Environmental Engineering Reference
In-Depth Information
time 1
time 5
time 10
time 30
1.0
0.6
0.20
0.20
0.5
0.8
0.15
0.15
0.4
0.6
0.3
0.10
0.10
0.4
0.2
0.05
0.2
0.05
0.1
0.0
0.0
0.00
246810
246810
246810
246810
0.14
0.20
0.20
0.20
0.12
0.15
0.15
0.15
0.10
0.10
0.10
0.10
0.08
0.05
0.05
0.05
0.06
246810
246810
246810
246810
Fig. 9.3 Proportions of individuals (
y
-axis) in ten age classes (
x
-axis) at four time points for two
different initial distributions of ten individuals. For the upper panel the simulation was initiated
with all individuals being in the first age class, whereas in the lower panel the same number of
individuals were uniformly distributed over all age classes. In both cases the same stable age
distribution was reached after a few time steps
the population growth rate (Caswell 1978,
2001
; de Kroon et al. 2000). According to
Caswell (1978), the elasticities for each entry
l
ij
of the matrix L are evaluated as:
e
ij
¼
@
ð
log
l
1
Þ
l
ij
l
1
@
l
1
@
l
ij
l
1
v
i
w
j
~
¼
l
ij
¼
v
T
@
log
l
ij
w
~
with
l
1
. An important characteristic
of the elasticities is that they sum up to unity (de Kroon et al. 1986). For specific life
history parameters they thus can indicate the relative importance for the population
dynamics. Elasticity analysis decomposes the population growth rate into contribu-
tions made by the different life history parameters; e.g. growth, survival, reproduc-
tion and, therefore, points to the parameters where management measures should
~
v
and
~
w
being the left and right eigenvectors of
