Stability and Instability in the Management of Mediterranean Desertification - Environmental Modelling: Finding Simplicity in Complexity - page 393

trajectories of change. As was shown earlier, these trajec-
tories are crucial in determining the final state, especially
if they carry the system across thresholds between quite
different attractor basins. The path to equilibrium need
be neither steady nor smooth. This problem often
occurs in mathematical modelling of change when, by
setting the rate of change equal to zero and solving
for equilibrium, the equilibrium state is obtained but
not the trajectories towards it. Kirkby (1971) used this
technique in his characteristic slope-formpaper, in which
the characteristic forms are derived from the values
of the parameters in the Musgrave erosion equation
(Musgrave, 1947). The equilibrium forms obtained are
in fact attractors in the m - n space though neither the
trajectories nor the relaxation times to equilibrium were
obtained in the analysis. This example appears to be
one of the earliest analytical applications of dynamical
systems analysis in the field of geomorphology, though
it is rarely recognized as such.
Another approach (Pianka, 1978) is to provide the
fitness distribution for plants in a climate phase space
and then judge the climatic change impacts by shifting
the curves in the phase space (Figure 24.9) and adjusting
the fitness response.

approached heuristically using the dynamical systems
approach. A novel treatment of the gradient problem
was developed by Pease et al . (1989). They envisaged the
production of the vegetation at ground level by the inter-
section of an atmospheric gradient with a planar surface
over which the change is passing (Figure 24.10). The
response at equilibrium is controlled by Lotka-Voltera
competitive growth (cf. Thornes, 1988) whereby not only
is there competition between the plants, but also the sta-
bility of the outcomes can be specified from a study of the
modelled interactions, as described below (Lotka, 1925;
Voltera, 1926). As the climate changes, the atmospheric
zones shift across the land, generating new vegetation
in response to this change. The species, in fact, track
the moving environment by adjustment of the patches
through a critical patch-size dynamic, in which the pop-
ulation within the patch is counter-balanced by dispersal
into an unsuitable habitat (Skellam, 1951). The dispersal
changes the mean phenotype and, because of the environ-
mental gradient, natural selection will favour different
phenotypic values in different parts of the environment.
The model also incorporates additive genetic variance.
Changing the climate also changes the carrying capac-
ity ( V cap ) in the logistic equations for plant response
(see below). But soil moisture is the control over the
above-ground biomass. Hence the cover and soil mois-
ture content is not a simple function of mean annual
rainfall, but also of the magnitude and frequency of rain-
fall and the evapotranspiration conditions forced mainly
by temperature. So rainfall ( R ) and temperature ( T )

24.4 Climate gradient and climate
change

Using the tools described earlier, we consider in this
section how the gradient-change problem can be

Figure 24.9 Pianka's representation of the hypothesized impact of climate change on plant fitness.