system

x′ =0.06x− 0.0004yx, y′ = − .08y + 0.0002xy.

The system models the populations of rabbits x and foxes y in a hypothetical nature pre-

serve, where time t is measured in months.

Begin your report with an introduction of the problem. In later paragraphs, deal with the

following questions and tasks.

Step 1. Produce a picture showing the direction field and phase portrait of the system. Plot

a selection of solution curves (trajectories) including the one referred to in Step 2 below. Also,

plot the equilibrium point. You will need to experiment with the minimum and maximum

populations in order to obtain a window that shows the equilibrium point and a good view of

1

trajectories. Explain the meanings of the direction arrows, the trajectories, and the equilibrium

point (in terms of rabbits and foxes). How can one tell from the picture which population is

the predator and which is the prey? In other words, how can we tell whether foxes eat rabbits

or rabbits eat foxes?

Step 2. Assume that at time 0, the preserve is stocked with 200 rabbits and 50 foxes. For

a second picture, obtain a plot of both the rabbit and fox populations versus time (using the

graph* menu of the display screen). Discuss the behavior of the populations over time. Your

discussion should include estimates of the minimums and maximums of each population and

when they occur. What is the period of oscillation of the rabbit and fox populations? Is it

reasonable that such populations would be cyclical? If, at a low point in the fox population, an

illness would wipe out the remaining foxes, what would then happen to the rabbit population

according to the model? Would it still be cyclical?

Step 3. Assume you are managing the preserve discussed in Step 2. You recognize that the

rabbit and fox populations are quite unstable, with populations swinging dramatically between

high and low values, so you plan to trap and remove a number of foxes. Based on the pictures,

at what point should you take this step and about how many foxes should you remove in order

to achieve the goal of making the population somewhat more stable? Discuss options and

observe what happen if the timing or the number is wrong. At what point on the trajectory

would the removal of foxes have the most destabilizing effect?

2

Grading Rubric for Projects

Mathematical Writing assignments involve writing skills as well as mathematical skills, so

the evaluation criteria include both. The criteria for this assignment are:

1. Comprehensiveness (30%):

(a) all aspects of the assignment are covered;

(b) all questions stated in the assignment are answered;

(c) explanations are thorough;

(d) supporting arguments are included.

2. Mathematical Correctness (30%):

(a) appropriate mathematical and computer methods are used;

(b) mathematical steps are correct;

(c) mathematical logic is clear.

3. Expository Quality (30%):

(a) organization is logical and includes an introduction and conclusion;

(b) transitions from one idea to the next are smooth;

(c) mathematical work and computer output are smoothly integrated into the text;

(d) mathematical statements are presented within complete sentences;

(e) any figures or computer output are described in the exposition;

(f) grammar, spelling and punctuation are correct.

4. Extras (10%):

(a) the overall impact is positive;

(b) creativity and innovation are apparent.

3