**Introduction to eigenvalues and eigenvectors:**

** **

As we saw in the simple succession model, after a long
enough time period, the fraction of the landscape that each vegetation type
occupies approaches some constant. In the simplest model, all the land moved to
the "climax" state - so the vector that described the landscape was
[0 0 1] for the case of 3 landscape types where the last type is the climax
state. However, when we added disturbance (fire) to the system, the state of
the system again approached some fixed fraction of each vegetation type, but
all types were present. What we want to do now is define a mathematical method
to find these "final" states - the jargon for this is that we are
finding an "eigenvector" for the system. This

means a vector that characterizes the system after a long
time period has elapsed. It is a vector the elements of which tell us what
fraction of the landscape each vegetation type will have after a long time.
This is also called the long-term equilibrium state of the vegetation.

One way to find an eigenvector is numerically. We can use
Matlab to find for any transition matrix **P**,
**P**^{n} , where n is a
large number (say 100), then multiply this times the initial vector for the
landscape, **x**_{0}, to get
a numerical answer for the long-term state of the landscape. If the initial
vector **x**_{0} contained
the fraction of the landscape in each vegetation type, then **P**^{100} **x**_{0} will be a vector giving the long-term
fraction of the landscape in each vegetation type. If the initial vector **x**_{0} rather represents the number of
hectares or acres of each type, then divide each term in **P**^{100} **x**_{0} by the sum of the components of this
vector to get the long-term fraction of the landscape in each state (the
eigenvector is specified only up to a constant multiple).

A second way to get the eigenvector is to realize that it
arises when the long-term structure of the system doesn't change. This is
expressed as **P y** = l **y** where l is a constant that
represents how the vector of vegetation types increases or decreases through
one time period. In our case of a fixed landscape, land area is neither created
nor destroyed, so there is no change from one time period to another and so l = 1.
This means that to find the eigenvector all we need to do is find a vector **y** that satisfies **P y** = **y**.
This is easy to do using simple algebra for small matrices, but for larger ones
the mathematics becomes more difficult. In this case you either use the theory
of determinants, or else use Matlab to find the answer.

Consider the 2x2 matrix

and lets look at the equations arising from **P y** = l **y** where

This holds if a y_{1} + b y_{2} = l y_{1} and c y_{1} + d y_{2} =
l
y_{2}. In order for these to both hold, we need

So the only way these can both hold is if y_{1}=y_{2}=0
(which is not interesting) or if

This is a quadratic equation in lambda: l^{2}
- (a+d)l
+ ad - bc = 0 and this equation is called the characteristic equation for this
matrix. We call a+d the Trace of the matrix **P** ( Tr(**P**) ) and ad-bc the
determinant of the matrix **P**

( Det(**P**) ). By
solving this quadratic for the roots, we find the eigenvalues l
(there will be two in general for a 2x2 matrix). To find the eigenvector, we
plug in one of these l values to find the ratio of y_{1} to y_{2},
and this gives us the eigenvector (it is only known up to a constant multiple).

Note that in the case of our succession model where the
landscape area doesn't change, we have l = 1, and so we must have
a+d = ad-bc. You should check for yourselves that this holds true because the
columns of the matrix **P** sum to one (so
a+d=1 and b+c=1). So then

gives the ratio of the elements of the eigenvector. Then
the long-term fraction of the landscape in each of the two vegetation classes
is obtained by normalizing the vector **y**
so itŐs components sum to 1.