Environmental Engineering Reference
In-Depth Information
trader is higher or lower than their cost of hunting (which includes opportunity
cost, direct costs of equipment and the cost of the effort expended on the hunting
trip). Each hunter kills one animal per hunting trip. The hunted population has
logistic growth (Chapter 1).
In this model, we have two kinds of box—one that represents parameters and
variables of the model (rectangle) and one that represents decisions (oval). The
parameters which are constant throughout have one symbol (
K, r
max
, s, p
), while
the variables, which vary from year to year, have a subscript representing time (
N
t
,
C
t
,
P
t
,
H
t
)
.
Finally, the individual cost,
c
i,t
, kill,
h
i,t
, and profitability,
B
i,t
, have two
subscripts, meaning that they vary both between individuals,
i
, and between years,
t
.
The double lines around the boxes for individual variables indicate that the process
has to be repeated for each individual hunter and then the number killed summed
to get the total. This model just shows the process for one year, but the arrow
onwards shows where the next year starts. For clarity of exposition we have the
biological component of the model in the top half of the diagram, and the
economic component in the bottom half.
The next step is to turn the conceptual model into equations which can be
parameterised from data and used in a model. This should be an easier task
because we have already introduced notation into the conceptual model. It's very
helpful to use standard notation if you can (e.g.
N
or
x
are most commonly used for
population size,
t
is commonly used for time, and
i
and
j
for age, stage or sex classes).
Use single symbols if at all possible, with subscripts for different components (i.e.
N
t
is population size at time
t
), rather than names or two-letter symbols, because
these can be confusing. This may all sound petty, but it's easy to make mistakes with
sloppy notation, as well as making it hard for others to follow what you're doing.
5.3.2.1 Deer mathematical model
For demographic models, the classic approach is to use a
Leslie matrix
. Getz and
Haight (1989) provide a particularly accessible introduction to matrix modelling
for harvested species. The Leslie Matrix summarises the transitions in the flow dia-
gram and uses matrix algebra to get the next year's population size as a function of
this year's. The equation is:
N
t
1
AN
t
where
N
t
is a vector of the number of individuals in each class, and
A
is the transi-
tion matrix:
00
P
S
1
0
0
0
S
2
S
A
(1
H
)
3 because there are three age classes in the model. The top
row is the contribution to fecundity—here only the adult class contributes to
The matrix is 3
