Environmental Engineering Reference
In-Depth Information
Y
2
Y
1
K
aY
1
f
1
ð
Y
Þ¼
rY
1
1
(7.10)
þ
bY
1
1
Y
2
g
2
Y
2
aY
1
f
2
ð
Y
Þ¼
e
mY
2
Y
3
(7.11)
h
2
Y
2
1
þ
bY
1
1
þ
Here, the prey grows logistically with growth rate
r
and carrying capacity
K
. The
consumption of prey by the specialist predator
Y
2
is modelled with the so-called
Holling-type II functional response, which assumes a linear relation between
prey density and prey consumption at low prey densities, but saturates if the prey
becomes abundant. This takes into account that there is maximum value of prey
biomass that each predator can consume in a given time. This maximum value is
given by
a
/
b
, the ratio of search rate
a
and prey handling time
b
. The parameter
e
1 is the predator's conversion efficiency and
m
its mortality. The constant top
predator is assumed to be a generalist, described by a Holling-type III functional
response. This functional response saturates at
g
2
/
h
2
, but assumes a lower than
linear consumption at low prey densities. This reflects that the generalist predator
Y
3
switches to a significant consumption of the specialist predator only when
Y
2
becomes abundant. If the top predator is absent, that is
Y
3
¼
<
0, model (
7.10
and
7.11
) reduces to the classical Rosenzweig-MacArthur predator-prey model (1963).
In this reduced model, the unique stationary point where both population densities
are strictly positive can be stable or unstable. In the unstable case, the equilibrium is
surrounded by a stable limit cycle, which corresponds to periodically varying
population densities. As we will see, the form of the spatiotemporal patterns that
can be observed in the full reaction-diffusion model greatly depends on whether the
spatially homogeneous system given by (
7.10
and
7.11
) is in the parameter range of
stationary or periodic dynamics.
7.4.1 Turing Patterns
Turing patterns are perhaps the most famous spatial patterns arising from reaction-
diffusion systems (Turing 1952). These stationary patterns appear after diffusive
instability of a stable, spatially uniform population distribution. For them to arise,
the diffusion coefficients of the two species need to be sufficiently different, i.e.
D
2
D
1
, and the growth terms have to obey certain conditions. For two interacting
species, these conditions are called activator-inhibitor (Gierer and Meinhardt 1972)
or destabilizer-stabilizer (Segel and Jackson 1972) relations. Because of their often
striking polarity and symmetry, Turing had thought them as a possible mechanism
of forming physiological gradients in biomorphogenesis. Applications in population
dynamics soon followed (Segel and Jackson 1972). Three simulation results of (
7.10
and
7.11
) for different initial conditions are shown in Fig.
7.4
.
