Mathematical Modeling of Social Phenomena

Dr. Courtney Brown

Assignment #11

Use Phaser for this assignment. You are to examine, evaluate, unravel, discover, experiment with, and understand three versions of a differential equation systems model. The model is the Lanchester Combat Model with two armies (x and y). There are three relevant concepts related to these models. These are (1) the operational loss rate, (2) the combat loss rate, and (3) the reinforcement rate. The operational loss rate is casualties that are noncombat related (e.g., due to disease, accidents, desertion, etc.). The combat loss rate can result from conventional or guerilla warfare on the part of either or both of the armies. The conventional rate (for conventional forces engaged in combat) is proportional to the total number in each army. The parameter for this rate is called the "combat effectiveness coefficient." The assumption is that the army is out in the open and that each soldier is within the killing range of the opposing forces. For a guerilla force, the combat loss rate is proportional to the product xy, reflectinig the required interaction of the forces on a soldier by soldier basis. The reinforcement rate is typically a variable input in the model, but you can treat it as a constant.

Thus, the three versions of the model are:

I. Two conventional armies.
dx/dt = -ay - ex + f
dy/dt = - bx - jy + g
where f and g are the reinforcement rates. Parameters e and j are operational loss rates, while the parameters a and b are the combat loss rates.

II. One guerilla army and one conventional army, where x=guerilla army, and y=conventional army
dx/dt = -axy - ex + f
dy/dt = -bx - jy + g

III. Two guerilla armies
dx/dt = -axy - ex + f
dy/dt = -bxy - jy + g

You are to compare the dynamics of these three models. Utilize all of the skill and vocabulary that you learned in the previous assignment. Pick heuristically useful parameter values and initial conditions. Locate basins, attractors, zero vectors, separatrices (if any), etc. Create vector field diagrams for each model as well as interesting flow diagrams where trajectories are allowed to continue. Then ... and this is important ... you are to try to interpret the substantive implications of the dynamics that you discover. Enjoy!