 ## Combat Models: Who Will Win The Battle?

### Guerilla Combat Model

(continued from last page...)

In this situation we suppose that both of the armies involved are waging guerilla warfare on the other. A possible model for the scenario may be the following system of differential equations:

x′(t) = -0.1 x(t) y(t)
y′(t) = -0.2 x(t) y(t)

x(0) = xo,  y(0) =yo

where x(t) represents one guerilla force strength at time t, call them the Xanudian Liberation Front, and y(t) represents the other guerilla force strength at time t, say the Yarovian People's Organization. Furthermore, xo is the initial Xanudian strength, and yo is the initial Yarovian strength. Let's suppose that all quantities are in thousands of troops. Carefully consider the relationships of the variables.

In a combat model, x′(t) and y′(t) are called the combat loss rates of the two armies. If you consider the two combat loss rates in the model above, they seem to obey common sense principles. The quantity, x(t) y(t), in both differential equations is a measure of the total number of inter-force encounters possible. The rates are negatively influenced by encounters with the opposing force, since when these forces encounter one another, they attempt to destroy their opponents. It would appear that in the current model the Xanudians are the better fighters, since they inflict a heavier combat loss rate on the Yarovians.

Now for our analysis of the model using Mathematica. I will assume in the following discussion that you have already done the previous two DE-systems-applications labs, and that you are therefore needing less step by step help with the details. Switch to Mathematica by clicking on the button at left. This will open up a fresh notebook for you. Then return here immediately for further instructions.

1. Let's begin by considering a situation in which both forces have an equal number of troops, say x(0) = 25, y(0) = 25, where both are measured in thousands. Use the NDSolve command, just as you did in the previous two labs, to solve the above system of differential equations, along with the initial conditions just given. Use the interval 0 ≤ t ≤ 30, and read the solution into the variable guerilla1.

2. That shouldn't have taken too long. Now make a phase-plane diagram by using ParametricPlot, just like you did when analysing predator-prey problems in the previous lab. Remember, you'll be plotting {x[t],y[t]}/.guerilla1[] this time, and your t interval is from 0 to 30. Include a PlotRange->All option in your command, and also an AxesOrigin->{0,0} option to force the axes to cross in the location you are used to, and assign the graph to the variable guerillaplot1.

3. Hmm, just about the most boring graph that we've seen in this course. Anyway, now for the important question: Who won?

Remember that your initial condition was (25, 25). This means that you start your solution (with respect to time) in the top right corner of the graph, and as time passes you move down and to the left along the solution line. You can determine who won by observing where this solution "curve" ended up. It looks like it stops at about (12.5, 0) on my plot. That would be 12,500 Xanudians, and zero Yarovians. I'd say that the Xanudians won! This is in agreement with our initial observation that the Xanudians are better fighters.

By the way, a couple of remarks: The model ceases to make sense if your solution curve ventures out of the first quadrant. (Why?) And, in general, what does the intersection of the solution curve with an axis (in standard position) mean in the "real world" for this model?

Well, maybe we should let the Yarovians have more troops to start with. Redo steps 1 and 2 above, but this time with initial conditions x(0) = 25, y(0) = 50. Of course, name your results guerilla2, and guerillaplot2 this time.

4. Who won the battle now? (You should have found that the two forces completely wiped each other out, kind of along the lines of USA/USSR nuclear arms policy during the cold war.) This result probably didn't surprise you. Let's redo steps 1 and 2 again, this time with initial conditions of x(0) = 25, y(0) = 75. In other words we're giving the Yarovian force an even larger army than in the last scenario. Rename your variables guerilla3, and guerillaplot3 this time.

5. At last, the Yarovians score a victory (about 25,000 of their troops survived)! What seems to be the basic requirement for a Yarovian victory in this model? It looks like they have to have more than twice the number of troops as the Xanudians, probably due to the fact that the Xanudians fight twice as effectively.

6. Finally, use the Show command to display all three graphs on the same coordinate system. (Is there any wonder that guerilla vs. guerilla warfare is said to obey the linear combat law?)

OK, that takes care of the guerilla vs. guerilla scenario. Let's move on to a slightly more interesting situation... 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.

ODE Laboratories: A Sabbatical Project by Christopher A. Barker

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