Environmental Engineering Reference
In-Depth Information
Table 9.1 Life table of the red deer (Lowe 1969) and the resulting
projection matrix
Age (years)
Birth rate
Mortality rate
Survival rate
1
0
0.14
0.86
2
0
0.10
0.90
3
0.31
0.11
0.89
4
0.28
0.12
0.88
5
0.30
0.14
0.86
6
0.40
0.16
0.84
7
0.48
0.19
0.81
8
0.36
0.50
0.50
9
0.45
0.67
0.33
10
0.28
0.37
0.63
0
@
1
A
0
0
0
:
31
0
:
28
0
:
30
0
:
40
0
:
48
0
:
36
0
:
45
0
:
28
0
:
86
0
:
90
0
:
89
0
:
88
L
¼
0
:
86
0
:
84
0
:
81
0
:
5
0
:
33
0
:
63
maximal eigenvalue
1.047
A further result is that the population will finally reach an equilibrium state called
stable age distribution, given by the right eigenvector
l
1
¼
~
w
1
of
l
1
. This can be derived
from the characteristic equation L
~
w
1
¼
l
1
~
w
1
by considering the second to
n
th row
of this matrix, setting
w
11
¼
1 and solving for successive values, finally arriving at:
1
T
. Appropriately scaled, this stable
age distribution gives the proportion of individuals in the different age classes and
does not depend on the initial distribution. Only for the sake of completeness it should
be noted that the appropriate left eigenvector v
1
(sometimes called the reproductive
value vector of the population) can be interpreted as contribution values of the age
classes to future generations.
As an example, consider the life table of the red deer (adopted after Lowe 1969,
cited in Begon et al. 1996) (Table
9.1
). This life table results in the Leslie matrix
given on the lower panel of the table. Entries which are not explicitly given are zero.
In this example we are dealing with a Leslie model with ten age classes; i.e., 1-year-
old animals, 2-year-old animals and so on, up to animals 10 years old and older.
Based on this matrix a projection for the next 30 years was done, starting with an
initial population of 10 individuals, all belonging to the first age class or uniformly
distributed over all 10 age classes, respectively.
w
1
¼
S
1
l
1
1
S
n
1
l
nþ
1
1
;
;
;
S
1
S
2
...
