Mathematics & Science Learning Center
Computer Laboratory

Applications of Systems of Differential Equations

Combat Models: Who Will Win The Battle?

Mixed Combat Model

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In this model we have a guerilla force fighting against a conventional army. A possible model for this situation might be this system of differential equations:

where g(t) represents the guerilla force strength at time t, and c(t) represents the conventional force strength at time t, g_o is the initial guerilla strength, and c_o is the initial conventional force strength. All numbers are measured in thousands. Once again, carefully consider the relationships of the variables.

This time the combat loss rates of the two forces aren't symmetric with respect to one another. The guerilla combat loss rate is proportional to the number of guerilla-conventional encounters, whereas the conventional combat loss rate is proportional only to the number of guerillas. Think about their differing styles of warfare, and you might see why this should be so. Guerillas don't choose to fight a conventional force openly, but favor tactics such as ambush, and sneak raids on army bases.

1. Solve the mixed combat system above with the initial conditions g(0) = 10, c(0) = 15. Use the interval 0 ≤ t ≤ 10, and read the solution into the variable mixed1. Follow this up with a phase-plane diagram, which you assign the name mixedplot1.

2. So who won this time? It looks like the solutions ends around (0, 5). Since the first coordinate represents guerillas, and the second represents the conventional army, it would appear that the guerillas were all killed, and 5,000 conventional troops survived.

   Redo the problem using each of the following ordered pairs as initial conditions of the form (g_o, c_o): initial conditions (8, 15), (9, 15), (11, 15), (12, 15), (13, 15). Name your solutions and plots appropriately. (Oh, on the last one, stop t at 5 instead of at 10—this will prevent a Mathematica error.)

3. Do the guerillas ever win? If so, under what conditions? Use Show to display all six of your mixedplots simultaneously.

   Note: The fact that some of your solutions go into the fourth quadrant is meaningless here. The numbers represent troop strengths, and hence they must be non-negative. You need to be smart enough to recognize the limitations of your model.

Let's now move on to consider the most tradional situation...

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