Environmental Engineering Reference
In-Depth Information
fecundity, so its cell is the only non-zero one. The sub-diagonal shows the propor-
tion of an age-class moving to the next age-class through survival (
S
1
and
S
2
), the
diagonal shows the proportion staying in the same age-class
S
A
(1
H).
Leslie Matrices have useful properties. It is possible to calculate values associated
with the matrix using a programme like Matlab or R; the dominant eigenvalue,
which is the population growth rate
, and the left eigenvector, which is the stable
age distribution when the population is at equilibrium.
The next issue is whether each of the cells of the matrix is just a number or a
function itself. Density dependence is an important part of population dynamics,
which can only be ignored for very small populations which are growing at their
maximal rate, not limited by resources (see Sections 1.3.1.1 and 2.4.4). In order to
incorporate density dependence, you need to decide which of the
vital rates
, i.e.
the cells of the matrix, are involved—is it juvenile survival
S
1
that is affected when
resources become limiting, or do adults have fewer offspring,
P
? In Milner-
Gulland
et al
. (2004), the form of density dependence was determined empirically
by fitting curves to data, and the shape for fecundity, for example, was:
1
P
t
e
(
a
b
.
N
t
)
1
i.e. the fecundity in a given year is a non-linear function of the total population size
in that year, where
a
and
b
are constants. Once density dependence is involved, the
model becomes more realistic but the eigenvalues and eigenvectors are no longer
easily calculable. This means that instead of finding an analytical solution using the
Leslie Matrix, a simulation approach is often used instead. The model is coded and
run as a series of vectors, cells of which are consecutively multiplied by the appro-
priate vital rate (see below). The Leslie Matrix is still a useful way to present the
model structure, however.
5.3.2.2 Hunting mathematical model
The hunting model has a number of components. First the biology:
N
t
1
N
t
P
t
H
t
N
t
K
P
t
r
max
N
t
1
This says that the population size next year is the population size last year plus the
productivity (which is represented by a logistic equation) minus the number killed
by hunting.
Next the economics:
B
i
,
t
p
c
i
,
t
If
B
i
,
t
0
,
h
i
,
t
1, else
h
i
,
t
0
H
t
i
[
h
i
,
t
]
