Biology Reference
In-Depth Information
The sum of the entries of the vector above is 1060.81, and
⎛
⎝
⎞
⎠
⎛
⎝
⎞
⎠
855
.
80
0
.
806
108
.
41
0
.
102
1
1060
37
.
31
0
.
053
=
.
17
.
79
0
.
017
.
81
.
.
18
05
0
017
.
.
23
45
0
022
The entries for the normalized vectors for
k
200 are actually equal to
6 decimal places, illustrating the convergence established by Eqs. (
7.10
) and (
7.11
).
Furthermore, based on Eq. (
7.8
), if this distribution is the same as the eigenvector
associated with the dominant eigenvalue, we should be able to multiply it by either
A
or
=
100 and
k
=
λ
and obtain the same vector. When we do this, we find
⎛
⎝
⎞
⎠
⎛
⎝
⎞
⎠
⎛
⎝
⎞
⎠
⎛
⎝
⎞
⎠
.
806
.
955
.
806
.
955
.
102
.
121
.
102
.
121
.
.
.
.
053
042
053
042
A
=
,
and 1
.
184
=
,
.
.
.
.
017
199
017
199
.
017
.
020
.
017
.
020
.
022
.
026
.
022
.
026
so the equilibrium state is the normalized eigenvector for the dominant eigenvalue.
7.9
DETERMINING POPULATION GROWTH RATE
AND THE STABLE DISTRIBUTION
Now that we have explained the mathematics behind determining the stable distri-
bution of individuals across stages, we can begin to interpret the biological meaning
of eigenvalues and eigenvectors. As shown above, the normalized long-range stable
distribution of individuals across states is the same as the normalized eigenvector
associated with the dominant eigenvalue. This eigenvector is also sometimes called
the
right eigenvector
. Therefore, if we can calculate this eigenvector from a projection
matrix and normalize it so that the sum of the entries is 1, we get the proportion of
individuals in each stage when the population is in a stable state.
The dominant eigenvalue itself also provides useful biological information; it is
an estimate of
population growth
. If the eigenvalue is larger than 1, the population is
predicted to grow, and if it is less than 1, the population will diminish. This estimate
is useful when trying to understand the future of a population, but it assumes that the
projection matrix stays constant over time, which is biologically unrealistic. As a pop-
ulation grows, competition and resource availability are likely to change survival and
reproduction rates, resulting in changes in the projection matrix. In addition, a con-
stant projection matrix ignores any changes in the environment from year to year. The
estimate of growth provided by the dominant eigenvalue, therefore, is more a measure
of what the population is capable of, given its current state, than a measure of what
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