Environmental Engineering Reference
In-Depth Information
0 with
the resulting isocline in the plane as shown in the earlier
section (and by the 'hump' in upper part of Figure 24.11)
and the trajectories calculated. If now
This equation can then be solved for
dV
/
dt
=
R
1
T
1
R
2
T
2
R
3
T
3
θ
is reduced by
interception, then
d
v
t
1
Ecotone 1
v
t
2
Ecotone 2
v
t
3
Ecotone 3
dt
=−
k
4
(
R
,
T
)
−
k
5
θ
+
k
6
R
.
(24.4)
R
1
T
1
R
2
T
2
R
3
T
3
is assumed to increase as a linear function of rainfall
and decreases in relation to water content
θ
(by drainage
proportional to conductivity,
k
5
) and vegetation
V
(by
water consumption and interception). The consumption
rate is also assumed to be a function of
T
, themean annual
ambient air temperature(for evapotranspiration). Setting
d
θ/
d
t
=
0 the isoclines for a given value of
T
can be
obtained. As a consequence, the isoclines run from high
θ
at low
V
to low
θ
at high
V
(Figure 24.11). The separate
lines T
1
-T
3
represent the equilibrium between
θ
R
1
T
1
R
2
T
2
R
3
T
3
s
2
s
1
s
3
and
V
and they can be represented on the same diagram as the
V
isocline. The family of R-T isoclines now intersect the
V
isocline at points A and B, where there are joint equilibria
corresponding to vegetation cover, rainfall, soil moisture
and air temperature. As before, the system dynamics can
be obtained for any point on the
V
-
θ
1
2
−
d
z
−
−
s
2
z
d
2
z
s
2
d (In
n
)
G d (In
ω
)
=
+
+
+
d
x
2
d
t
d
x
d
z
z
= Mean phenotype
plane. For the
plane, points above the isocline will have
V
decreasing
until they reach the isocline and because the system is
moving towards the isocline from both higher and lower
values of
V
,the
V
isocline is stable on the right hand
limb. Once the system comes to lie on the isocline, there
can be no further change. At the intersection of the two
isoclines (points A and B) there is a joint equilibrium of
V
and
θ
G = Additive genetic variance in character
How migration changes phenotype
1
2
Adjustment of adaptive capacity
Figure 24.10
The formulation of Pease
et al
. (1989) of a climate
change moving across a landscape. Upper: on a flat plane;
middle: on a topographically varying surface; lower: on a
surface with different soil types. Box and text- equations used
for modelling behaviour.
whose behaviour is resultant of the behaviour
of both (Figure 24.11). Intersections on the rising limb
are unstable equilibria. Perturbations from equilibrium
A will be driven away from A. By contrast, perturba-
tions from equilibrium B will be driven back towards
B, which is an attractor. This point is shown by the
time graphs in the lower part of Figure 24.11. Moreover,
an oscillatory damped approach to equilibrium charac-
terizes perturbations spiralling into B, whereas there will
be an undamped explosive behaviour spiralling out from
A. In practice, fluctuations about A and B are likely to
occur at random, so that there is a bivariate probabilistic
distribution of values about A and B of bifurcation points.
θ
variations offer a good approximation to plant cover
(Eagleson, 1979), given all other co-linearities. Different
coefficients of water use efficiencies may also determine
the main functional vegetation type (bare, grass, shrubs
or trees) (Tilman, 1982). Using the logistic equation
k
1
V
1
dV
dt
=
V
V
cap
−
(24.2)
with
V
cap
=
200 and
k
1
=
6
.
1, the equilibrium biomass is
higher and reached later than if
V
cap
=
150 and
k
1
=
.
1
as indicated in Figure 24.2. This expression can be re-
written to allow for positive growth, as
6
(soil moisture)
increases linearly with coefficient
k
3
as follows:
θ
24.5 Implications
dt
=
k
2
V
1
−
dV
V
V
cap
The implications are that shifts in either
θ
or
V
iso-
clines not only identify changes in equilibrium that
+
k
3
θ.
(24.3)
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