Environmental Engineering Reference
In-Depth Information
a
III
), these are stable equilibria. If the system starts on
the equilibrium, it is impossible for it to move away and,
in fact, if it is artificially displaced (e.g. by grazing) the
dynamics of the system force it to return to the isocline.
A few general comments are in order here. First, the
equilibria we have shown here are both unstable and
stable. In unstable equilibria, slight perturbations carry
the system away from the equilibrium, sometimes far
away and even a tiny perturbation can cause a big change.
This situation is like a ball balanced on the nose of a sea
lion. A tiny wobble (perhaps caused by a breeze) can easily
upset the balance. In the stable equilibrium, considerable
effort is needed to drive the system away from the isocline
and if it is driven away, then the complex behaviour
forces it back to the equilibrium, like a marble displaced
from its resting place at the bottom of a tea cup and
then rolling back to the lowest point. Stable equilibria
that draw in the system after perturbations are called
attractors. Figure 24.6 shows another metaphor for this
behaviour. Ball A is in unstable equilibrium and a slight
perturbation will push it into local attractor basins at
B or C. Further perturbations can force it into D or E.
F is called the global equilibrium because the ball will
remain stable for all perturbations and, following a set of
sequential perturbations, the system (ball) will end up at
F. There is another point here. The deeper the attractor
basin, the bigger the perturbation needed to displace the
system from the equilibrium.
In Figure 24.7, the combined effect of both the erosion
and vegetation components are shown. The points where
E
and
V
intersect are the joint equilibria - the point
where neither erosion nor vegetation is changing. At A,
both vegetation and erosion are zero. At B vegetation is
at a maximum and erosion is 0. At C, erosion is at a
maximum and vegetation is 0. The behaviour at D is a
mixture of unstable and stable. If the system is perturbed
in the areas with vertical lines, it will move in towards D, so
now the point D is acting as a repeller. In the vertical ADB,
C
X
E
max
E
= 0
D
Y
E
B
A
V
'Safe' zone for erosion system
always goes to
V
max
where
E
is close to zero
Figure 24.7
Combined effects of vegetation and erosion. The
intersection of the isoclines is where both variables are
unchanging. The shaded area is the 'safe' zone for erosion.
the system behaviour will carry it to the attractor at B,
which is very attractive for soil conservationists, because
it has zero erosion and maximum vegetation cover. In
the vertically lined area ADC, the system is forced to
the attractor at C, which is very undesirable because it
represents zero vegetation cover and maximum erosion.
In fact we can go further. The line A-D (if we can locate
it) separates all trajectories to attractor C from those to
attractor B. In conservation terms, this is a most desirable
line. Ideally, we want to keep the system to the right of
this line, or move it there by management actions. The
triangle ABD is the area we wish our system to be in. Any
combined values of
E
and
V
in this area take the system
to B. Conversely, in ACD the system will move to the
attractor (stable) equilibriumat Cwithmaximumerosion
and minimum vegetation cover and it will then require
a great effort to move it away. At D, it could go either
way and A, C and B are possible destinations. Another
feature emerges. If the system is at point X according to
its values of
E
and
V
, then it has a long way to go before
the system dynamics take over. In this sense, point X is
fairly stable. By contrast, if it is in condition (state) Y,
then quite a small perturbation will carry it to C, D or B.
So, in some cases, very small adjustments to the system
can quickly take the result to stable conditions (states)
or to unstable conditions. In complex systems, a little
change goes a long way, principally because of the positive
feedbacks that were described above. The difficulty is
to know (i) Where are the isoclines (thresholds) over
which the behaviour changes very rapidly with only small
perturbations? (ii) Where are the attractors and are they
stable or unstable? (iii) What are the equilibria and
are they stable or unstable or saddles? Thornes (1990)
examined these questions with more elaborate systems,
including those with stony soils.
Local
Unstable
Stable
A
ED
Global
B
C
F
Figure 24.6
Metaphor for different types of equilibria: local,
stable, unstable and global.
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