R/simecol Simulation Model for the Battle of Iwo Jima

José Romero

6 years ago

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It was about 71 years ago today that the Second World War on the Asian front ended. After the tragic Little Boy and Fat Man detonations over Hiroshima and Nagasaki on the 6th and 9th of August 1945, respectively, the Japanese surrendered in August 14th.

When I taught the Simulation and Modeling course for the Systems Engineering department at the UNEFA, we used to do an exercise in class on simulating the battle of Iwo Jima, one of the most decisive and sanguinary battles in the Pacific. This simulation was based on a System Dynamics model derived from the Lanchester Laws of combat and was implemented in R using the simecol/deSolve/FME packages. Prior to the class, we had covered the topic of Forrester diagrams and System Dynamic models in general1, the Lanchester Laws of combat and of course a bit of background on the use of the simecol and related packages for dynamic model fitting and simulation.

In this post, I will present a brief summary on the Lanchester Laws of Combat, some historical background on the battle of Iwo Jima and then go into the simulation model itself, detailing its implementation and data fitting in R through the said packages. Subsequently, I will contrast the model’s output with the real historical data and derive some interpretations. To conclude the post, I will pose some “what if …” questions on alternative hypothetical scenarios and offer some thoughts about possible applications of simulation models like this one for teaching history in classrooms.

The Lanchester Models of Combat

Frederick Lanchester was a British engineer well-known for his inventions and contributions to the budding automotive industry at the end of the nineteenth century (he’s the guy that invented the accelerator pedal, for example). Lanchester also made some contributions to the theory of aerodynamics and being concerned over the ways in which the aeronautical industry could change warfare (he was more of a pacifist, really), he set out to study the dynamics of warfare and develop mathematical models of battle through systems of differential equations, now known as the Lanchester Laws of Combat. These differential equation models are fairly abstract – they leave out many details that we commonly associate with battles and focus on the way in which number of combat units of two armies decrease over time.

Without loss of generality, we will represent in each time instant \(t\) the number of troops or combat units of the red army as \(R(t)\) or \(R\) and of the green army as \(G(t)\) or simply \(G\). At the start of the battle, at \(t=0\), \(R(0)\) and \(G(0)\) represent the number of combat units of each army. When \(R(t)\) or \(G(t)\) reach 0 at any given time, the corresponding army is left without troops and so the battle ends with the opposing army as victorious. Lanchester proposed two different scenarios for battle, according to the warfare technology used:

First Lanchester Law

In the first Lanchester Law, two armies confront each other in hand-to-hand combat. At each time instant, the number of casualties of each army is proportional to the number of confrontations between the combatant units and the effectiveness or lethality coefficient of the enemy army. This is represented mathematically by the following differential equations:

\[\begin{align*}\dfrac{dR}{dt} &= -g \cdot R \cdot G \\ \dfrac{dG}{dt} &= -r \cdot R \cdot G\end{align*} \]

Derivatives \(\tfrac{dR}{dt}\) and \(\tfrac{dG}{dt}\) indicate the way in which the battle unfolds and represent the rate of attrition of the red and green armies, respectively. The \(r\) y \(g\) parameters are the effectiveness coefficients of each army and represent the number of casualties on the enemy side inflicted during one time instant at each confrontation (represented by \(R \cdot G\)). This model of combat is also referred to as the the model of “unaimed fire”, the word “fire” in this case not necessarily referring to artillery fire but rather to the directionality of discharge of the lethal armament used in combat. In fact, this model, being applied mainly to battle scenarios with no artillery can also be used for battles with low-precision artillery.

If we divide one equation by the other and isolate the red and green terms on each side, we obtain \(r \cdot dR = g \cdot dG\). In turn, if we integrate (with respect to \(t\)) this last equation we can see that at each time instant, the expression \(r\cdot R(t) – g \cdot G(t)\) remains constant. The terms \(r\cdot R\) and \(g \cdot G\) are called the fighting strengths of the reds and greens, respectively. The fighting strengths, which could also be considered as the number of troops eliminated the the red and green armies, determine result of the battle. Since \(r\cdot R(t) – g \cdot G(t)\) is constant throughout the whole time, the reds will win if \(r\cdot R(t) – g \cdot G(t) > 0\), the greens win if \(r\cdot R(t) – g \cdot G(t) < 0\) and if \(r\cdot R(t) – v \cdot G(t) = 0\), then both sides will be wiped out simultaneously. Since the fighting strengths are linear in \(R\) and \(G\), the first Lanchester Law is also known as the linear law of combat.

We can infer some characteristics of combat scenarios based on this linear law of combat:

An army’s efficiency parameter and its number of troops are equally important factors to its fighting strength (ie. if you double the number of troops but decrease the efficiency to one half, the fighting strength will be the same).
If you manage to divide the enemy in two contingents and confront each contingent successively, this will not alter the result of the battle.

Second Lanchester Law

This law supposes a scenario where both sides use aimed artillery fire, hence the casualty infliction mechanism no longer depends on the number of hand-to-hand combat confrontations being produced, but rather on the number of troops firing upon the enemy:

\[\begin{align*} \dfrac{dR}{dt} &= -g \cdot G \\ \dfrac{dG}{dt} &= -r \cdot R\end{align*} \]

Again, dividing one equation by the other, isolating the terms corresponding to each army and integrating, we obtain the following constant expression: \(r\cdot R^2 -g \cdot G^2=0\). This time, the fighting strengths \(r\cdot R^2\) and \(g \cdot G^2\) are quadratic on \(R\) and \(G\), hence the alternative name of “square law of combat” being applied to Lanchester’s Second Law. In two armies with equal number of troops, the more efficient one will prevail. If however, one army manages to divide the enemy in two or more contingents and confront each contingent successively, then it will certainly win over the other. This is what Von Clausewitz and Sun Tzu meant by advising generals not to divide their troops because each division would significantly reduce an army’s fighting strength. In other words, if an army is numerically superior, then it must combat undivided against a more efficient enemy. As an example of this principle, Hannibal confronted a qualitatively superior enemy at Cannas by dividing him and prevailed. Conversely, the American strategy of flooding the island of troops at Iwo Jima proved effective in managing to establish a beach head.

Applications of the Lanchester combat models and further elaborations

Lanchester originally applied his model to the study of Nelson’s naval tactics in the Battle of Traffalgar, that famous naval battle of the Napoleonic Wars. In more recent literature, Lanchester models were applied to the Kursk battle in 1943 involving panzers in the German and Soviet sides, the battle of Ardennes (again in World War II) and the naval conflict in the Atlantic in which the German U-Boats inflicted damage on the Allied convoys. In the Engel 1954 paper we find an application to the Battle of Iwo Jima, based on the data compiled by Cliiford and Moorehouse. On this last point, I must remark that the model fits to these battles has been partially successful due to the fact that under normal battle conditions, it is quite difficult to accurately compile day-to-day data of casualties on both sides.

The Lanchester models themselves can be further generalized to include reinforcement like so:

\[\begin{align*}\dfrac{dR}{dt} &= f_r(t) – g \cdot G \\ \dfrac{dG}{dt} &= f_g(t) -r \cdot R \end{align*} \]

Here, \(f_r(t)\) and \(f_g(t)\) represent the reinforcement rates of the red and green armies respectively. This is the model we will use for our Battle of Iwo Jima simulation, as the American side brought in reinforcements on several occasions, while, as we shall see, the Japanese had none.

