Calculus: Project 34

Project 34: Lanchester's Combat Models

Frederick William Lanchester was an English engineer who died after the end of World War II in 1946. He had a long time interest in aircraft and was one of the first to realize the extent to which the use of air power would alter warfare. The first combat model here was proposed in his 1916 book Aircraft in Warfare published during World War I. His book predicted that "the number of flying machines eventually to be utilized by ...the great military powers will be counted not by hundreds but by thousands ...and the issue of any great battle will be definitely determined by the efficiency of the aeronautical forces." This project investigates some of Lanchester's work on modeling combat.

Two forces numbering x[t] and y[t] are fighting each other. Lanchester assumed that if the forces were fighting a conventional war then the combat loss rates would be proportional to the total number of enemy forces. Expressed as a set of differential equations, this becomes

where a and b are positive constants representing the effectiveness of the y and x forces. Initial conditions would represent the initial strength of each force.

The Basic Combat Model
Begin your project with an explanation of the way these differential equations model combat. In particular, explain why the parameters a and b represent the effectiveness of the armies.

: A Solution of the Combat Equations

Figure 34.2 illustrates the course of combat between two forces of equal effectiveness, a=b=0.1, and different initial strengths. One force starts at 60,000 and the other at 50,000. After 12 units of time, the smaller force is annihilated and the larger force loses about 25,000 men.

You need to be careful with numerical results for these equations. Negative values of the variables won't make any sense to the model. With this in mind, perform some experiments to get a feel for the behavior of the model.

Explicit Experiments
Use the computer program AccDEsol to solve these differential equations for a larger value of tfinal >12. (a=b=0.1 and x₀=60,000, y₀=50,000.) Explain the odd behavior of the solutions for larger values of t. At what point do the numerical results fail to make sense in the combat model? Experiment with different values of the initial troop strengths, x₀ and y₀. What happens if the two forces are of equal effectiveness and equal strength?

Experiment with various values of the parameters a and b.

You need to organize the results of your experiments with the explicit time solutions of the equations. Hint 34.3 should give you some ideas about the behavior of the model but at the same time not be easy to summarize. It will be helpful to take another view of the solutions that makes the comparison more natural by plotting both forces on one graph.

Figure 34.1: The Flow of Battle

Flow Experiments
The differential equations of the basic combat model are autonomous. Why? This means that you can use the Flow2D computer program to experiment with the behavior of your model. Modify that program to solve the equations with a=b=0.1 and use all the initial conditions that you experimented with in the explicit solution experiments (Hint 34.3) in a single flow. At the very least, this will give you a way to compare all the results of those experiments at once. Experiment with the flow solution of the basic combat model when , using the same initial troop strengths as before. Compare the results obtained by explicit solution with the flow animations.

A basic kind of comparison is to compute with a fixed-size x force and different initial sizes of the y force. You could make a list of initial conditions, for example, using Mathematica's Table[.] command: initconds = Table[{ 40000,y} , { y,0,80000, 10000} ]

34.1 The Principle of Concentration

If one force can concentrate its entire strength on a portion of an opposing army and destroy it, it may then be able to attack the remaining portion of the opposing army with its remaining strength and destroy it. Lanchester believed air power would be so effective because of this. An air force can be deployed in full force against select portions of an opposing army.

Divide and Conquer
1) First consider two forces x and y of equal effectiveness. Suppose the x-force starts with 50,000 men and the y-force with 70,000 men. Use the rkSoln or the Flow2D program to determine the outcome of the battle if all 50,000 x-troops faced all 70,000 y-troops.
2) Let the x and y forces have equal effectiveness and let the x-force start at 50,000 and the y-force start at 70,000. Suppose now, however, that the x-force can concentrate its entire strength on 40,000 opposing troops before facing the remaining 30,000 y-force men in a second battle. How many men are left at the end of the first battle? What is the outcome of the second battle in this instance?

3) Suppose now that an x-force of 50,000 troops is outnumbered by 120,000 y-troops. Assuming that the x-force can strike a portion of the y-force in full strength and then face the other portion with its remaining men, is there any way for the x-force to win the battle? Justify your answer.

It would be convenient to have a simple analytical method to use to compute answers to questions like those in the exercise above. Computer simulations can probably give you the answers you need, but there is an easier way

34.2 The Square Law

Consider the case where the two forces have differing combat effectiveness. (In particular, we will use a=0.0106 and b=0.0544 for experiments. These coefficients have been used to model the course of the battle of Iwo Jima in World War II.) The outcome of the battle in this case is not as clear as with the battles above. A force of superior numbers may still lose to a smaller force if the combat effectiveness of the smaller force is high. We can still determine the outcome of a battle, however, by using the differential equations above to derive an integral invariant similar to the energy in a spring or the epidemic invariant or the predator-prey invariant from the CD text Section 24.4. Note that since

we can take the quotient of

and

to obtain

Now we can separate variables and integrate to get a relationship between x and y. (Review the invariants in CD Section 24.4 of the core text.)

