Noise-induced phenomena in environmental systems - Noise-induced Phenomena in the Environmental Sciences - page 104

The deterministic counterpart of these dynamics is a system that is either always
unstressed or always stressed, depending on whether the constant level q of available
resources is greater or smaller than the minimum value θ required for survival. Thus

the dynamics become d B / d t = f 1 ( B ) if in the deterministic system q >θ (always

unstressed conditions, i.e., P 1 = 1), or d B / d t = f 2 ( B ), otherwise (always stressed,

=

i.e., P 1 0). In the specific case of the example in Fig. 4.1 [Eqs. ( 4.2 )] both of

these deterministic dynamics have only one stable state, B m

=

1 [from the condition

0], and the deterministic system converges
to either of them, depending on whether the (constant) level of resources q is above
or below the threshold

f 1 ( B m ) = 0] or B m = 0 [from f 2 ( B m ) =

. Thus the deterministic dynamics are not bistable, and it is
the random driver that induces bistability [i.e., bimodality in p ( B )] in the stochastic
dynamics of B (Fig. 4.1 ).
The emergence of bistable dynamics has important implications for the way ecosys-
tems respond to changes in environmental conditions and disturbance regimes (e.g.,
Holling , 1973 ; Gunderson , 2000 ). In fact, the existence of alternative stable states is
often associated with the possible occurrence of abrupt and highly irreversible shifts
in the state of the system (see Box 4.1), with consequent important limitations to its
resilience (e.g., Walker and Salt , 2006 ). Ecosystem bistability is often induced by
positive feedbacks between the state of the system and environmental drivers such as
resource dynamics or disturbance regime (see Box 4.1).
This section showed that bistability may also result as an effect of noise ( Hors-
themke and Lefever , 1984 ). In this case, the randomness of the switching between
stressed and unstressed conditions is able to profoundly affect the dynamical properties
of the system by inducing bifurcations that did not exist in the underlying deterministic
dynamics: As the noise intensity (i.e., the variance) exceeds a critical threshold, new
ordered states are observed to emerge. In this chapter we present other examples show-
ing how the ability of random environmental fluctuations to induce bistability can be
found in continental-scale land-atmosphere interactions, landscape-scale vegetation-
fire dynamics, hill-slope-scale soil development, population dynamics, and in a few
other contexts.
The following subsection shows that the opposite effect may also occur: Noise
can turn a deterministic bistable system into a stochastic system with only one stable
state ( D'Odorico et al. , 2005 ). In either case, noise does not merely induce random
fluctuations about the stable states of the system; rather, it creates order by determining
the number of stable and unstable states.

θ

4.2.2 Noise-induced stability in dryland ecosystems
In the previous section we analyzed an example of bistable ecosystem dynamics
driven by dichotomous Markov noise. In that system bistability emerged as a
noise-induced effect, resulting from the interaction of the random driver with the