If we are willing to relax our assumptions that the battle is based wholly on either an aimed or an unaimed fire combat model, then we get Bracken’s generalized Lanchester Model:

\[\begin{align*} \dfrac{dR}{dt} &= -g \cdot R^q \cdot G^p \\ \dfrac{dG}{dt} &= -r \cdot R^p \cdot G^q\end{align*} \]

If we define \(\alpha=1+p-q\), we get \(gG^\alpha-rR^\alpha\) as the constant quantity and we can easily see how the first and second Lanchester models are particular cases of this model. However (see MacKay, 2005), when fitting this model to any particular battle there is an ambiguity in the estimates for \(p\) and \(q\), as several best fitting estimates could be obtained.

Some historical background on the Battle of Iwo Jima2

The island of Iwo Jima3 is a tiny island of some 8 square miles situated about 750 miles south from Tokyo in the Japanese mainland and halfway between Saipan and Tokyo. Were it not for this last fact, the place would not have been the stage of a major battle in the Second World War and would probably remain largely unknown today. By the summer of 1944, the Americans had gained control of Saipan and the Mariana islands, from where they staged bombing raids to mainland Japan with long range B-29 Superfortress bombers. However, the losses inflicted upon these bomber planes by Japanese fighter planes deploying from the two airfields in Iwo Jima were unsustainable to the Americans and furthermore, the island’s radio station relayed early warning reports to the mainland to ready themselves for the bombing raids. For these reasons, it was of strategic importance to the Americans on their northward multi-pronged move towards Japan to seize control of Iwo Jima. And, naturally, the Japanese knew this.

For the Japanese, Iwo Jima was more than just strategically significant. Iwo Jima was sacred Japanese soil. On account of this, an eventual American invasion of Iwo Jima would be a heavy blow to the Japanese morale as for more than four thousand years, no foreign country had invaded Japan. Therefore, the Japanese were not ready to surrender the island without first inflicting heavy losses upon the American forces. Emperor Hirohito intrusted Lietenant General Tadamichi Kuribayashi4 with this mission.

Kuribayashi drew up his plan for the defense of the island upon the following premises:

The Japanese would engage in combat from underground, not on the surface of the island.
The battle would be defensive, the idea being to seriously erode the American forces.
No Japanese soldier would survive.
Every Japanese was to kill 10 Americans before being killed.

Kuribayashi’s strategy stood apart from the usual Japanese tactics to which the Americans had become accustomed. He specifically forbade the famous “Banzai” suicide attacks to repel the American disembarkation on the island, which besides, were completely futile to prevent it, since the Japanese would not be receiving air or naval support, and hence no reinforcements. Instead, he began construction of a gallery of underground tunnels connecting a network of bunkers and pillboxes. By the time the Americans arrived, the Japanese had completed over 11 miles of tunnels and placed large artillery pieces protected by thick steel walls from within Mount Suribachi.

For their part, Americans were also preparing for what they designated as Operation Detachment (the invasion of Iwo Jima). According to their intelligence reports, there were some 14,000 Japanese troops on the island (there were really over 21,000). Before D-Day, and based on their experience at Saipan, Tarawa and Peleliu, the Marines requested for 10 days of previous bombings. The Navy finally acquiesced and on the 16th of February, 1945 began a 3-day bombing. Previously, though, B-29s had completed a 72 day bombing campaign.

The bombing campaign on Iwo Jima had been devastating … but only on the surface. Inexplicably, not only had the Japanese defensive fortifications not diminished, according to an American intelligence report, but they had grown in number from 450 to 750 3 months after the bombings began. For the unsuspecting American forces, the real surprise lay waiting for them underground…

On D-day (February 19th), the Americans disembarked 54,000 troops. During the first moments, many thought that no Japanese had survived the bombings and that the estimates on the number of Japanese troops were exaggerated. Then, as the troops made their way 300 yards into the beach and the spot became congested with troops and equipment (what Kuribayashi was waiting for), all hell broke loose. There was crossfire from Mount Suribachi and some hills to the north of the island. The beach offered no protection, as the fine and ashy volcanic sands made it impossible to dig trenches. As Japanese fire impacted upon the disembarkation boats, the amphibious vehicles got stuck in the fine sands. The beaches of Iwo Jima became an infernal chaos in the carnage that ensued.

During the first days of battle, it seemed like the Americans had made no progress at all. They were fighting an enemy they couldn’t see – an enemy who fought from under the ground and usually at night. Upon burning underground burrows where Japanese were thought to be with flamethrowers, Americans were surprised to see Japanese troops re-emerge from burrows that were thought to be “cleansed” days ago. What the unsuspecting Americans didn’t know is that these holes and burrows were part of a vast network of underground tunnels. In the meantime, from the 20th to the 23rd day of February, the Marines attempted to take Suribachi. A small group of Marines eventually managed to raise the American flag on the top of Mount Suribachi, a moment immortalized for posterity in the famous historical photograph by Joe Rosenthal5.

Fighting continued for 36 days after the start of the battle. Once Suribachi was taken, the Americans continued to slowly and painfully advance to the north of the island, encountering fierce pockets of Japanese resistance along the way. One can only surmise about the fierceness of the combat that took place on Iwo Jima by reading the names of certain places on a map of the island, such as “the meat grinder”. During the whole of Operation Detachment, the Americans made two more disembarkation of troops: one of 6,000 men and another of 13,000 men on the third and sixth day of the battle, respectively. In total, the Americans sustained 28,000 casualties including about 8,000 dead. The cost of taking this small island was much more than they expected. As for the Japanese, over 20,000 died and about 1000 were taken prisoners. The last two holdouts6 did not surrender until 1951. It is said that Kuribayashi himself died in a suicide raid on the southern part of the island disguised as a common soldier without his rank of General insignia.

Data for the simulation study

The data for this simulation study is based on a report titled “The Iwo Jima Operation” written by Captain Clifford Moorehouse, an American officer. This report contains day-to-day records on the number of American casualties. Together with the data on the number of American troops put ashore on the three disembarkations, this data serves to reconstruct the number of active American troops on the given days of the 36-day battle. Of course, we must assume an estimate number of 21,500 Japanese troops at the start of the battle and 0 active Japanese troops at the end, in between which no estimates exist. The data for the number of active troops on each side is summarized in the following table, in which the “time” column indicates the day of the battle and the “ame” and “jap” columns indicate the active troops for the Americans and the Japanese, respectively:

time	ame	jap	time	ame	jap	time	ame	jap
0	0	21500	13	59549	NA	25	53347	NA
1	52839	NA	14	59345	NA	26	53072	NA
2	50945	NA	15	59081	NA	27	52804	NA
4	56031	NA	16	58779	NA	28	52735	NA
5	53749	NA	17	58196	NA	29	52608	NA
6	66155	NA	18	57259	NA	30	52507	NA
7	65250	NA	19	56641	NA	31	52462	NA
8	64378	NA	20	54792	NA	32	52304	NA
9	62874	NA	21	55308	NA	33	52155	NA
10	62339	NA	22	54796	NA	34	52155	NA
11	61405	NA	23	54038	NA	35	52155	NA
12	60667	NA	24	53938	NA	36	52140	0

