Imagine a reserve park with animals from a species that we are
trying to protect.
The park doesn’t have a fence and so animals cross the boundary,
both from the inside out and in the other direction.
Every year, 10% of the animals from inside of the park leave, and
1% of the animals from the outside
find their way in.
We can ask if we can find a stable level of population for this park:
is there a population that, once established, will stay constant over time,
with the number of animals leaving equal to the number of animals entering?

To answer that question, we must first establish the equations.
Let the year

${displaystyle n}$

population in the park be

${displaystyle p_{n}}$

and
in the rest of the world be

${displaystyle r_{n}}$

.

${displaystyle {begin{array}{rl}p_{n+1}&=.90p_{n}+.01r_{n}\r_{n+1}&=.10p_{n}+.99r_{n}end{array}}}$

We can set this system up as a matrix equation (see the Markov Chain topic).

${displaystyle {begin{pmatrix}p_{n+1}\r_{n+1}end{pmatrix}}={begin{pmatrix}.90&.01\.10&.99end{pmatrix}}{begin{pmatrix}p_{n}\r_{n}end{pmatrix}}}$

Now, “stable level” means that

${displaystyle p_{n+1}=p_{n}}$

and

${displaystyle r_{n+1}=r_{n}}$

, so that the
matrix equation

${displaystyle {vec {v}}_{n+1}=T{vec {v}}_{n}}$

becomes

${displaystyle {vec {v}}=T{vec {v}}}$

.
We are therefore looking for eigenvectors for

${displaystyle T}$

that are associated with
the eigenvalue

${displaystyle 1}$

.
The equation

${displaystyle (I-T){vec {v}}={vec {0}}}$

is

${displaystyle {begin{pmatrix}.10&-.01\-.10&.01end{pmatrix}}{begin{pmatrix}p\rend{pmatrix}}={begin{pmatrix}0\0end{pmatrix}}}$

which gives the eigenspace: vectors with the restriction that

${displaystyle p=.1r}$

.
Coupled with additional information,
that the total world population of this species is
is

${displaystyle p+r=110,000}$

, we find that the stable state is

${displaystyle p=10,000}$

and

${displaystyle r=100,000}$

.

If we start with a park population of ten thousand animals,
so that the rest of the world has one hundred thousand, then every year
ten percent (a thousand animals) of those inside will leave the park,
and every year one percent (a thousand) of those from the rest of
the world will enter the park.
It is stable, self-sustaining.

Now imagine that we are trying to gradually build
up the total world population of this species.
We can try, for instance, to have the world population grow at a rate
of 1% per year.
In this case, we can take a “stable” state for the park’s population to
be that it also grows at 1% per year.
The equation

${displaystyle {vec {v}}_{n+1}=1.01cdot {vec {v}}_{n}=T{vec {v}}_{n}}$

${displaystyle ((1.01cdot I)-T){vec {v}}={vec {0}}}$

, which gives this system.

${displaystyle {begin{pmatrix}.11&-.01\-.10&.02end{pmatrix}}{begin{pmatrix}p\rend{pmatrix}}={begin{pmatrix}0\0end{pmatrix}}}$

The matrix is nonsingular, and so the only solution is

${displaystyle p=0}$

and

${displaystyle r=0}$

.
Thus, there is no (usable) initial population that
we can establish at the park
and expect that it will grow at the same rate as the rest of the world.

Knowing that an annual world population growth rate of 1% forces an
unstable park population,
we can ask which growth rates there are that would
allow an initial population for
the park that will be self-sustaining.
We consider

${displaystyle lambda {vec {v}}=T{vec {v}}}$

and solve for

${displaystyle lambda }$

.

${displaystyle 0={begin{vmatrix}lambda -.9&-.01\-.10&lambda -.99end{vmatrix}}=(lambda -.9)(lambda -.99)-(.10)(.01)=lambda ^{2}-1.89lambda +.89}$

A shortcut to factoring that quadratic is our knowledge that

${displaystyle lambda =1}$

is an eigenvalue of

${displaystyle T}$

, so the other eigenvalue is

${displaystyle .89}$

.
Thus there are two ways to have a stable park population (a population that
grows at the same rate as the population of the rest of the world, despite
the leaky park boundaries): have a world population that is does not
grow or shrink, and have a world population that shrinks by 11% every year.

So this is one meaning of eigenvalues and eigenvectors— they give a
stable state for a system.
If the eigenvalue is

${displaystyle 1}$

then the system is static.
If the eigenvalue isn’t

${displaystyle 1}$

then the system is either growing or
shrinking, but in a dynamically-stable way.

## Exercises

Problem 1

What initial population for the park discussed above
should be set up in the
case where world populations are allowed to decline by 11% every year?

Problem 2

What will happen to the population of the park in the event of
a growth in world population of 1% per year?
Will it lag the world growth, or lead it?
Assume that the park population is ten thousand, and the
world population is one hundred thousand,
and calculate over a ten year span.

Problem 3

The park discussed above is partially fenced so that now,
every year, only 5% of the animals from inside of the park leave (still,
about 1% of the animals from the outside
find their way in).
Under what conditions can the park maintain a stable population now?

Problem 4

Suppose that a species of bird only lives in Canada, the United States,
or in Mexico.
Every year, 4% of the Canadian birds travel to the US, and 1% of them
travel to Mexico.
Every year, 6% of the US birds travel to Canada,
and 4% go to Mexico.
From Mexico, every year 10% travel to the US, and 0% go to Canada.

1. Give the transition matrix.
2. Is there a way for the three countries to have constant
populations?
3. Find all stable situations.

Solutions