Environmental Engineering Reference
In-Depth Information
Y(x, t)
1
0.5
0
6
3
5
0
10
10
1
, all
Fig. 7.2 Spatial propagation of a logistically growing population for
r
¼
3,
K
¼
1,
D
¼
parameters given in arbitrary units (a.u.), no-flux boundary conditions
Y
K
f
ð
Y
Þ¼
rY
1
1
(7.6)
where
K
_ is the minimal viable population density. The system is bistable, with
extinction as well as carrying capacity as stable steady states. This changes the
dynamics (not only locally), such that, in the absence of noise, the initial condition
determines the final steady state. With diffusion, the same initial, smooth density
distribution as in Sect.
7.2.2
will not necessarily grow and propagate towards
capacity but can also break down. The front moves back towards total extinction, i.e.
until the population has died out everywhere. The two stable steady states introduce a
critical size of the spatial extent of a population (Malchow and Schimansky-Geier
1985). Population patches greater than the critical size will survive, while the others
will go extinct. In spherically symmetric coordinates, the temporal dynamics of the
radius
R
of a population patch is:
dR
dt
¼
1
R
k
1
R
2
D
(7.7)
where
R
k
is the critical radius that has to be exceeded in order to survive, even if the
local density is greater than
K
_. This is a superposition of two critical size problems,
which is demonstrated in Fig.
7.3
. It is important to understand that one can find
moving fronts in single population systems. In the long run however, the spatial
population distribution on a finite domain will be uniform.
