Environmental Engineering Reference
In-Depth Information
1.5
1.8
0.8
0.1
age
class 1
age
class 2
age
class 3
age
class 4
0.5
0.6
0.7
Fig. 9.1 Illustration of a life cycle with four age classes. The
solid line arrows
indicate survival
rates and the
dotted line arrows
indicate fertilities
The life cycle of a population with four age classes is given in Fig.
9.1
. The solid
line arrows represent the survival rates to the next age class, whereas the dotted line
arrows indicate the reproduction of the different age classes. The first age class is
assumed to be non-reproductive, so there is no dotted line arrow starting from this
class. All individuals surviving to or beyond age class 4 are aggregated in that age
class. Individuals of the first age class will survive with a probability of 0.5 and will
then be in age class 2. Likewise, individuals in age class 2 will survive to age class 3
with a rate of 0.6, and those of class 3 will be in age class 4 with a 70% probability.
10% of the individuals in age class 4 will survive and will remain in age class 4.
Individuals of age classes 2-4 contribute to the first age class with rates of 0.8, 1.8
and 1.5, respectively. Therefore, if
x
i
(
t
) denotes the number of individuals in age
class
i
at time
t,
the following equations can be set up to evaluate the number of
individuals at the next time step
t
+1:
x
1
ð
t
þ
1
Þ¼
0
:
8
x
2
ð
t
Þþ
1
:
8
x
3
ð
t
Þþ
1
:
5
x
4
ð
t
Þ
x
2
ð
t
þ
1
Þ¼
0
:
5
x
1
ð
t
Þ
x
3
ð
t
þ
1
Þ¼
0
:
6
x
2
ð
t
Þ
x
4
ð
t
þ
1
Þ¼
0
:
7
x
3
ð
t
Þþ
0
:
1
x
4
ð
t
Þ
Summarizing these equations into a matrix formulation one can write:
0
@
1
A
¼
0
@
1
A
0
@
1
A
x
1
ð
t
þ
1
Þ
00
5
0
:
50 0 0
00
:
81
:
81
:
x
1
ð
t
Þ
x
2
ð
t
þ
1
Þ
x
2
ð
t
Þ
x
3
ð
t
þ
1
Þ
60 0
000
:
x
3
ð
t
Þ
x
4
ð
t
þ
1
Þ
:
70
:
1
x
4
ð
t
Þ
~
~
consist of the number of individuals in the four age
classes at time points
t
and
t
+ 1, respectively. The matrix on the right hand side of
the equation is the so-called projection or Leslie matrix. In the first row of this
matrix the age-specific fertility rates are written and the sub-diagonal contains the
respective survival rates.
The vectors
x
ð
t
Þ
and
x
ð
t
þ
1
Þ
