Environmental Engineering Reference
In-Depth Information
individuals in the adult age-class, generate a random number between 0 and 1. If the
random number is less than or equal to 0.8, the individual survives. If it is greater
than 0.8, it dies. Note that this procedure doesn't make the model individual-based,
because you don't track individuals from class to class over time.
Environmental variation
can be important at any population size. The rele-
vance of environmental variation depends on the relative magnitude of the mean
and variance of the parameter concerned—if the variance is substantial in relation
to the mean (say, as a rough guide, more than 10%), it should be included.
Environmental variation can be incorporated in two main ways:
●
If the main issue is not day-to-day variation but occasional
catastrophes
(e.g. periodic fires that completely transform plant demographic rates, Morris
et al
. 2006), you can develop several matrices, for example one for good years
and one for catastrophe years, and then use one or other each year (for example,
if the chance of a catastrophic winter is 0.1, then if your random number
is
0.1 the catastrophe matrix is used).
●
For more usual types of environmental variation, one approach is to develop
the type of model that is used for regression-based sensitivity analyses, in
which a particular year's vital rate is picked from a
distribution
. This could be
a uniform distribution (in which each value within the feasible range has the
same chance of being picked) or a more complex distribution such as a
Normal distribution (see Appendix for code to generate a Normal distribu-
tion). If there are data showing that environmental variation leads to
correlated variation in vital rates (i.e. in bad years, survival is poor in all age
classes and reproduction is also low, and the opposite in good years; Ezard
et al
. 2006), then you can model these correlations explicitly. A simple way to
do this is to use correlated Normal distributions. This was done in the example
deer model (Milner-Gulland
et al
. 2004). For two correlated vital rates
A
and
B
with mean values
x
and variances
s
, a stochastic value,
, for vital rate
A
is
first drawn from a Normal distribution by using a Normal deviate,
z
(this is a
random number that is picked from a Normal distribution of mean zero and
standard deviation 1; see the Appendix for some code to calculate one):
A
x
A
s
A
z
A
If
r
is the correlation between the vital rates, a random draw for vital rate
B
is then
given by:
r
2
B
x
B
z
B
s
B
1
z
A
r
Stochastic models can become quite complex and challenging to programme.
However, it is important to include variability in the model if populations are
small or variation is large. This is because, for non-linear systems like biological
populations, stochastic models do not produce the same mean outputs as deter-
ministic models. A deterministic model overestimates the mean population size in
a stochastic system. This is due to a mathematical phenomenon called the lack of
