Biology Reference
In-Depth Information
b.
The algebraic dimension, and thus the geometric dimension, of the eigenspace
of
r
is 1.
c.
There is an expression of the eigenvector of
r
in which all of the entries are
positive.
In summary, the theorem states that positive, square matrices have a single positive
eigenvalue that is larger than the others, and associated with that eigenvalue is a
single eigenvector with only positive entries. We can use this eigenvalue and its
associated eigenvector to calculate the stable distribution of individuals across stages
for a projection matrix after an infinite number of time intervals. This distribution and
its eigenvalue provide information about the possible future growth and structure of
a population.
The assumptions of the theorem are not always met bymodels built frombiological
examples, but the theorem still applies if certain additional assumptions are met.
Projection matrices are structured to have the same number of columns and rows,
because we allow for the possibility that every stage could transition to every other
stage. Therefore, the assumption of a squarematrix is met. Amatrix
A
is
positive
when
(
0 for all
i
and
j
. (In other words, a matrix is positive when every element in
the matrix is greater than 0.) Our example matrix (
7.2
) is clearly not a positive matrix;
it has a number of elements that are 0. Fortunately, the Perron-Frobenius theorem still
holds if the matrix is nonnegative, irreducible, and primitive.
A
nonnegative
matrix may contain zeroes but has no negative elements; that is
A
)
i
,
j
>
(
0 for all
i
and
j
. Models constructed from biological data do not contain neg-
ative values because organisms can only contribute neutrally or positively to growth;
therefore, we expect all the matrices we construct to be nonnegative.
Amatrixis
irreducible
if there is a path through the life cycle that connects every
stage to every other stage, or more generally, beginning at any stage
i
it is possible
to get to the stage
j
in a finite number of steps. If a matrix contains a stage that acts
only as a sink and does not contribute to reproduction or any other life stage, the
matrix is
reducible
. Reducible matrices are uncommon but they can occur when a
post-reproductive life stage is included in the model (Figure
7.3
). (See [
1
]formore
information on working with reducible models.) Our ginseng example is irreducible
because every individual has the potential to contribute to reproduction, either directly
or by passing through other life stages first.
A nonnegative, irreducible matrix must also be
primitive
for the Perron-Frobenius
theorem to still hold true. A primitive matrix is a matrix that, when raised to a suffi-
ciently high power, contains only positive elements. In other words, there is a positive
integer
m
for which
A
)
i
,
j
≥
A
m
0 for every entry in the matrix
A
. Again, it is uncom-
mon for a projection matrix constructed from biological data to fail to be primitive,
but it can occur if organisms only reproduce after they reach a certain stage and then
die before reproducing again (Figure
7.4
). If a matrix contains a self-loop (that is,
individuals in a stage can stay in that stage in the following time step), it is primi-
tive. We can also raise the matrix to a higher power and see whether the elements all
become positive to determine whether it is primitive. Reducible matrices will also fail
(
)
i
,
j
>
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