The data on the three American disembarkations can be summarized by the following stepwise function definition:

\[f(t) = \left\{ \begin{array}{cl} 54000 & 0 \leq t \lt 1 \\ 0 & 1 \leq t \lt 2 \\ 6000 & 2 \leq t \lt 3 \\ 0 & 3 \leq t \lt 5 \\ 13000 & 5 \leq t \lt 6 \\ 0 & 6 \leq t \leq 36 \end{array} \right.\]

Engel (1954) estimated the effectiveness coefficients for the Americans and the Japanese to be 0.0106 and 0.0577 respectively. We can take these values to simulate the model and compare its fit to the actual observed number of active troops according to the Moorehouse data. Then we will fit these parameters using simecol’s fitOdeModel function and see if we obtain an even better fit. I must remark that when using simecol, it is not necessary to know in advance the values of your model’s parameters, as the fitOdeModel takes care of this problem for you. After all, estimating a model’s parameters is one of the main reasons why we build simulation models in the first place.

R/simecol implementation for the simulation model of the Battle of Iwo Jima

We have come to the core of this blog entry: the implementation of the simulation model for the Battle of Iwo Jima using R and simecol. First off, let me briefly explain what simecol is. It is an R package for the simulation of ecological models through R’s S4 object class system using a state-space representation. What this means (the S4 object class part) is that you can work with several model objects in memory having different parameter values, etc. and simulate them, fit their parameters separately and so on, using an object-oriented methodology that makes it easy for researchers without extensive knowledge of R programming to get their simulation models up and running quickly using a general purpose programming language like R, which provides them with flexibility in modeling while at the same time access to all the powerful statistical/analytical tools available through the base R and extensive package system.

The state-space representation part mentioned in the last paragraph means that the various types of models one can implement in simecol- 1) dynamic system models using differential equations (or simecol odeModel class objects. This is the type we will deal with here), 2) cellular automata models and 3) individual agent based models – are viewed as subtypes of a more general family of models in which you have: 1) a set of variables describing the state of the model at a given time, 2) rules governing the way these state variables change over time, 3) a set of model parameters 4) a specification for the time instants at which you wish to observe the state variables, 5) externalities (called inputs), which are not part of the state variables of the system interrelated amongst each other but which affect them and 6) a specification for a solver function, which updates the state variables based on the rules that describe their change over time.

< svg height="220" viewbox="0 0 550 220" width="450"> Schematic diagram of simecol odeModel objects < defs> < marker id="arrow" markerheight="10" markerunits="strokeWidth" markerwidth="10" orient="auto" refx="0" refy="3"> < path d="M0,0 L0,6 L9,3 z" fill="#000"> < rect fill="#ffffff" height="40" stroke-width="1px" stroke="#000000" width="140" x="180" y="2"> < text style="fill: black; -size: 10pt;" x="200" y="25">external inputs < rect fill="#ffffff" height="70" stroke-width="1px" stroke="#000000" width="180" x="160" y="70"> < text style="fill: black; -size: 10pt;" x="180" y="85">differential equations < text style="fill: black; -size: 10pt;" x="180" y="105">parameters < text style="fill: black; -size: 10pt;" x="180" y="125">state variables < rect fill="#ffffff" height="45" stroke-width="1px" stroke="#000000" width="120" x="390" y="80"> < text style="fill: black; -size: 10pt;" x="400" y="95">output < text style="fill: black; -size: 8pt;" x="400" y="110">(state variables) < rect fill="#ffffff" height="45" stroke-width="1px" stroke="#000000" width="120" x="2" y="80"> < text style="fill: black; -size: 10pt;" x="12" y="95">initial state < text style="fill: black; -size: 8pt;" x="12" y="110">(state variables) < rect fill="#ffffff" height="45" stroke-width="1px" stroke="#000000" width="180" x="160" y="160"> < text style="fill: black; -size: 10pt;" x="170" y="175">solver (rk4, lsoda, etc.) < text style="fill: black; -size: 10pt;" x="170" y="195">t ⇦ t + Δt

< polyline fill="#ffffff" marker-end="url(#arrow)" points="365,105 365,182 350,182" stroke-width="1" stroke="#000000"/> < polyline fill="#ffffff" marker-end="url(#arrow)" points="160,182 140,182 140,118" stroke-width="1" stroke="#000000"/> < circle cx="140" cy="105" fill="#000000" r="4"/> < circle cx="365" cy="105" fill="#000000" r="4"/>

In the case of dynamic system models with differential equations, the rules describing the way the system changes over time are obviously characterized by a system of differential equations. These differential equations usually have parameters that further characterize the dynamic behavior of the system and which the researcher is usually interested in fitting from observed data. In this battle of Iwo Jima example, the parameters would be the effectiveness coefficients of the Americans and the Japanese. In our Iwo Jima model we also have the American reinforcements- those would be the inputs (or externalities) which are included in the differential equation describing how the number of American troops changes over time. Finally, the solver function is a specification of a particular numerical integrator that solves for the state variables using the differential equations, for example, 4th-order Runge-Kutta, Euler, LSODA, etc. In this sense, simecol serves as a wrapper for another package called deSolve, which includes the compiled FORTRAN numerical integration routines for fast execution in R.

Without further ado, here’s the R script for the simulation of the Battle of Iwo Jima using the simecol library:

