In this video, we’ll find the eigenvalues
of the conflict model at its fixed point,
and thus determine the type and stability
of the fixed point.
So we’re looking at
z-prime equals A times z, where
A equals negative alpha k
Ell negative beta
There was a constant term, representing the
grievance, but we’ve rewritten it as per
the last video.
Our fixed point is now at the origin, we take
A minus lambda I, take its determinant, that
is to say the determinant of this matrix,
which is this, and set it equal to zero.
When we expand this multiplication and simplify,
we get
Lambda squared plus (alpha plus beta) lambda
plus alpha beta minus k ell equals zero.
We the quadratic formula to find the eigenvalues.
Negative (alpha plus beta) plus or minus the
square root of (alpha plus beta) squared minus
four times (alpha beta minus k ell) divided
by two.
What can we say about these eigenvalues?
Let’s start our analysis by looking at this
term under the square root.
This is the discriminant, it tells us how
many roots the quadratic has,,
We’ll FOIL this out and distribute this
multiplication over subtraction.
Combine these terms, and refactor this as
(alpha minus beta)^2 plus four k ell. k and
ell are positive constants, and anything squared
is nonnegative, so this is greater than zero.
Remembering how to interpret the discriminant,
this means there are two real eigenvalues,
and they are not equal.
A second piece of information: One of the
eigenvalues is negative.
A negative number minus a positive number
is still negative, a negative number divided
by two is negative.
The other eigenvalue can be positive or negative,
depending on the parameters.
First, suppose alpha beta minus k ell is less
than zero.
Then (alpha plus beta)^2 minus 4 times (alpha
beta minus k ell) is greater than (alpha plus
beta) squared; we’re subtracting a negative
number here, which increases it.
So
So the square root of this term is greater
than the square root of this term; the square
root is monotonic.
So we have negative (alpha plus beta) plus
something greater than alpha plus beta; that’s
positive, and when we divide by two it’s
still positive.
So the eigenvalue is positive.
The same logic shows that if alpha beta minus
k ell is greater than zero, the eigenvalue
is negative.
In summary, then, if alpha beta minus k ell
is less than zero, we have a positive eigenvalue
and a negative eigenvalue, giving us an unstable
saddle.
Mutual coexistence is impossible in the long
term.
The assumption is that if both countries’
military preparedness goes to infinity, they
engage in war with one another, this is not
sustainable.
If alpha beta minus k ell is greater than
zero, the countries can exist in an armed
peace; the fixed point is an asymptotically
stable node, and war need not occur.
We have considered Richardson’s theory of
conflict.
Our consideration is somewhat superficial;
Richardson spent much of his life on this
topic, and we have considered his simplest
model.
We have already quoted him on this, but it
is worth repeating again that nobody thinks
that you can look at two countries, compute
a few parameters, and decide whether they’re
going to go to war.
To end with the same quote we began with,
“The process described by the equations
is not to be thought of as inevitable.
It is what would occur if instinct and tradition
were allowed to act uncontrolled.”
