Noise-induced phenomena in environmental systems - Noise-induced Phenomena in the Environmental Sciences - page 106

ν ν

ν

V, V,

V,

v

1 v

1 v

1 v 0 0

0

1

0.8

0.6

0.4

I II III

0.2

R 1 R 2

0

500 R 100 200 300 400

Figure 4.2. Deterministic stable (solid thick lines) and unstable (dashed thick lines)
states of Eqs. ( 4.4 ) with R 1 =

360 mm. Noise-induced statistically
stable states of the stochastic dynamics are shown as a dotted curve (analytically
calculated). Numerically calculated values of the modes of

260 mm and R 2 =

v are shown with crosses

( σ R = 0 . 4 R ) and squares ( σ R = 0 . 6 R ). The agreement between numerical and

c + is a suitable mathematical approximation for

analytical solutions suggests that c =

the calculation of the modes of . The plots in the top inset represent the shapes of the

deterministic ( V ) and stochastic (

v

) potentials associated with the different regimes
of the deterministic (solid line) and stochastic (dotted line) dynamics, respectively.
These potentials are calculated for the cases of

V

R = 150, 310, and 410 mm, with

σ R = . Changes in the shapes of these potentials between the deterministic

and stochastic dynamics indicate the emergence of noise-induced stability (after
D'Odorico et al. , 2005 ).

0 . 6 R

heterogeneities: Following D'Odorico et al. ( 2005 ), we start from a minimalist model
of the bistable vegetation dynamics and show how random interannual climate (rain-
fall) fluctuations typical of arid climates (e.g., Noy-Meir , 1973 ; Nicholson , 1980 )may
lead to the emergence of an intermediate statistically stable configuration between the
two stable states of the underlying bistable deterministic dynamics.
We express the dynamics of dryland vegetation as

− v

3

(if R < R 1 ) (4 . 4a)

d d t d V

d

,

=− =

v

v − v v − ≥ .

(1 )( c ) (if R R 1 ) (4 4b)

where 1 in the case of

completely vegetated region), R is a fluctuating limiting resource (e.g., annual pre-
cipitation), and V is the potential function (see inset in Fig. 4.2 ). As an effect of
the vegetation-soil-moisture feedbacks, these dynamics can be either monostable or
bistable, depending on the value of annual precipitation R : For small values of R

v is the normalized vegetation biomass (0 ≤ v ≤ 1; v =