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library(simecol)
iwojima_engel <- new("odeModel",
    main = function (time, init, parms, ...) {
        x <- init
        p <- parms
        f <- approxTime1(inputs, time, rule = 2)["reinforcements"]
        dame <- f - p["j"] * x["jap"]
        djap <- - p["a"] * x["ame"]
        list(c(dame, djap))
    },
    parms = c(j=0.0577, a=0.0106),
    times = c(from=0, to=36, by=0.01),
    init = c(ame=0, jap=21500),
    inputs = as.matrix(data.frame(
                day = c(0, 0.999, 1, 1.999, 2, 2.999, 3, 4.999, 
                        5, 5.999, 6, 36),
                reinforcements = c(54000, 54000, 0, 0, 6000, 6000, 0, 0, 
                            13000, 13000, 0, 0))),
    solver = "lsoda"
) 
#copy the iwojima Engel model onto another for fitting later
iwojima_fitode <- iwojima_engel
#now read the actual data...
obs <- read.csv2("iwo_jima.csv")
#the weightdf dataframe assigns equal weight to all observations,
#except for the NA values in the japanese column (those get 0 weight)
weightdf <- data.frame(ame=rep(1, nrow(obs)), jap = rep(0, nrow=(obs)))
weightdf[1,"jap"] <- 1
weightdf[nrow(obs),"jap"] <- 1
#return the ssq of the Engel model (a measure of goodness of fit)
ssqOdeModel(parms(iwojima_engel),
  iwojima_engel,
  obstime = obs$time,
  yobs=obs[2:3],
  weights=weightdf)
#and now we fit the model parameters with simecol's fitOdeModel
#including our weightdf as weights makes the PORT routine faster
result_fit <- fitOdeModel(iwojima_engel,
  obstime=obs$time,
  yobs=obs[2:3],
  fn=ssqOdeModel,
  weights=weightdf,
  method="PORT",
  scale=c(1/0.1, 1/0.01),
  lower=c(j=0,a=0))
result_fit
#we update the new model parameters
parms(iwojima_fitode) <- result_fit$par
ssqOdeModel(parms(iwojima_fitode),
  iwojima_fitode,
  obstime = obs$time,
  yobs=obs[2:3],
  weights=weightdf)
#finally we simulate and plot the graphs
iwojima_engel <- sim(iwojima_engel)
iwojima_fitode <- sim(iwojima_fitode)
svg("iwojima_engel.svg")
matplot(x=iwojima_engel@out,
  main = "Model1 : Engel(1954)",
  xlab = "Day",
  ylab = "Number of active troops",
  type = "l", lty = c("solid", "solid"), 
  col = c("blue","red"))
points(obs$time,obs$ame,col="blue",pch=19)
points(obs$time[c(1,36)],obs$jap[c(1,36)],col="red",pch=19)
graphics.off()
svg("iwojima_fitode.svg")
matplot(x=iwojima_fitode@out, 
  main = "Model2 : fitOdeModel",
  xlab = "Day",
  ylab = "Number of active troops",
  type = "l", lty = c("solid", "solid"), 
  col = c("blue","red"))
points(obs$time,obs$ame,col="blue",pch=19)
points(obs$time[c(1,36)],obs$jap[c(1,36)],col="red",pch=19)
graphics.off()

Let’s go over the code above. First, in line 1 we load the simecol library. In lines 2-20, we define an odeModel class object called “iwojima_engel”. Notice how the “main” slot serves to specify the system of differential equations as a function returning the differentials on each state variable as a list. The other simecol methods will “know” that these differentials will correspond to the “ame” and “jap” variables because they are given as a list in the same order as we specify them in the “init” slot on line 13. Notice that the f function used to represent the American reinforcements is given as a function of the “inputs” slot defined in lines 14-18 (we will later explain what that approxTime1 function does). The “times” slot in line 12 is given in very small increments of 0.01 days so that the solver function (specified as “lsoda” in line 19) can do the numerical integration in small steps.

In line 24 we read in the Moorehouse data stored in a csv file (if you want to do this simulation exercise yourself, just take the Moorehouse data from the table above and put it in a csv file in the same directory as the R script (the separation character is a semicolon). Lines 27 to 29 simply create a data frame of the same dimensions as the “obs” data frame with 1 for valid values and 0 for NA values on that data frame. This simply serves to assign equal weights to each observed non-NA value so as to make the parameter fitting faster later on.

Lines 31-35 simply invoke the ssqOdeModel function on the iwojima_engel object to compute a measure of goodness of fit (consult the help page for ssqOdeModel to see how it is calculated). Lines 38-46 do the actual fitting of the iwojima_engel model object, producing a better-fitting set of parameters (remember, the American and Japanese effectiveness parameters). To do this, the fitOdeModel invokes a non-linear optimization routine using the PORT algorithm to obtain the set of parameter values that minimize the ssqOdeModel measure. In fact, this produces a better fit of the model (indicated by a lower ssqOdeModel value).

In line 48 we simply feed the new parameters into the iwojima_fitode, which was simply a copy of the iwojima_engel object and finally, in lines 55-76 we plot both model objects. Notice how the svg graphics device is used instead of pdf or png graphics. The svg graphics device produce better looking, non pixelating graphics that look great no matter how much you zoom in the page on your browser.

