Math 175: Numerical Analysis II Lecturer: Jomar F. Rabajante AY IMSP, UPLB

Simple Romeo and Juliet Conflict Model

Two Famous Classes of Conflict Models Lanchester models (combat or attrition models) Richardson models (arms race models)

LANCHESTER MODELS Here, x force and y force are engaged in battle. The nonnegative variables x(t) and y(t) denote the strength of the forces at time t, where t is measured in days since the start of the conflict. For ease, let the strength be the number of soldiers. (This is an approximation since the number of soldiers is a discrete integer)

A TYPICAL LANCHESTER MODEL The rate at which soldiers are added or lost: OLR – operational loss rate (loss due to diseases, desertions and other noncombat mishaps) CLR – combat loss rate RR – reinforcement rate

THREE VERY TRADITIONAL LANCHESTER MODELS 1. Conventional Combat (CONCOM) OLR For example, a & d can be the average per soldier death or desertion rate due to noncombat mishaps

THREE VERY TRADITIONAL LANCHESTER MODELS 1. Conventional Combat (CONCOM) CLR; b & c are the combat effectiveness coefficient

THREE VERY TRADITIONAL LANCHESTER MODELS The combat efficiency coefficient is difficult to measure. One approach is to set (let’s just discuss b) r y is the firing rate (shots/combatant/day) of the y force p y is the probability that a single shot kills an opponent

THREE VERY TRADITIONAL LANCHESTER MODELS 2. Guerilla Combat (GUERCOM) CLR

THREE VERY TRADITIONAL LANCHESTER MODELS Examples of Guerilla tactics: ambushes, sabotage, raids, the element of surprise, and extraordinary mobility to harass a larger and less- mobile traditional army, or strike a vulnerable target, and withdraw almost immediately. Usually, Guerilla tactics are used by smaller- numbered force to attack larger-numbered force. Unlike CONCOM (which is one-on-one), the guerilla model is used in area-fire situations (concentrated firepower).

THREE VERY TRADITIONAL LANCHESTER MODELS Imagine that a force occupies some region R. The enemy fires into R. Under this circumstances, the loss rate for the force is proportional to their number in region R. So do not be concentrated in one area!

THREE VERY TRADITIONAL LANCHESTER MODELS The combat efficiency coefficient is difficult to measure. One approach is to set (let’s just discuss g) A ry is the area of effectiveness of a single y shot A x is the area occupied by the guerillas

THREE VERY TRADITIONAL LANCHESTER MODELS 3. Mixed Guerilla-Conventional Combat (VIETNAM)

What do you think is the applicable Lanchester model for the simple Romeo and Juliet Conflict Model?

It is a CONCOM model without OLR and no replacements. Actually, this model is called the SQUARE LAW!!! The attrition rate of each belligerent is proportional to the size of the adversary.

Richardson’s Model Read the pdf file arms race model uploaded in our yahoogroup. This might help you in your project.

INTERACTING POPULATION MODELS You can model: Neutralism Amensalism Commensalism Competition Mutualism Predation Parasitism, etc…

Lotka-Volterra Predator-Prey Model This is the grandpa of all predator-prey models. This is a “bad” model but is very historical. Let N(t) be the number (or density) of prey Let P(t) be the number (or density) of predators Prey will grow exponentially in the absence of predator. The loss of prey and the growth of predators are proportional to N and P. In the absence of prey, predators die out.

Epidemics and Rumor models Will be discussed in the laboratory.

Simple Models of Electrical Circuit Read the pdf file RLC circuit uploaded in our yahoogroup. I will use these models in one of your laboratory classes. So study how the DEs are derived.