Environmental Engineering Reference
In-Depth Information
GAUSS
5e-03
EXP
GEOM
5e-04
5e-05
5e-06
0
100
200
300
400
Distance from origin (x)
Fig. 16.2 Examples of dispersal kernels: Gaussian function (“GAUSS”; thin-tailed), exponential
function (“EXP”) and geometric function (“GEOM”; fat-tailed). The
y
-axis is on a logarithmic
scale for easier recognition of the shape of kernel tails
kernels are leptokurtic, with dispersal to both short and long distances occurring
more frequently than they would under a Gaussian kernel (Fig.
16.2
). Moreover,
different types of tails exist among leptokurtic dispersal kernels such as thin-tailed
kernels (quicker decrease than an exponential), exponential-like kernels (long-
range decrease similar to an exponential) and fat-tailed kernels (slower decrease
than an exponential). These dispersal kernels generate different colonization pat-
terns (Clark et al. 2001), mixing propagules and gene flow at long distances
(Devaux et al. 2007).
The kernel matrix K describes dispersal events occurring during each demo-
graphic transition in the stage-structured integro-differential model. Each term
k
ji
(row
j
, column
i
) of matrix K corresponds to the dispersal kernel for dispersal
events occurring during the transition from stage
i
to stage
j
. The absence of
dispersal is modelled by a Dirac delta function
(
x
) which (very roughly speaking)
d
is zero if
x
0, and integrates to 1. In our two-stage
example (Fig.
16.1
), dispersal only occurs when seeds are released from mature
plants, i.e. during the transitions F
6¼
0, is infinite when
x
¼
!
S (incorporation of dispersed seeds into the
seed bank) and F
!
F (germination of dispersed seeds). The kernel matrix thus
d
ð
x
Þ
k
ð
x
Þ
writes: K
¼
d
ð
x
Þ
k
ð
x
Þ
Model Output
As in non-spatial stage-structured population models, the asymptotic population
growth rate
l
in the integro-differential stage-structured model corresponds to the
