Imperial powers frequently find themselves in a situation where they have defeated the military force of a nation, and now want to occupy the land of that nation. However, the people of that nation are hostile and resent the presence of the occupiers. The natives thus form resistance movements to make life unpleasant and dangerous for the occupiers. The native insurgent forces have no chance of forcing the occupiers out of the land; instead their goal is simply to cause problems for the occupiers in the hope that eventually public opinion or a loss of will causes the occupiers to withdraw.
We will model the numbers o(t) of occupiers and i(t) of insurgents using a simple pair of equations.
i' = o(α - Fi),
o' = βi + (C - ρo).
We can interpret these equations in this way. The number of insurgents tends to increase so long as occupiers are present. The parameter α represents the resentment and antagonism that the occupiers excite among the native population, and is positive. On the other hand, when insurgents interact with the occupiers, they may be killed, captured, or otherwise neutralized; or again, they may be bought off or convinced in some other way not to continue their activities. On the other hand, they may score military victories or take sympathetic losses that cause others to join them. The term F o i describes the interaction of the insurgents with the occupiers, and F describes the effectiveness of the occupiers in dealing with insurgents; it may be positive, negative, or zero.
The second equation indicates that the number of occupiers tends to rise as long as there are insurgents to deal with, but the occupying force has some desired number of forces that it wishes to attain. The willingness of the occupiers to increase their forces in response to insurgent action is described by β > 0, while the desired number of occupying forces is C, and the ability and willingness of the occupiers to make changes in forces is given by ρ > 0. The struggle these equations describe is sometimes called "asymmetric warfare", and that asymmetry is reflected in the equations. The first equation is nonlinear, and looks like one of the equations from the traditional predator-prey system used to model phenomena such as this. The second equation, however, is linear, and reflects the fact that the occupiers decide how many forces to put in the occupied country. The insurgents cannot really do much damage to the occupiers, so we neglect any reduction of occupying forces due to actual interaction with the insurgents.
The occupying country typically wants to reduce its forces to some maintenance level C. The question at hand is how to accomplish that.
Exercise 1: Find the equilibrium points of these equations, and interpret them in terms of the numbers of insurgents and occupiers. In other words, which of the equilibria are interesting in view of the assumptions about insurgents and occupiers?
Exercise 2: For each equilibrium point found in Exercise 1, linearize the differential equation around that point and discuss the stability of the equilibrium.
Evidently the parameter F is critical to the analysis of this system. If we could be sure that every time one occupier met one insurgent, then the insurgent would be killed or taken prisoner or persuaded not to be an insurgent any more, then F would be 1. Unfortunately for the occupiers, the insurgents are usually hard to recognize, and they melt into the ambient population during fights. Worse yet, in their search for insurgents the occupiers sometimes antagonize non-insurgents, causing others to become insurgents. If the occupiers were so ineffective as to detain four non-insurgents for every actual insurgent detained, and if half of those improperly detained became insurgents or had relatives that became insurgents in outrage, then F would be -1 (one insurgent detained, two new ones created).
Exercise 3: Describe the change in the phase portrait for this system as F goes from positive to negative values. I.e. sketch a sample phase portrait for F > 0 and for F < 0. Discuss the case when F = 0.
Exercise 4: A set of parameters
for the system is given in the applet below.
Plot several phase portraits using different parameter
values, and discuss the following questions.
How sensitive are the results to the choice of β?
How sensitive are the results to the value of ρ?
Under what circumstances does the number of occupiers actually approach the desired maintenance force size C?
for mathematical applications. There are
illustrating how we might use interactive web objects to
help students learn Calculus.