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128.074219 L 129.011719 127.453125 L 129.121094 126.832031 L 129.332031 125.589844 L 129.441406 124.96875 L 129.652344 123.726562 L 129.761719 123.105469 L 129.867188 122.488281 L 130.078125 121.246094 L 130.1875 120.625 L 130.398438 119.382812 L 130.507812 118.761719 L 130.71875 117.519531 L 130.828125 116.898438 L 131.039062 115.65625 L 131.148438 115.035156 L 131.464844 113.171875 L 131.574219 112.550781 L 131.785156 111.308594 L 131.894531 110.6875 L 132.105469 109.445312 L 132.214844 108.824219 L 132.425781 107.582031 L 132.535156 106.960938 L 132.851562 105.097656 L 132.960938 104.476562 L 133.171875 103.234375 L 133.28125 102.613281 L 133.492188 101.371094 L 133.601562 100.75 L 133.8125 99.507812 L 133.921875 98.882812 L 134.238281 97.019531 L 134.347656 96.398438 L 134.558594 95.15625 L 134.667969 94.535156 L 134.773438 93.914062 L 134.878906 93.289062 L 134.988281 92.667969 L 135.199219 91.425781 L 135.308594 90.804688 L 135.414062 90.183594 L 135.519531 89.558594 L 135.625 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107.53125 L 216.375 107.570312 L 216.480469 107.613281 L 216.691406 107.691406 L 216.800781 107.734375 L 217.011719 107.8125 L 217.121094 107.855469 L 217.332031 107.933594 L 217.441406 107.976562 L 217.652344 108.054688 L 217.761719 108.09375 L 217.867188 108.136719 L 218.078125 108.214844 L 218.1875 108.257812 L 218.398438 108.335938 L 218.507812 108.375 L 218.613281 108.417969 L 218.71875 108.457031 L 218.828125 108.496094 L 219.039062 108.574219 L 219.148438 108.617188 L 219.464844 108.734375 L 219.574219 108.777344 L 219.785156 108.855469 L 219.894531 108.894531 L 220 108.933594 L 220.105469 108.976562 L 220.214844 109.015625 L 220.425781 109.09375 L 220.535156 109.132812 L 220.746094 109.210938 L 220.851562 109.253906 L 220.960938 109.292969 L 221.171875 109.371094 L 221.28125 109.410156 L 221.492188 109.488281 L 221.601562 109.53125 L 221.8125 109.609375 L 221.921875 109.648438 L 222.238281 109.765625 L 222.347656 109.804688 L 222.558594 109.882812 L 222.667969 109.921875 L 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112.285156 L 229.28125 112.324219 L 229.492188 112.402344 L 229.601562 112.4375 L 229.8125 112.515625 L 229.921875 112.550781 L 230.238281 112.667969 L 230.347656 112.703125 L 230.558594 112.78125 L 230.667969 112.820312 L 230.773438 112.855469 L 230.878906 112.894531 L 230.988281 112.933594 L 231.09375 112.96875 L 231.199219 113.007812 L 231.308594 113.046875 L 231.414062 113.082031 L 231.625 113.160156 L 231.734375 113.195312 L 231.945312 113.273438 L 232.054688 113.308594 L 232.265625 113.386719 L 232.375 113.421875 L 232.480469 113.460938 L 232.585938 113.496094 L 232.691406 113.535156 L 232.800781 113.574219 L 232.90625 113.609375 L 233.011719 113.648438 L 233.121094 113.683594 L 233.332031 113.761719 L 233.441406 113.796875 L 233.546875 113.835938 L 233.652344 113.871094 L 233.761719 113.910156 L 233.867188 113.945312 L 234.078125 114.023438 L 234.1875 114.058594 L 234.292969 114.097656 L 234.398438 114.132812 L 234.507812 114.171875 L 234.613281 114.207031 L 234.71875 114.246094 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136.53125 L 312.375 136.554688 L 312.585938 136.601562 L 312.691406 136.628906 L 312.800781 136.652344 L 313.011719 136.699219 L 313.121094 136.722656 L 313.226562 136.75 L 313.332031 136.773438 L 313.441406 136.796875 L 313.652344 136.84375 L 313.761719 136.867188 L 313.867188 136.894531 L 314.078125 136.941406 L 314.1875 136.964844 L 314.398438 137.011719 L 314.507812 137.035156 L 314.613281 137.0625 L 314.71875 137.085938 L 314.828125 137.109375 L 315.039062 137.15625 L 315.148438 137.179688 L 315.464844 137.25 L 315.574219 137.273438 L 315.679688 137.296875 L 315.785156 137.324219 L 315.894531 137.347656 L 316.105469 137.394531 L 316.214844 137.417969 L 316.425781 137.464844 L 316.535156 137.488281 L 316.851562 137.558594 L 316.960938 137.582031 L 317.171875 137.628906 L 317.28125 137.652344 L 317.492188 137.699219 L 317.601562 137.722656 L 317.8125 137.769531 L 317.921875 137.792969 L 318.238281 137.863281 L 318.347656 137.886719 L 318.558594 137.933594 L 318.667969 137.957031 L 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139.296875 L 325.171875 139.34375 L 325.28125 139.363281 L 325.492188 139.410156 L 325.601562 139.433594 L 325.707031 139.453125 L 325.8125 139.476562 L 325.921875 139.496094 L 326.132812 139.542969 L 326.238281 139.5625 L 326.347656 139.585938 L 326.453125 139.609375 L 326.558594 139.628906 L 326.667969 139.652344 L 326.773438 139.675781 L 326.878906 139.695312 L 326.988281 139.71875 L 327.09375 139.738281 L 327.199219 139.761719 L 327.308594 139.785156 L 327.414062 139.804688 L 327.519531 139.828125 L 327.625 139.847656 L 327.734375 139.871094 L 327.839844 139.894531 L 327.945312 139.914062 L 328.054688 139.9375 L 328.160156 139.957031 L 328.265625 139.980469 L 328.375 140 L 328.585938 140.046875 L 328.691406 140.066406 L 328.800781 140.089844 L 328.90625 140.109375 L 329.011719 140.132812 L 329.121094 140.152344 L 329.226562 140.175781 L 329.332031 140.195312 L 329.441406 140.21875 L 329.546875 140.238281 L 329.652344 140.261719 L 329.761719 140.28125 L 329.867188 140.304688 L 329.972656 140.324219 L 330.078125 140.347656 L 330.1875 140.367188 L 330.292969 140.390625 L 330.398438 140.410156 L 330.507812 140.433594 L 330.613281 140.453125 L 330.71875 140.476562 L 330.828125 140.496094 L 330.933594 140.519531 L 331.039062 140.539062 L 331.148438 140.558594 L 331.253906 140.582031 L 331.359375 140.601562 L 331.464844 140.625 L 331.574219 140.644531 L 331.679688 140.667969 L 331.785156 140.6875 L 331.894531 140.707031 L 332 140.730469 L 332.105469 140.75 L 332.214844 140.773438 L 332.425781 140.8125 L 332.535156 140.835938 L 332.640625 140.855469 L 332.746094 140.878906 L 332.851562 140.898438 L 332.960938 140.917969 L 333.066406 140.941406 L 333.171875 140.960938 L 333.28125 140.980469 L 333.386719 141.003906 L 333.492188 141.023438 L 333.601562 141.042969 L 333.707031 141.066406 L 333.8125 141.085938 L 333.921875 141.105469 L 334.027344 141.128906 L 334.238281 141.167969 L 334.347656 141.191406 L 334.558594 141.230469 L 334.667969 141.253906 L 334.878906 141.292969 L 334.988281 141.316406 L 335.199219 141.355469 L 335.308594 141.375 L 335.414062 141.398438 L 335.625 141.4375 L 335.734375 141.457031 L 335.839844 141.480469 L 335.945312 141.5 L 336.054688 141.519531 L 336.160156 141.539062 L 336.265625 141.5625 L 336.375 141.582031 L 336.585938 141.621094 L 336.691406 141.644531 L 336.800781 141.664062 L 337.011719 141.703125 L 337.121094 141.722656 L 337.226562 141.746094 L 337.332031 141.765625 L 337.441406 141.785156 L 337.652344 141.824219 L 337.761719 141.847656 L 338.078125 141.90625 L 338.1875 141.925781 L 338.292969 141.945312 L 338.398438 141.96875 L 338.507812 141.988281 L 338.71875 142.027344 L 338.828125 142.046875 L 339.039062 142.085938 L 339.148438 142.109375 L 339.464844 142.167969 L 339.574219 142.1875 L 339.785156 142.226562 L 339.894531 142.246094 L 340.105469 142.285156 L 340.214844 142.308594 L 340.425781 142.347656 L 340.535156 142.367188 L 340.851562 142.425781 L 340.960938 142.445312 L 341.171875 142.484375 L 341.28125 142.503906 L 341.492188 142.542969 L 341.601562 142.5625 L 341.8125 142.601562 L 341.921875 142.621094 L 342.238281 142.679688 L 342.347656 142.699219 L 342.558594 142.738281 L 342.667969 142.757812 L 342.878906 142.796875 L 342.988281 142.816406 L 343.199219 142.855469 L 343.308594 142.875 L 343.625 142.933594 L 343.734375 142.953125 L 343.945312 142.992188 L 344.054688 143.011719 L 344.160156 143.027344 L 344.265625 143.046875 L 344.375 143.066406 L 344.691406 143.125 L 344.800781 143.144531 L 345.011719 143.183594 L 345.121094 143.203125 L 345.226562 143.21875 L 345.332031 143.238281 L 345.441406 143.257812 L 345.652344 143.296875 L 345.761719 143.316406 L 345.867188 143.335938 L 345.972656 143.351562 L 346.078125 143.371094 L 346.1875 143.390625 L 346.398438 143.429688 L 346.507812 143.449219 L 346.613281 143.464844 L 346.71875 143.484375 L 346.828125 143.503906 L 347.039062 143.542969 L 347.148438 143.558594 L 347.464844 143.617188 L 347.574219 143.636719 L 347.679688 143.652344 L 347.785156 143.671875 L 347.894531 143.691406 L 348 143.710938 L 348.105469 143.726562 L 348.214844 143.746094 L 348.425781 143.785156 L 348.535156 143.800781 L 348.851562 143.859375 L 348.960938 143.875 L 349.171875 143.914062 L 349.28125 143.933594 L 349.386719 143.949219 L 349.492188 143.96875 L 349.601562 143.988281 L 349.707031 144.003906 L 349.8125 144.023438 L 349.921875 144.042969 L 350.027344 144.058594 L 350.238281 144.097656 L 350.347656 144.113281 L 350.558594 144.152344 L 350.667969 144.167969 L 350.878906 144.207031 L 350.988281 144.222656 L 351.199219 144.261719 L 351.308594 144.277344 L 351.519531 144.316406 L 351.625 144.332031 L 351.734375 144.351562 L 351.839844 144.367188 L 351.945312 144.386719 L 352.054688 144.40625 L 352.160156 144.421875 L 352.265625 144.441406 L 352.375 144.457031 L 352.585938 144.496094 L 352.691406 144.511719 L 352.800781 144.53125 L 352.90625 144.546875 L 353.011719 144.566406 L 353.121094 144.582031 L 353.332031 144.621094 L 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358.238281 145.417969 L 358.347656 145.4375 L 358.558594 145.46875 L 358.667969 145.488281 L 358.878906 145.519531 L 