math 175: numerical analysis ii
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Math 175: Numerical Analysis II. Lecturer: Jomar F. Rabajante AY 2012-2013 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. - PowerPoint PPT PresentationTRANSCRIPT
Math 175: Numerical Analysis II
Lecturer: Jomar F. RabajanteAY 2012-2013
IMSP, UPLB
Simple Romeo and Juliet Conflict Model
0,
ba
bRdtdJ
aJdtdR
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 rateRR – reinforcement rate
RRCLROLRdtdx
)(
THREE VERY TRADITIONAL LANCHESTER MODELS
1. Conventional Combat (CONCOM)
)(
)(
tQdycxdtdy
tPbyaxdtdx
OLR
For example,a & d can bethe average per soldierdeath or desertion rate due tononcombat mishaps
THREE VERY TRADITIONAL LANCHESTER MODELS
1. Conventional Combat (CONCOM)
)(
)(
tQdycxdtdy
tPbyaxdtdx
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)
ry is the firing rate (shots/combatant/day) of the y force
py is the probability that a single shot kills an opponent
yy prb
THREE VERY TRADITIONAL LANCHESTER MODELS
2. Guerilla Combat (GUERCOM)
)(
)(
tQdycyxdtdy
tPgxyaxdtdx
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)
Ary is the area of effectiveness of a single y shotAx is the area occupied by the guerillas
x
ry
yy
AA
p
prg
where
THREE VERY TRADITIONAL LANCHESTER MODELS
3. Mixed Guerilla-Conventional Combat (VIETNAM)
)(
)(
tQdycxdtdy
tPgxyaxdtdx
What do you think is the applicable Lanchester model for the simple Romeo and Juliet Conflict Model?
0,
ba
bRdtdJ
aJdtdR
It is a CONCOM model without OLR and no replacements. Actually, this model is called the SQUARE LAW!!!
0,
ba
bRdtdJ
aJdtdR
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
mPbNPdtdP
cNPrNdtdN
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.