The Combat Invariant
Perform the integration above, remembering that you will introduce an arbitrary constant while integrating. The solutions x[t] and y[t] will stay on the integral invariant for all time. Prove this once you have computed the formula as in CD Section 24.4.

The initial values of x and y will determine the value of the arbitrary constant introduced by the integration. What kind of curves are the integral invariants? For given values of a and b and initial strengths of the x and y forces, how can you use the integral invariant to determine who will win the battle?

Once you have found the invariant formula, you can derive Lanchester's "square law." What is the outcome of the battle if the arbitrary constant you introduced by integrating is 0? What is the outcome if the constant is negative? positive?

The Square Law
Complete the following sentences and derive the square law:

1. If bx₀²<ay₀², then ....
2. If bx₀²=ay₀², then ....

3. If bx₀²>ay₀², then ....

Consider the specific case of a=0.0106 and b=0.0544. Use the Flow2D program to solve the differential equations for several initial conditions. In particular, include an initial condition for which the two forces annihilate each other. Also include initial conditions for which the x force wins and the y force wins. What region of the x and y plane do the initial conditions have to be in for the x force to win? for the y force to win? for the forces to mutually annihilate each other? What is the connection between the square law and the flow?
34.3 Guerrilla Combat
Lanchester's combat equations have been modified to model combat in which one force is a conventional force and one force is a guerrilla force. The modification takes into account the fact that the fighting effectiveness of a guerrilla force is due to its ability to stay hidden. Thus, while losses to the conventional force are still proportional to the number of guerrillas, the losses to the guerrilla forces are proportional to both the number of the conventional force and its own numbers. Large numbers of guerrillas cannot stay hidden and, therefore, derive little advantage from guerrilla combat.
If x is a guerrilla force and y a conventional force, the modified combat equations are:

where here again a and b are constants representing the combat effectiveness of each force.
Just as above, we can calculate an integral invariant to determine how solutions to the differential equation behave. Take the ratio of and to obtain and integrate to obtain an expression involving x and y and an arbitrary constant. What is the shape of these integral invariants? Derive another combat law:
- Consider the case in which both forces are guerrilla forces. Then the combat equations become:
  
  Find the integral invariants in this case (they are the simplest of the three models you've seen). What do the invariants represent? Finally, derive another combat law that will enable you to predict the outcome of a battle given certain initial conditions. Use the Flow2D program to demonstrate your combat law.
  34.4 Operational Losses (Optional) Our final modifications of Lanchester's combat model will take into account operational losses. These are losses that occur through accidents, "friendly" fire, or disease. We will assume that the rate of these losses is proportional to the size of the force. The new combat model then takes the form:
  
  where a and b are the usual combat effectiveness constants and e and f are operational loss rates. It seems plausible that these should be small in comparison to a and b. This is a linear system like the ones we studied in the text.
- Suppose that a, b, e, and f are positive. Show that the characteristic roots of the linear dynamical system above are real and of opposite sign.
  In this instance, it is not possible to find an integral invariant as above. Notice, however, that the expressions describing rates of change are linear in x and y. If this is the case, it can be shown that if e and f are small compared to a and b then the solutions to these equations behave exactly like the solutions to the combat model without operational losses. You showed above that if the initial conditions are on a certain line through the origin, then the forces mutually annihilate each other. A unit vector that is drawn in the direction of this line is called a characteristic vector (eigenvektor in German). The key to extending the combat law to this case is in finding an invariant direction or unit vector in the direction.
  The computer can find these vectors and enable you to derive a combat law for the model with operational losses. To do this we first write the differential equation in matrix form.
  
  For example, in Mathematicawe can define a matrix in the computer as a list of lists,
  
  The command Eigenvectors[m] will then return two unit vectors. One will be in the wrong quadrant, but the other will give the direction of the line along which the initial conditions must lie in order for the two forces to mutually annihilate each other. For example, if a=b=0.1 and e=f=0.001, then the two eigenvectors are given in the list of lists,
  
  The second list is the vector
  
  which points along the line "y=x" in the first quadrant.
  The computer also has a command Eigenvalues[m] that returns the characteristic roots of the linear dynamical system. These are the same roots we studied in Chapter 23 of the core text,
  
  The roots are approximately r₁=0.099 and r₂=-0.101 in this case. The single command Eigensystem[m] returns the eigenvalues and eigenvectors. Eigenvectors and eigenvalues are related by the equation
  
  or
  
  In particular,
  
  for any constant x₀. In this case,
  
  is a solution of the differential equations for any initial vector
  
  In other words, if x₀=y₀, then x[t]=y[t]=x₀e^-0.101t
- - Consider the model above for two forces of equal combat effectiveness, a=b=0.1, and operational loss rates e=0.00001 and f=0.0001. Use the Eigensystem command to find the eigenvector-eigenvalue pairs of the matrix above. Once you have the direction of the line of mutual annihilation, use the Flow2D program to plot initial conditions for which the x force wins, for which the y force wins, and for which the forces annihilate one another. What is the combat law for these particular parameters? How large can the operational losses become before the eigenvector associated with the negative eigenvalue no longer lies in the first quadrant? Is the model useful if this happens?
    
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