Environmental Engineering Reference
In-Depth Information
5.4
Simulation
modeling for
population viability
analysis
Simulation models provide an alternative, more specifi c way of gauging viability.
These encapsulate survivorships and reproductive rates in age-structured popula-
tions. To see what is involved let's consider the example of a seabird population
(Morris et al., 1999). The fi rst step is to arrange the available information on birth
and death rates into a matrix where the columns represent three different age classes
(from left to right - juveniles, subadults and adults) and the rows represent the
probability over 1 year that an individual in one age class will progress to the next.
Thus, in the matrix M below there is a probability of 0.665 that a juvenile will
survive to be a subadult (i.e. 43.5% of juveniles die as juveniles), a probability of
0.724 that a subadult will survive to be an adult and a probability of 0.95 that an
adult will still be alive the following year (i.e. only 5% of adults die each year). The
fi rst row contains the crucial birth rate information: only adults have babies, and
each adult only produces 0.054 surviving juveniles per year.
From:
Juveniles
Subadults
Adults
Juveniles
Subadults
Adults
0
0.
0
0
0 724
0 054
0
095
.
TO
:
665
=
M
0
.
.
Next we construct an initial population 'vector', which consists simply of numbers
in each age class in year 1 (
N
t
): for our seabird population this is 40 juveniles, 50
subadults and 60 adults. Now the matrix M is multiplied by the vector, as shown
below, to give us new numbers for each age class in the following year (
N
t
+1
; 3.2
juveniles, 26.6 subadults and 93.2 adults). This new population vector is then mul-
tiplied by the matrix M to give numbers in the third year, and so on. Some taxa,
plants for example, may be better represented by size classes rather than age classes,
and there may be from two to many classes in the model, but the procedure is the
same.
M
×
N
=
N
t
t
+
1
0
0
0 054
.
40
50
60
0
×
40
)
+×
0
50
)
+
0 054
.
×
60
32
26 6
93 2
.
(
(
(
)
0 665
.
0
0
×
=
0 665
.
×
40
)
+×
0
50
)
+×
0
60
=
.
.
(
(
(
)
0
0 724
.
0 95
.
040
×
)
+
07
.24
×
50
)
+
0 95
.
×
60
(
(
(
)
)
refl ecting the birth and death rates of age classes in the population in question. But
to better represent reality, random variations in the rates in the matrix are intro-
duced to denote the impact of environmental variation. This can be done each year,
for example, by drawing each matrix entry from a continuous range of possible
values whose mean and variance have been estimated from the available data. The
models can be extended to include disasters with a specifi ed frequency (one 'random'
year in each century, for example) and intensity (refl ected in the extent to which
birth and death rates are affected). Density dependence can be introduced where
required, as can population harvesting or supplementation. In the more sophisti-
cated models, every individual is treated separately in terms of the probability, with
its imposed uncertainty, that it will survive or produce a certain number of offspring
Ultimately the simulated population will take on a constant rate of growth (
λ
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