Biology Reference
In-Depth Information
7.7
THE STABLE DISTRIBUTION OF INDIVIDUALS
ACROSS STAGES
In some cases, it might be helpful to predict the number of individuals in each stage
one time interval into the future, but most of the time we want to make predictions
further into the future. From the projection matrix we can compute the predicted
average number of members at any later time. For example, if we wanted to predict
the number of individuals in each stage after two time intervals, we could calculate
the number in each stage after one interval and then multiply the resulting vector by
the transition matrix again. Mathematically, we can write
A
An
t
0
)
=
A
2
n
n
(
t
0
+
2
)
=
An
(
t
0
+
1
)
=
(
(
t
0
).
The vector of the number of individuals in each stage at time
t
0
is multiplied by
the projection matrix once for each time interval; in this case, since there are two time
intervals, the projection matrix is raised to the second power.
To predict the distribution of individuals over stages at any time, we multiply the
initial vector of the number of individuals in each stage class by the projection matrix
to a power equal to the number of time intervals. Therefore, for any positive integer
k
A
k
n
n
(
t
0
+
k
)
=
(
t
0
).
(7.7)
After each generation, the population will grow or shrink in overall size, but the
proportion
of individuals in each stage at time
t
0
+
k
becomes very similar to the
proportion
at time
t
0
+
1 for large values of
k
. For example, if we predict the number
of individuals in each stage class for a population with projection matrix (
7.2
) and the
initial distribution of vector (
7.3
) after 10 years, we get the following distribution:
k
+
⎛
⎝
⎞
⎠
4609
.
7
591
.
5
204
.
2
n
(
t
0
+
10
)
=
.
.
94
6
.
91
1
125
.
6
For the same population and initial distribution, we predict the following distribution
after 11 years:
⎛
⎝
⎞
⎠
5406
.
0
691
.
4
241
.
2
n
(
t
0
+
11
)
=
.
114
.
0
109
.
8
147
.
7
To compare these two vectors in terms of the
relative
distribution of individuals across
stages, we can standardize by dividing each value in the vector by the total number
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