Environmental Engineering Reference
In-Depth Information
There are two catches to this optimistic picture. First,
descent to the attractor is not smooth; the approach is
very fuzzy in the sense that that there may be a variabil-
ity that 'captures' the system for the attractor before it
gets there. If the 'external' forces perturbing the system
are also noisy (vary a lot in their strength), then the
system may be forced onto a trajectory from an unex-
pected position. Thornes and Brandt (1993) explored the
behaviour of the erosion-vegetation dynamical system for
stochastic (unpredictable in time) variations in rainfall
for a system in the dry Mediterranean in south east Spain.
Although this fuzziness produced some quite unexpected
behaviour, the vegetation showed a strong oscillation of
high and low vegetation covers and corresponding low
and high erosion rates. This kind of behaviour in dynam-
ical systems usually indicates that there is no strongly
preferred attractor and that the system is moving between
attractors. An alternative explanation is that the system
vegetation-moisture-erosion is a 'hypercycle' (Eigen and
Schuster, 1979) that can also produce nonlinear oscilla-
tions. This alternative explanation has been explored in
the context of grazing (Thornes, 2005). The results of
Thornes and Brandt also showed that occasionally there
would be overshoots. With a lot of rainfall, the vegetation
grew well in excess of carrying capacity and in subsequent
years had to 'die' back to the carrying capacity through
negative feedback. It was also found that the stability of
the system, as formulated, was significantly affected by
the stone content.
Note that two controlling influences have been iden-
tified in this section: internal or intrinsic and external
or extrinsic. The intrinsic influences are thresholds and
behaviour within the system itself, the trajectories, thresh-
olds and equilibria (repellers and attractors) that deter-
mine the courses of behaviour, their final destinations
(unless they go off the graph, far away from equilibrium)
and their stability. Extrinsic forces are those that perturb
the system away from the internal controls towards new
equilibria. If we had a good understanding of both sets
of mechanisms, and there was not too much fuzziness
or randomness, then we might be in a position to use
this knowledge to determine the future trajectories or
the reactions to deliberate perturbations, such as 'over-
grazing' or torrential rainfall. Obtaining absolute values
for this behaviour, such as the trajectories in space and
time, requires not only an understanding of the system
couched in this framework, but also a good knowledge of
the parameters of the system and the initial conditions.
(What are the values of
E
and
V
at the start?) These are
demanding requirements, though if some of the many
plot experiments were reconfigured, the data could be
easily obtained.
Because there are several (sometimes many) thresholds
in the system separating attractor basins, an important
question is how close to the threshold can we get before
the systemis captured by an attractor basin. In Figure 24.7,
how close can we get to the A-D line before we go to either
complete cover and no erosion (attractor B) or no cover
and very high erosion (attractor C)? Obviously this is a
crucial question for management. If we accidentally tip
the system across the boundary, the resulting outcome
could be the exact opposite of what is required. The
judgement as to what is a good outcome and a bad
outcome is independent of the dynamics. But in the case
of the management implications, it might be a question
of how many kg m
−
2
of biomass should be harvested or
animals should be allowed to graze. This decision will
determine the horizontal movement in Figure 24.7 and
hence depends on two things: (i) how far the system is
away from the boundary and (ii) how 'noisy' the process
is. This last point is very important.
24.3.1 Spatial variability
Because the value of
E
depends on the soil and rain-
fall characteristics over space, the runoff prediction is
extremely variable. On two adjacent soil patches the ero-
sion rate could be quite different, putting us either side
of the threshold. So bare and strongly vegetated patches
could co-exist and indeed they do! Through time, the
intensity varies changing the overland flow production.
Although one generally uses a mean intensity to fix the
position in the space
E
-
V
, if this variation is very large
then it is likely to straddle the threshold. In fact, the
distribution of intensity is exponential (Kosmas
et al
.,
1999). So if the mean is located on the threshold between
two attractors, we can define a 'belt' either side of the
threshold, in which there is a risk of the system moving
to another attractor. The larger the deviation, the larger
the risk. The 'noisier' the system the larger the risk. This
'noise' varies spatially, as does the runoff production,
vegetation cover and erosion. Consequently, spatial out-
comes could be quite varied and it is interesting to ask
where the thresholds exist in geographical space. In a
patchy landscape with different soil types and different
rates of runoff production, the system is differentially
located with respect to the thresholds. Just crossing a soil
boundary or the edge of a vegetation patch could trip us
towards another attractor. Human-induced patchiness is,
in this respect, critically important in trying to foresee the
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