Environmental Engineering Reference
In-Depth Information
dispersal kernels K(
x
,
y
)). The integral term represents the sum of all the contribu-
tions originating from all locations
y
.
The Demographic Component: A Transition Matrix
The matrix A governs the demographic transitions occurring between each devel-
opmental stage at each time step. Each term
a
ji
(row
j
, column
i
) of matrix A
corresponds to the demographic parameter of the transition from stage i to stage j.
For example, let us consider a simple two-stage model for an annual plant species
with a soil seed bank where stage S corresponds to seed bank seeds and stage F to
mature plants (Fig.
16.1
). The population is censused just before seed release at the
end of the reproductive period.
Let us assume that we consider demography on location
x
. The transition matrix
A allows computation of the density of individuals in both stages S and F at time
t
þ
1 (population vector n(
x
,
t
þ
1)) from the density at time
t
through the relation
. The matrix A can be made space-
sp
gg
n(
x
,
t
þ
1)
¼
An(
x
,
t
), with A
¼
dependent, for example to model spatial heterogeneity in habitat quality (i.e. direct
dependence on location
x
), or dependent on the density n(
x
,
t
) at location
x
to
include density-dependence, such as in the integro-difference equation above. See
Chap. 9 and Caswell (2001) for more details on transition matrices and stage-
structured population models.
The Dispersal Component: A Matrix of Dispersal Kernels
A dispersal kernel is a probability density function
k
that gives the probability
k
(
x
,
y
)
that an individual (or a particle, e.g. pollen) moves from location
x
to location
y
.
This general notation of
k
does not make any assumption about the spatial homo-
geneity regarding dispersal. If the space is homogeneous, then dispersal from
location
x
to location
y
only depends on the distance separating
x
and
y
(
d
(
x
,
y
),
say |
y
-
x
| in a one-dimensional habitat) and thus
k
(
x
,
y
)
¼
k
(
d
(
x
,
y
)). Most dispersal
fp
s
S
fg
F
g
Fig. 16.1 A two-stage life cycle diagram. Stages: seed bank seeds (S) and mature plants (F).
Demographic parameters: annual survival in the seed bank (
s
), germination (
g
), rate of seed
incorporation in the seed bank (
p
) and seed production (
f
)
