Biology Reference
In-Depth Information
As before, it is important to note that this distribution does not give us the proportion
(relative distribution) of individuals at each of the life stages. To obtain the relative
distribution at the steady state, each of the elements of the vector
c
1
v
1
will have to be
divided by the sum of all the elements.
From a practical perspective, Eq. (
7.11
) states that to find the relative distribution
at the steady state, one needs to compute the dominant eigenvalue of the projection
matrix, calculate an eigenvector for the dominant eigenvalue, and then normalize it
by dividing each of its components by the sum of its elements.
To illustrate the convergence fromEq. (
7.11
), we turn again to the ginseng example
with projection matrix
A
from Eq. (
7.2
) and distribution vector
n
from Eq. (
7.3
).
Based on Eq. (
7.10
), we expect that if we calculate the dominant eigenvalue
(
t
0
)
λ
1
for
1
λ
A
k
n
the matrix and evaluate the expression
for large values of
k
, we will obtain
approximations for both the steady state distribution of individuals across stages and
of the eigenvector associated with the dominant eigenvalue.
Using software for the computations (see next section for the appropriateMATLAB
andR commands), we can find that the dominant eigenvalue for matrix
A
is
(
t
0
)
1
184.
We can then compute the proportions of individuals across stages
k
years into the
future from:
λ
=
1
.
A
k
n
(
t
0
)
.
k
λ
If we let
k
=100, we find
⎛
⎝
⎞
⎠
848
.
37
107
.
47
A
100
n
(
t
0
)
36
.
98
=
.
100
17
.
63
λ
17
.
89
23
.
25
In order to compare this vector with others, we must normalize it to a sum of 1. The
sum of the entries of the vector above is 1051.59, so therefore
⎛
⎞
⎛
⎞
848
.
37
0
.
806
⎝
⎠
⎝
⎠
107
.
47
0
.
102
1
1051
36
.
98
0
.
053
=
.
17
.
63
0
.
017
.
59
17
.
89
0
.
017
23
.
25
0
.
022
Letting
k
=200 gives
⎛
⎞
855
.
80
⎝
⎠
108
.
41
A
200
n
(
t
0
)
37
.
31
=
.
17
.
79
λ
200
18
.
05
23
.
45
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