Environmental Engineering Reference
In-Depth Information
To generalize, a Leslie model describing the population dynamics of a popula-
tion with
i
1, ...,
n
age classes, age-specific survival rates
S
i
and age-specific
fertility rates
F
i
can be written in the form:
¼
0
@
1
A
0
@
1
A
0
@
1
A
,
x
ðt þ
1
Þ¼
Lx
ðtÞ
x
1
ð
t
þ
1
Þ
F
1
F
2
F
n
x
1
ð
t
Þ
x
2
ð
t
þ
1
Þ
S
1
0
0
x
2
ð
t
Þ
.
.
.
.
x
n
ð
.
.
x
n
ð
.
.
¼
0
S
2
.
.
.
.
.
.
.
.
.
.
.
t
þ
1
Þ
t
Þ
0
0
S
n
1
S
n
The Leslie matrix L contains the life-history parameters of the population, i.e.
survival and reproduction rates. Obviously some of the
F
i
-values in the Leslie
matrix are usually zero, because in natural populations only certain age classes
will be reproductive ones. Starting with a given initial population, this equation can
be used to update the population vector for future points in time.
Other criteria such as size classes or different stages of the life cycle can also be
considered, without changing the structure or behaviour of the model.
Long-Term Behaviour
Several characteristics concerning the long-term behaviour of the population can be
derived from the projection matrix by means of some well-known results of matrix
algebra (see e.g. Kaw 2008; Meyer 2000; Searle 2006). First the model equation can
be written as:
L
2
L
tþ
1
~
x
ð
t
þ
1
Þ¼
L
~
x
ð
t
Þ¼
~
x
ð
t
1
Þ¼...¼
~
x
ð
0
Þ
Second, it is known that any Leslie matrix L is similar to a diagonal matrix with
the eigenvalues
t+1
V
with
W
and
V
being composed of the right and left eigenvectors, respectively.
Hence the model can be rewritten as:
l
i
as diagonal entries. Hence L
t+1
can be transformed to
W
L
X
X
n
i¼
1
l
n
i¼
1
l
t
V
t
i
~
v
i
~
t
i
~
~
x
ð
t
Þ¼
W
~
x
ð
0
Þ¼
w
i
~
x
ð
0
Þ¼
w
i
c
i
L
Third, it is known that for each projection matrix L there exists one positive real
eigenvalue
l
1
of modulus greater than any other. This dominant eigenvalue there-
fore determines the asymptotic behaviour of the population as can be seen from the
above model representation. If
l
1
>
1 the population will grow exponentially,
whereas in case of
l
1
is often
called the growth rate of the population and is related to the intrinsic rate of increase
obtained from Lotka's equation via
r
l
1
<
1 the population will decrease. For this reason
¼
ln(
l
1
).
