 ## Combat Models: Who Will Win The Battle?

### Conventional Combat Model

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In this model we have two conventional forces fighting against each other. A possible model for the situation might be this system of differential equations:

b′(t) = -b(t) - 4 f(t) + 5
f ′(t) = -3b(t) - 2 f(t) + 5

b(0) = bo,  f(0)) = fo

where b(t) represents the British army strength at time t, and f(t) represents the French army strength at time t. bo is the initial British army strength, and fo is the initial French army strength. All quantites are measured in tens of thousands. As usual, carefully consider the relationships of the variables.

The British army's rate of change is dependent on three terms: -b(t) is called the operational loss rate—it accounts for losses incurred due to ordinary attrition not attributable directly to the enemy (examples might be accidents, plane crashes, deaths due to disease, etc.); - 4f(t) is the combat loss rate due to the enemy's attacks; 5 is a fixed reinforcement rate. A similar analysis may be made of the French army's rate of change.

In executing your Mathematica analysis of this model use the skills that you learned in the preceding two analyses. Follow these guidelines:

1. Use sensible naming conventions for your results.

2. The values of b(t) and f(t) change extremeley rapidly in this model. Because of this, you may find that using an extremely short domain for t is prudent. I have suggested such domains next to each problem.

3. Using the plotting option PlotRange->{{0,20},{0,20}} is helpful, giving you a view of the portion of the solution curve that is really relevant.

Which army, if any, wins under each of the following initial conditions, given in the form (bofo)? Estimate the number of troops left in the victorious army after the battle.

1. (20, 20), using the interval 0 ≤ t ≤ 4
2. (21, 20), using the interval 0 ≤ t ≤ 1
3. (20, 21), using the interval 0 ≤ t ≤ 1

Finish up your analysis by showing all three of your solution curves on the same plot by employing the Show command.

If you're finished with the above analysis, you may go on to the next page for an overview of the results you should have found. 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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