358.988281 145.539062 L 359.199219 145.570312 L 359.308594 145.589844 L 359.519531 145.621094 L 359.625 145.640625 L 359.734375 145.65625 L 359.839844 145.671875 L 359.945312 145.691406 L 360.054688 145.707031 L 360.265625 145.738281 L 360.375 145.757812 L 360.691406 145.804688 L 360.800781 145.824219 L 361.011719 145.855469 L 361.121094 145.871094 L 361.226562 145.890625 L 361.332031 145.90625 L 361.441406 145.921875 L 361.546875 145.9375 L 361.652344 145.957031 L 361.761719 145.972656 L 362.078125 146.019531 L 362.1875 146.039062 L 362.398438 146.070312 L 362.507812 146.085938 L 362.613281 146.101562 L 362.71875 146.121094 L 362.828125 146.136719 L 363.039062 146.167969 L 363.148438 146.183594 L 363.253906 146.199219 L 363.359375 146.21875 L 363.464844 146.234375 L 363.574219 146.25 L 363.785156 146.28125 L 363.894531 146.296875 L 364.105469 146.328125 L 364.214844 146.347656 L 364.425781 146.378906 L 364.535156 146.394531 L 364.851562 146.441406 L 364.960938 146.457031 L 365.171875 146.488281 L 365.28125 146.507812 L 365.492188 146.539062 L 365.601562 146.554688 L 365.8125 146.585938 L 365.921875 146.601562 L 366.238281 146.648438 L 366.347656 146.664062 L 366.558594 146.695312 L 366.667969 146.710938 L 366.878906 146.742188 L 366.988281 146.757812 L 367.199219 146.789062 L 367.308594 146.804688 L 367.625 146.851562 L 367.734375 146.867188 L 367.945312 146.898438 L 368.054688 146.914062 L 368.265625 146.945312 L 368.375 146.960938 L 368.691406 147.007812 L 368.800781 147.023438 L 369.011719 147.054688 L 369.121094 147.070312 L 369.226562 147.082031 L 369.332031 147.097656 L 369.441406 147.113281 L 369.652344 147.144531 L 369.761719 147.160156 L 370.078125 147.207031 L 370.1875 147.222656 L 370.292969 147.234375 L 370.398438 147.25 L 370.507812 147.265625 L 370.71875 147.296875 L 370.828125 147.3125 L 370.933594 147.328125 L 371.039062 147.339844 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316.878906 L 123.148438 316.910156 L 123.253906 316.9375 L 123.464844 317 L 123.574219 317.03125 L 123.679688 317.058594 L 123.785156 317.089844 L 123.894531 317.121094 L 124 317.148438 L 124.105469 317.179688 L 124.214844 317.210938 L 124.320312 317.242188 L 124.425781 317.269531 L 124.535156 317.300781 L 124.640625 317.332031 L 124.746094 317.359375 L 124.851562 317.390625 L 124.960938 317.421875 L 125.066406 317.449219 L 125.171875 317.480469 L 125.28125 317.511719 L 125.386719 317.539062 L 125.492188 317.570312 L 125.601562 317.601562 L 125.707031 317.632812 L 125.8125 317.660156 L 125.921875 317.691406 L 126.027344 317.722656 L 126.132812 317.75 L 126.238281 317.78125 L 126.347656 317.8125 L 126.453125 317.839844 L 126.558594 317.871094 L 126.667969 317.902344 L 126.773438 317.929688 L 126.878906 317.960938 L 126.988281 317.992188 L 127.09375 318.019531 L 127.199219 318.050781 L 127.308594 318.082031 L 127.414062 318.109375 L 127.519531 318.140625 L 127.625 318.167969 L 127.734375 318.199219 L 127.839844 318.230469 L 127.945312 318.257812 L 128.054688 318.289062 L 128.265625 318.351562 L 128.375 318.378906 L 128.691406 318.472656 L 128.800781 318.5 L 129.011719 318.5625 L 129.121094 318.59375 L 129.332031 318.65625 L 129.441406 318.6875 L 129.652344 318.75 L 129.761719 318.777344 L 130.078125 318.871094 L 130.1875 318.90625 L 130.398438 318.96875 L 130.507812 319 L 130.71875 319.0625 L 130.828125 319.09375 L 131.039062 319.15625 L 131.148438 319.191406 L 131.464844 319.285156 L 131.574219 319.316406 L 131.679688 319.351562 L 131.785156 319.382812 L 131.894531 319.414062 L 132 319.449219 L 132.105469 319.480469 L 132.214844 319.511719 L 132.320312 319.546875 L 132.425781 319.578125 L 132.535156 319.609375 L 132.640625 319.644531 L 132.746094 319.675781 L 132.851562 319.710938 L 132.960938 319.742188 L 133.066406 319.777344 L 133.171875 319.808594 L 133.28125 319.84375 L 133.386719 319.875 L 133.492188 319.910156 L 133.601562 319.941406 L 133.8125 320.011719 L 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322.027344 L 140 322.0625 L 140.105469 322.101562 L 140.214844 322.136719 L 140.425781 322.207031 L 140.535156 322.246094 L 140.746094 322.316406 L 140.851562 322.355469 L 140.960938 322.390625 L 141.066406 322.425781 L 141.171875 322.464844 L 141.28125 322.5 L 141.492188 322.570312 L 141.601562 322.609375 L 141.8125 322.679688 L 141.921875 322.71875 L 142.238281 322.824219 L 142.347656 322.863281 L 142.558594 322.933594 L 142.667969 322.972656 L 142.878906 323.042969 L 142.988281 323.078125 L 143.09375 323.117188 L 143.199219 323.152344 L 143.308594 323.1875 L 143.414062 323.222656 L 143.519531 323.261719 L 143.625 323.296875 L 143.734375 323.332031 L 143.839844 323.367188 L 143.945312 323.40625 L 144.054688 323.441406 L 144.160156 323.476562 L 144.265625 323.515625 L 144.375 323.550781 L 144.585938 323.621094 L 144.691406 323.660156 L 144.800781 323.695312 L 145.011719 323.765625 L 145.121094 323.800781 L 145.226562 323.839844 L 145.332031 323.875 L 145.441406 323.910156 L 145.546875 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127.878906 L 291.359375 127.902344 L 291.464844 127.929688 L 291.574219 127.953125 L 291.679688 127.980469 L 291.785156 128.003906 L 291.894531 128.03125 L 292 128.054688 L 292.105469 128.082031 L 292.214844 128.105469 L 292.320312 128.132812 L 292.425781 128.15625 L 292.535156 128.183594 L 292.746094 128.230469 L 292.851562 128.257812 L 292.960938 128.28125 L 293.066406 128.308594 L 293.171875 128.332031 L 293.28125 128.359375 L 293.492188 128.40625 L 293.601562 128.433594 L 293.707031 128.457031 L 293.8125 128.484375 L 293.921875 128.507812 L 294.027344 128.53125 L 294.132812 128.558594 L 294.238281 128.582031 L 294.347656 128.609375 L 294.558594 128.65625 L 294.667969 128.683594 L 294.878906 128.730469 L 294.988281 128.757812 L 295.09375 128.78125 L 295.199219 128.808594 L 295.308594 128.832031 L 295.414062 128.855469 L 295.519531 128.882812 L 295.625 128.90625 L 295.734375 128.929688 L 295.839844 128.957031 L 295.945312 128.980469 L 296.054688 129.003906 L 296.160156 129.027344 L 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142.421875 L 373.386719 142.433594 L 373.492188 142.449219 L 373.601562 142.460938 L 373.8125 142.484375 L 373.921875 142.496094 L 374.027344 142.507812 L 374.132812 142.523438 L 374.238281 142.535156 L 374.347656 142.546875 L 374.558594 142.570312 L 374.667969 142.582031 L 374.773438 142.597656 L 374.878906 142.609375 L 374.988281 142.621094 L 375.199219 142.644531 L 375.308594 142.65625 L 375.414062 142.667969 L 375.519531 142.683594 L 375.625 142.695312 L 375.734375 142.707031 L 375.945312 142.730469 L 376.054688 142.742188 L 376.265625 142.765625 L 376.375 142.777344 L 376.585938 142.800781 L 376.691406 142.816406 L 376.800781 142.828125 L 377.011719 142.851562 L 377.121094 142.863281 L 377.332031 142.886719 L 377.441406 142.898438 L 377.652344 142.921875 L 377.761719 142.933594 L 378.078125 142.96875 L 378.1875 142.980469 L 378.398438 143.003906 L 378.507812 143.015625 L 378.71875 143.039062 L 378.828125 143.050781 L 379.039062 143.074219 L 379.148438 143.085938 L 379.464844 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143.773438 L 386.078125 143.808594 L 386.1875 143.816406 L 386.398438 143.839844 L 386.507812 143.847656 L 386.71875 143.871094 L 386.828125 143.878906 L 387.039062 143.902344 L 387.148438 143.910156 L 387.359375 143.933594 L 387.464844 143.941406 L 387.574219 143.953125 L 387.679688 143.964844 L 387.785156 143.972656 L 387.894531 143.984375 L 388 143.996094 L 388.105469 144.003906 L 388.214844 144.015625 L 388.320312 144.023438 L 388.425781 144.035156 L 388.535156 144.046875 L 388.640625 144.054688 L 388.746094 144.066406 L 388.851562 144.074219 L 388.960938 144.085938 L 389.066406 144.097656 L 389.171875 144.105469 L 389.28125 144.117188 L 389.386719 144.125 L 389.492188 144.136719 L 389.601562 144.148438 L 389.707031 144.15625 L 389.8125 144.167969 L 389.921875 144.175781 L 390.027344 144.1875 L 390.132812 144.195312 L 390.238281 144.207031 L 390.347656 144.214844 L 390.453125 144.226562 L 390.558594 144.234375 L 390.667969 144.246094 L 390.773438 144.257812 L 390.878906 144.265625 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145.660156 L 408.480469 145.664062 L 408.691406 145.679688 L 408.800781 145.6875 L 409.011719 145.703125 L 409.121094 145.707031 L 409.332031 145.722656 L 409.441406 145.730469 L 409.546875 145.738281 L 409.652344 145.742188 L 409.761719 145.75 L 409.972656 145.765625 L 410.078125 145.769531 L 410.1875 145.777344 L 410.398438 145.792969 L 410.507812 145.796875 L 410.71875 145.8125 L 410.828125 145.820312 L 410.933594 145.824219 L 411.039062 145.832031 L 411.148438 145.839844 L 411.253906 145.847656 L 411.359375 145.851562 L 411.464844 145.859375 L 411.574219 145.867188 L 411.679688 145.871094 L 411.785156 145.878906 L 411.894531 145.886719 L 412 145.894531 L 412.105469 145.898438 L 412.214844 145.90625 L 412.320312 145.914062 L 412.425781 145.917969 L 412.535156 145.925781 L 412.640625 145.933594 L 412.746094 145.9375 L 412.851562 145.945312 L 412.960938 145.953125 L 413.066406 145.957031 L 413.171875 145.964844 L 413.28125 145.972656 L 413.386719 145.976562 L 413.492188 145.984375 L 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Finally, for comparison, we include a little table with the main results used to compare the different parameter values (Engel’s estimations and the simecol estimations):

