Environmental Engineering Reference
In-Depth Information
by replacing the (stochastic) local term in (
6.20
) with its average. In dimensionless
form, Eq. (
6.20
) becomes
∂v
(
r
,
t
)
=−
(1
−
P
)
v
(
r
,
t
)
+
P
ηv
(
r
,
t
)[1
−
v
(
r
,
t
)]
∂τ
exp
−|
exp
r
−
−
2
−
|
r
|
r
−
(
r
,
t
)d
r
,
2
+
ζ
r
|
v
(6.21)
χ
2
with
b
1
d
1
α
1
r
d
1
;
b
2
b
1
;
d
2
d
1
;
η
=
α
2
α
1
,
τ
=
α
1
t
;
r
=
ζ
=
;
=
χ
=
(6.22)
where
1 represent the relative importance of local-spatial dynamics [see
Eq. (
6.15
)] and of competition-facilitation processes, respectively;
ζ
and
<
χ>
1 depends on
the ratio between the radii of root system and canopy footprints, and
expresses the
relative importance of logistic growth and stress-induced mortality. To simplify the
notation, in what follows we drop the tilde “
η
∼
” and indicate by the dimensionless
,
coordinate vector
r
(
x
y
). The homogeneous steady states are obtained as solutions
v
=
v
0
=
of (
6.21
)for
const:
−
(1
−
P
)
v
0
+
P
ηv
0
(1
−
v
0
)
+
ζv
0
exp
−|
2
−
exp
−|
2
d
r
=
r
−
r
−
2
r
|
r
|
/χ
0
.
(6.23)
In particular, we study the linear stability of the homogeneous stable state
if
P
1
η
+
v
=
0
≤
,
(6.24)
0
1
1
−
P
η
+
ζπ
η
2
)
v
0
=
1
−
P
(1
−
χ
(otherwise)
,
P
by seeking for solutions of (
6.21
) in the form of a sum of
v
0
with a perturbation
term
δ
v
,
ˆ
δ
v
e
γτ
+
i
k
·
r
v
=
v
0
+
δ
v
=
v
0
+
,
(6.25)
where
ˆ
δ
v
is the ampli
tude
of the perturbation,
γ
is its growth factor,
k
is the wave-
=
√
−
number vector,
i
1 is the imaginary unit, and “
·
” is the scalar-product operator.
v
=
v
0
would
indefinitely grow with time. To determine the relation between growth factor and
wave number in (
6.25
), we insert Eq. (
6.25
)into(
6.21
), and obtain (after a Taylor
expansion for small values of
0
is linearly unstable when
γ>
0, because any disturbance
δ
v
of
v
ˆ
δ
v
)
γ
(
k
)
=−
(1
−
P
)
+
η
P
(1
−
2
v
0
)
+
ζ
W
(
k
)
e
−
k
2
2
k
2
2
χ
2
e
−
=−
(1
−
P
)
+
η
P
(1
−
2
v
0
)
+
ζπ
−
χ
,
(6.26)
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