Noise-induced patterns in environmental systems - Noise-induced Phenomena in the Environmental Sciences - page 255

v st D

1

0.3

0.8

0.2

0.6

1 2 1

0.4

0.1

0.2

0

λ 0

1 λ 0

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0.2 0.4 0.6 0.8

(a) (b)

Figure 6.5. (a) Stable (solid curve) and unstable (dashed curve) steady states of
v st as functions of the noise parameter λ 0 ,for D = 0 . 30, α = 0 . 45, b /λ 0 =− 0 . 9,

ω 0 = 0001; (b) number of (stable) steady states of the system. Figure

taken from D'Odorico et al. ( 2007b ).

0 . 4, = 0 .

Fig. 6.3 . Moreover, the numerical simulations show that the intermediate steady state
in Fig. 6.3 (dashed curve) is unstable.
To investigate the dependence of multiple steady states on noise intensity, v st was

plotted as a function of the noise parameter λ 0 , maintaining a constant ratio b /λ 0 .The

results (Fig. 6.5 ) show that in the absence of fires (i.e., 0) the system has only

one steady configuration corresponding to landscapes completely dominated by trees.
For relatively large values of

λ = 0

0 fires prevent the establishment of woody vegetation
and the system has only one steady state at

λ

0. In intermediate conditions
the system undergoes phase transitions. The parameter b represents the “strength”
of the feedback between vegetation and fires. In the absence of such feedback (i.e.,
b

v = st

=

0, not shown) the system exhibits neither phase transitions nor pattern formation.
Figure 6.5 (b) shows the number of stable states of the system (i.e., of stable solutions
of the self-consistency equation) as a function both of

λ 0 and D . The disappearance of
multiple (statistically) steady states for low and high values of noise intensity suggests
that the phase transition is reentrant ( Porporato and D'Odorico , 2004 ). In the region
of the parameter space located between these two phase transitions, ordered states
emerge. Once the possibility of noise-induced bimodality and phase transitions has
been detected, the existence of patterned states can be assessed numerically. Examples
of vegetation patterns generated by the model (starting from random initial conditions)
are shown in Fig. 6.6 .
Patterns produced by this model are not stationary and emerge only in the transient
from the initial condition to the asymptotic state of uniform vegetation. Although
transient conditions may last for very long time - especially close to the bifurcation
point - the mechanism proposed in this study explains only the initiation of vegetation
patterns, and other processes need to be invoked to stabilize and maintain spatial
organization.