Noise-induced phenomena in environmental systems - Noise-induced Phenomena in the Environmental Sciences - page 133

1

E s L s cm d

0.8

0.7

______ a

0.03 b 0.50

............ a

0.03 b 0.60

0.6

____ a

0.03 b 0.30

0.5

0.4

0.3

0.2

0.1

s

0.2 0.4 0.6 0.8 1

Figure 4.20. Stable and unstable states of the deterministic process shown as in-
tersections of the loss curve (thick curve), E ( s ) + L ( s ), with the average input rate

α ∗ = α ∗

λ ( s ) ( a + bs ) for different values of b . The parameters are the same as in

Fig. 4.19 .

λ bs . The solid-curve plot shows very well-defined bimodal behavior. The

emergence of these two preferential states is important from a climatologic perspective
because it suggests that, because of soil-moisture-precipitation feedbacks, the system
is more likely to be in either “dry” or “wet” conditions, whereas long-term average
climatic conditions have the lowest probability of occurrence. Thus in some regions
the common use of long-term average parameters to characterize the hydroclimatic
regime would not provide meaningful information on soil-moisture dynamics.
In Chapter 3 we have also stressed that, because of the dependency of

( s ) = a +

on s ,
the WSN in Eq. ( 4.9 ) is multiplicative. Because multiplicative random drivers are
known for their ability to cause noise-induced transitions (see Chapter 3) we could
wonder whether the bistability (i.e., bimodality) found in p ( s )(Fig. 4.19 )emerges
as a noise-induced effect or as a result of the soil-moisture-precipitation feedbacks.
To address this point, we study the deterministic counterpart of Eq. ( 4.19 ), which we
obtain by replacing the noise term with its mean value

λ

α ∗ : λ

( s )

α ∗
nZ r λ

d s
d t =

( s ) − ρ ( s ) , (4.25)

α ∗ is the mean of h nZ r h [from Eq. ( 4.20 )], which in general might differ

where =

from 1 imposes a bound to the amount of water that

can infiltrate the ground. The equilibria of the deterministic dynamics are given
by the condition

α because the condition s ≤

= α ∗ /

ρ ( s ) nZ r λ ( s ), i.e., by intersections of the E ( s ) + L ( s )curve

α ∗ λ = α ∗ ( a

(Fig. 4.20 ) with the line ( s ) + bs )(Fig. 4.20 , thin straight line). In the

absence of feedbacks ( b 0) this line is horizontal and only one intersection (stable

state) exists. With suitable slopes of the

=

( s ) relation, multiple intersections may exist,
as shown in Fig. 4.20 (dotted line): In this case the deterministic dynamics already
exhibit three equilibria corresponding to two stable states separated by an unstable

λ