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Applications of Systems of Differential Equations
Predator-Prey Problems
(continued from last page...)
Exercises
In each of the following exercises the instructions are the same:
Produce the rabbit-fox graph of each Lotka-Volterra model given, on the
interval 0 ≤ t ≤ 10. Read the solution into the variable rabfox#,
and the plot itself into the variable rabfoxplot#, where
# represents the number of the exercise you are doing, i.e.
either 3, 4, 5, or 6. (We already did 1 and 2 while I was helping you.
When you have finished creating the required graph for each exercise you
should click on the "Check Answer" button to check your answer for
accuracy.
EXERCISE 3
| System |
Initial Conditions |
Check
Answer |
r′(t) = 2 r(t) - 0.01 r(t) f(t)
f′(t) = - f(t) + 0.01 r(t) f(t)
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r(0) = 120
f(0) = 200
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EXERCISE 4
| System |
Initial Conditions |
Check
Answer |
r′(t) = 2 r(t) - 0.01 r(t) f(t)
f′(t) = - f(t) + 0.01 r(t) f(t)
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r(0) = 100
f(0) = 200
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Note: this graph should be a single point! Unfortunately Mathematica doesn't manage to show this point using the standard ParametricPlot command. In order to make up for this shortcoming you could instead use the ListPlot command, as shown below. I plotted just one point by substituting a value of t into the ordered pair we've been plotting in all the other graphs. (I chose to plug in t=2, but the t-value I used is irrelevant, provided it lies in our domain of 0 ≤ t ≤ 10, since all values produce exactly the same point!)
rabfoxplot4 =
ListPlot[{{rabfox4[[1, 1, 2]],
rabfox4[[1, 2, 2]]}}/.t->2,
PlotStyle->PointSize[.01]]
What does
this single point graph mean as far as the dynamics of the two populations are concerned? What would the graphs of
rabbits vs. time, and foxes vs. time look like? |
EXERCISE 5
| System |
Initial Conditions |
Check
Answer |
r′(t) = 2 r(t) - 0.01 r(t) f(t)
f′(t) = - f(t) + 0.01 r(t) f(t)
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r(0) = 400
f(0) = 100
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EXERCISE 6
| System |
Initial Conditions |
Check
Answer |
r′(t) = 2 r(t) - 0.01 r(t) f(t)
f′(t) = - f(t) + 0.01 r(t) f(t)
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r(0) = 800
f(0) = 20
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Note: In this case Mathematica will not form a complete orbit unless you include the PlotRange->All option, so make sure you do! |
To finish off this laboratory, we're going to make one last picture...
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If you're lost, impatient, want an overview of this laboratory assignment, or
maybe even all three, you can click on the compass button on the left to
go to the table of contents for this laboratory assignment.
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