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```Math 175: Numerical Analysis II
Lecturer: Jomar F. Rabajante
AY 2012-2013
IMSP, UPLB
Simple Romeo and Juliet Conflict Model
 dR


aJ
 dt

dJ

  bR
 dt
a, b  0
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:
dx
  ( OLR  CLR )  RR
dt
OLR – operational loss rate (loss due to diseases,
desertions and other noncombat mishaps)
CLR – combat loss rate
RR – reinforcement rate
LANCHESTER MODELS
1. Conventional Combat (CONCOM)
For example,
a & d can be
the average
per soldier
death or
desertion rate
due to
noncombat
mishaps
dx
  ax  by  P ( t )
dt
dy
  cx  dy  Q ( t )
dt
OLR
LANCHESTER MODELS
1. Conventional Combat (CONCOM)
dx
  ax  by  P ( t )
dt
dy
dt
  cx  dy  Q ( t )
CLR; b & c are the combat effectiveness
coefficient
LANCHESTER MODELS
The combat efficiency coefficient is difficult to
measure. One approach is to set (let’s just
discuss b)
b  ry p y
ry is the firing rate (shots/combatant/day) of the
y force
py is the probability that a single shot kills an
opponent
LANCHESTER MODELS
2. Guerilla Combat (GUERCOM)
dx
  ax  gxy  P ( t )
dt
dy
dt
  cyx  dy  Q ( t )
CLR
LANCHESTER MODELS
Examples of Guerilla tactics: ambushes, sabotage,
raids, the element of surprise, and
extraordinary mobility to harass a larger and lessmobile traditional army, or strike a vulnerable
target, and withdraw almost immediately.
Usually, Guerilla tactics are used by smallernumbered force to attack larger-numbered force.
Unlike CONCOM (which is one-on-one), the guerilla
model is used in area-fire situations
(concentrated firepower).
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!
LANCHESTER MODELS
The combat efficiency coefficient is difficult to
measure. One approach is to set (let’s just discuss g)
g  ry p y
where p 
A ry
Ax
Ary is the area of effectiveness of a single y shot
Ax is the area occupied by the guerillas
LANCHESTER MODELS
3. Mixed Guerilla-Conventional Combat
(VIETNAM)
dx
  ax  gxy  P ( t )
dt
dy
dt
  cx  dy  Q ( t )
What do you think is the applicable
Lanchester model for the simple Romeo
and Juliet Conflict Model?
 dR


aJ
 dt

dJ

  bR
 dt
a, b  0
 dR


aJ
 dt

dJ

  bR
 dt
a, b  0
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
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.
dN
 rN  cNP
dt
dP
dt
 bNP  mP
Epidemics and Rumor models
• Will be discussed in the laboratory.
Simple Models of Electrical Circuit