	Engel (1954)	fitOdeModel
Japanese Effectiveness	0.0577000	0.05540003
American Effectiveness	0.0106000	0.01080927
ssqOdeModel	0.3167629	0.05121009

Interpretation of the simulation results and some what-if questions

Before I go on with the most fun part of modeling (playing with your models and interpreting them), I must clarify that simecol is a package originally intended for ecological modeling. It has also been used for engineering and economics modeling. However, because it is so versatile, I have used it for a military-related application such as simulating a battle using a Lanchester model, which is not an application for the package that its creators originally intended nor support. I do not mean by this that my view on this matter is different from the authors of simecol. In fact, I can see their wisdom in not supporting applications of simecol that they’re afraid of.

What motivated me to present this model of the Battle of Iwo Jima are mainly educational reasons. First of all, the topic of historic and real battles is usually interesting for the young audiences to whom I’ve given classes on modeling and simulation. For me as a teacher, this serves as a sugar-coating to make the mathematical modeling and computational implementation topic-pills go down easier. Another important reason for me as a teacher is that these type of models lend themselves to interesting interpretations and discussions in class about politics and history. It gives us, as we shall see, a scientific framework for discussing about history and other social topics. To quote one of the authors on the bibliography I consulted:

Lanchester models provide an excellent example of the strengths (and weaknesses) of simple mathematical modeling. Further, as we have seen, the basic model leads to many other ‘What if…?’ questions which can be easily investigated. Many more such questions can be asked, and of course once one begins numerical simulations the possibilities are endless. To model warfare can seem more politically challenging (not to say incorrect) than to use ecology or epidemiology, but a little understanding of how military planners arrive at their tactical conclusions can also strip away mystique and (for this author, at least) expose some of the subject’s limitations! (MacKay, 2005, p. 7)

Some possible ‘what-if’ questions or questions for interesting discussion:

Given the same effectiveness of the American and Japanese troops as those found in your model, with how many troops would the Japanese have won the Battle of Iwo Jima if the Americans had disembarked the same amount of troops on the island? Would it have been possible for the Japanese to amass this number of troops given the space and logistic limitations of Iwo Jima?
If the Japanese soldiers were 5 times more effective or lethal than the Americans, why did more Japanese die in Iwo Jima?
How many soldiers would the Japanese have needed to make the battle linger for, say, one year? How many American casualties would this have brought about? Would this have changed the outcome of the war?
Lanchester models are based on conventional warfare. Nowadays, this conventional warfare scenario is not as generalized as before. For example, it is said that in Venezuela there is a non-conventional war being waged, although there are no battles in the classical sense involving regular military forces. There are civilian casualties, such as those caused by one of the highest crime rates in the world. And now, there is an added element of extreme scarcity of foods and medical supplies which cause civilian casualties as well. How would you as a modern-day Lanchester propose a differential equation model for this war scenario? What state variables and parameters would you define?

