Environmental Engineering Reference
In-Depth Information
be no more erosion, so the maximum 'carrying capacity'
that can be reached (
K
s
) is the total soil depth. Unfor-
tunately, in soil-erosion formulae, such as the Universal
Soil Loss Equation (USLE: see Chapter 15), the erosion is
assumed to proceed to potential depth, even where this
might be deeper than the actual soil depth. Sometimes it is
assumed by the unwitting model user that bedrock can be
consumed at that rate! Claims for very high soil-erosion
rates based on the USLE should be treated with caution.
Although the logistic equation has not been validated for
soil erosion, the 'slowing down' effect is often recorded
from suspended sediment data and ascribed to soil 'deple-
tion'. This depletion is usually taken to indicate that all
the 'available' soil has been used up.
The logistic assumption is very important because it
underpins many ecological models developed by Lotka
and Voltera (Lotka, 1925; Voltera, 1926) and described
by Maynard Smith (1974) and May (1973). These models
form the basis of the following sections of this paper. A
more detailed discussion of the technique and the details
of the modelling approach which follows is contained in
Thornes (1985).
E
•
E
= 0
E
max
V
Isoline
Trajectories from
different starting
points
Figure 24.4
System behaviour in terms of erosion in the
absence of vegetation. The solid line is the isocline along which
there is no change in the erosion (
dE
/
dt
=
0).
E
a
III
24.3 Complex interactions
a
a
II
There are constraints to the logistic growth described
above and these give rise to some very interesting and
complex behaviour. The interactions can be set up as
competition between erosion and plant growth. As in
the above section, more plant cover means less erosion,
so the tendency to erosion is inhibited by vegetation
cover. Thinner soils mean less available soil moisture
and so less plant growth. Following Maynard Smith's
analysis (1974), Thornes (1985) set up the differential
growth equations for plant logistic growth and erosion
logistic growth, constrained by erosion and vegetation
respectively. The mathematical argument is not repeated
here but, by solving the partial differential equations, the
behaviour of the erosion-vegetation interactions can be
explored. By this procedure, we can start the system at
any value of
E
and
V
and follow the behaviour as solved in
time by the equations. Figure 24.4 shows how the erosion
behaves in the absence of vegetation. The line
d
dt
=
E
=
0
is called an isocline. It shows all the values along which
there is no change in the erosion rate (it is in equilibrium).
For a point below the isocline, the erosion rate increases
(moves up) until the isocline is reached. For points above
the isocline, their time trajectories move down to the line
of equilibrium. Figure 24.5 shows the equivalent diagram
a
I
V
V
man
Figure 24.5
Isocline for vegetation in the absence of erosion
(
dV
/
dt
=
0).
for the vegetation system behaviour in the absence of
erosion. In this case, the isocline is the point along which
dV
dt
=
V
0 and if the system is in any state in the plane
(given by
E
,
V
values), the trajectories within the isocline
carry the system towards higher
V
until they eventually
reach the isocline. Notice that, outside the isocline, the
systemmoves to the right until it again reaches the isocline
V
=
0. For the left-hand rising limb, the system always
flows away from the equilibrium. Thus, at point
a
,the
vegetation will move to full cover at point
a
I
if it is just to
the right of the isocline. If it is just to the left of the isocline,
it will move towards zero vegetation cover at point
a
II
. So,
all the points on the left hand, rising limb of the isocline
are unstable equilibria because a slight perturbation will
carry the system away from the equilibrium to either
zero or complete cover (nearly 100%). Because there is
convergence on the right hand set of equilibria (e.g. point
=
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