Biology Reference
In-Depth Information
The probability that a two-leaved plant is a three-leaved plant at time t 0 +
1is
0.41, therefore A 5 , 4 =
0
.
41.
The probability that a two-leaved plant is a four-leaved plant at time t 0 +
1is
0.05, therefore A 6 , 4 =
0
.
05.
If we continue to build the matrix in the same fashion from Figure 7.1 , we get the
complete projection matrix in tabular form, presented in Table 7.3 .
The same information can be described by the following projection matrix:
0
0
0
0
.
27
3
.
90
40
.
00
0
.
15
0
0
0
0
0
0
0
.
21
0
.
55
0
.
05
0
0
.
(7.2)
0
0
0
.
35
0
.
45
0
0
0
0
0
0
.
41
0
.
78
0
0
0
0
0
.
05
0
.
19
1
.
0
While the general model allows for individuals in all life stages to potentially
transfer to any other life stage within one time interval and reproduce offspring that
appear in any other life stage, the biological reality of our study species means several
transitions do not actually occur. Therefore, we have a number of zeroes in our matrix,
and only the first row reflects the creation of offspring. In addition, the probability
that a four-leaved plant stays a four-leaved plant is an interesting case; because no
four-leaved plants died or reverted to a smaller stage over the course of the study, the
probability is one, and it appears that the four-leaved plants are immortal. This cannot
happen in reality, but because these probabilities are our best estimates based on data
and we have no additional information from which to estimate the “true” number,
we will leave it at one. Gathering additional years of data from this population could
generate a more biologically realistic estimate.
It is also useful to point out that all matrix elements except those involving the
production of seeds are transition probabilities and are thus less than or equal to
one. However, for the first row, the values represent the average number of seeds
Table 7.3 The projection matrix for a population of American ginseng (data
from [ 10 ]).
Stage at time t 0
Seed Seedling One-
Two-
Three-
Four-
leaved
leaved
leaved
leaved
plant
plant
plant
plant
Stage Seed
0
0
0
0.27
3.90
40.00
at
Seedling
0.15
0
0
0
0
0
time
One-leaved plant
0
0.21
0.55
0.05
0
0
t 0 +1
Two-leaved plant
0
0
0.35
0.45
0
0
Three-leaved plant
0
0
0
0.41
0.78
0
 
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