As you can see, the subject lends itself to some very interesting discussions in class. I might add that mathematical modeling of this sort gives us a fresh new perspective on social issues and gives us the foundations for discussing about these issues in a more scientific and less dogmatic way. I think having this sort of discussion integrated into our educational system is much needed today, particularly in my country.

As a closing remark for the readers of this blog, let me say that I will post more about simecol and other related packages, such as FME (with more added functionalities for parameter fitting). And no, the other posts won’t be about military/warfare models! Stay tuned…

Acknowledgements

I wish to thank, first and foremost, you, dear reader, for putting up with such a long post. My thanks go out to the creators of the simecol package. Dr. Thomas Petzoldt’s kind help on the R-sig-dyn-mod mailing list has been invaluable in helping me come to grips with this powerful R modeling tool. I would also like to thank James Cagney for his historyanimated media on the Battle of Iwo Jima. It helped me to understand the military and historical background of the battle. Make sure to check out his historyanimated site! Finally, I would like to thank Professor Christine Lind for providing me with the Moorehouse data without which this post would not have been possible.

Notes

There are some excellent and thought provoking articles on the educational application of System Dynamics to school children without any Calculus background by the System Dynamics in Education Project, led by Jay Forrester himself. In a 1992 article, Forrester criticizes pre-college education as “poorly serving the needs of society”, because it does not take into account the complex interactions between people and phenomena. The system dynamics view, Forrester argued, would empower schoolchildren to deal with complexity. Implementation of these ideas in the classroom entails the use of software like STELLA II or Vensim (both commercial packages with GUI’s available both for Macintosh and Windows). It would be awesome if there was a such an application for R using the simecol and FME packages as backends for the actual simulation and fitting of SD models, but including a GUI with which schoolchildren could construct such models from Forrester diagrams… Oh! I’m daydreaming again.
There is an excellent series of animated presentations available at the Historyanimated site (see Cagney, 2008). I contacted the author himself, James Cagney, in 2011 for permission to use his material in class and for some class notes I was preparing and thinking about publishing on the web. He told me to add the following interesting note: “The animation for The Battle of Gettysburg (civilwaranimated) is used to train smoke jumpers in the western US (men who parachute from airplanes to fight forest fires) – they tell me that fighting a fire is just like fighting a battle.”. So there it is, James!
Iwo Jima, or Iō-tō (“sulphur island” in japanese) was first visited by a Westerner in 1543, a Spaniard by the name of Bernardo de la Torre, who called the island “Isla de Sufre” or Sulphur Island (Sufre was the archaic Spanish spelling for azufre or sulphur). It is an appropriate name, as the southernmost tip of the island hosts a volcanic vent (hence the sulphur) that produces the island’s most prominent geographic feature: Mount Suribachi. The fine volcanic ash sands of its beaches would prove a major nuisance for the americans as they tried to disembark amphibious vehicles during their invasion of the island. On account of this sulphur, no other scenario of any World War II battle could better resemble hell and in fact, ancient Japanese legend has it that the island was inhabited by demons.
A man descended from 5 generations of samurai, Koribayashi was also a veteran of the Manchurian and Chinese campaigns. He had previously served as military attaché in Washington during the 1920’s, where he acquired extensive knowledge of Americans. About the latter, he wrote: “The U.S.A. is the last country in the world against whom the Japanese should wage war” (Anonymous, 2006).

At this point I would digress in class and talk to my students about how this Pulitzer-winning photograph was an artistic marvel. I would also mention to them that in Venezuela, we also have our own “Joe Rosenthal”. His name was Héctor Rondon and his famous picture of a chaplain helping a wounded Venezuelan soldier in a street corner, amidst friendly and unfriendly crossfire, also won the Pulitzer Prize in 1963 and was featured in LIFE Magazine.

The events surrounding this “Pietá”-like picture were those of the Porteñazo, a military coup led by communist rebels backed by Cuba against the democratically elected president Romulo Betancourt in 1962. It is ironic that while today, the Chavista propaganda talks about US imperialism and intervention in our country before the so-called Socialist Revolution, the Cuban regime had tried on several ocassions to export their revolution to Venezuela (El Porteñazo was one such incident in which over 100 Venezuelan soldiers died). That was until Hugo Chavez came along and peacefully handed control of this country to the Cuban Castro regime without a single shot being fired. At any rate I leave you a small print of the photograph for your viewing.

On the subject of Japanese holdouts, there is the noteworthy case of Hiroo Onoda, who did not surrender in 1945 and spent 30 years in the jungles of the Philipines (if you didn’t know about Onoda, look him up. It’s one hell of a story). An english progressive rock band called Camel made a concept album called Nude inspired upon the true story of Hiroo Onoda (the name Nude derives from his family name Onoda).

Bibliographical References

ANONYMOUS. (2006). “Iwo Jima el Test Supremo”. Post in online forum. Retrieved March 2011, from http://www.forosegundaguerra.com/viewtopic.php?p=18492
Battle of Iwo Jima. (2016, August 2). In Wikipedia, The Free Encyclopedia. Retrieved August 5, 2016, from https://en.wikipedia.org/wiki/Battle_of_Iwo_Jima
CAGNEY , J. (2008). PacificWarAnimated.com – Historical Animation of the Battle of Iwo Jima. Nonprofit History Animated Foundation. Oregon, US. Retrieved March 2011 from http://pacificwaranimated.com/Iwo.html
ENGEL, J. H. (1954). “A Verification of Lanchester’s Law”. Journal of the Operations Research Society of America, 2(2), pp. pp. 163–171.
Iwo Jima. (2016, July 31). In Wikipedia, The Free Encyclopedia. Retrieved August 5, 2016, from https://en.wikipedia.org/wiki/Iwo_Jima
LIND, C. (2009). Verifying Lanchester’s Combat Model – Battle of Iwo Jima.
MACKAY, N. (2005). “Lanchester Combat Models”. Retrieved March, 2011 from http://arxiv.org/pdf/math/0606300.
PETZOLDT, T. and RINKE , K. (2007). “simecol: An Object-Oriented Framework for Ecological Modeling in R”. Journal of Statistical Software, 22(9), pp. 1–31. ISSN 1548-7660. http://www.jstatsoft.org/v22/i09.
R DEVELOPMENT CORE TEAM (2009). R: A Language and Environment for Statistical Computing. R Foundation for Statistical Computing, Vienna, Austria. ISBN 3-900051-07-0. http://www.R-project.org.

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