Noise-induced phenomena in environmental systems - Noise-induced Phenomena in the Environmental Sciences - page 145

p s

2.5

2

1.5

1

______

τ
τ

0.0 yr

0.5

n

0.5 yr

n

s

0.2 0.4 0.6 0.8 1

Figure 4.25. Effect of autocorrelation on the emergence of noise-induced transi-
tions in soil-moisture dynamics in systems forced by colored Gaussian noise (same
parameters as in Fig. 4.24 , with s gn = 1).

would change in the case of colored noise, i.e., if ξ gn ( t )inEq.( 4.50 ) is autocorrelated

as a result of memory in the synoptic forcings determining convective precipitation.
As noted in Chapter 2, to study the combined effect of intensity ( s gn )andauto-
correlation scale ( τ n ) on noise-induced transitions, we need to use expressions of the

probability distribution of the state variable that are valid also for relatively high values
of s gn and n . Thus we can use here an approximate expression based on the the united

colored-noise approximation (see Chapter 2). This expression provides a suitable
representation of the probability distribution of s in that condition ( 2.107 ) in Chap-
ter 2 is verified. The results of this analysis show (Fig. 4.25 ) that even relatively small-
correlation scales can have dramatic effects on the shape of the probability distribution
of s . In this specific case, an increase in

τ

τ

n may turn a stochastic bistable system into
a system with only one statistically stable state, suggesting that the memory of large-
scale synoptic forcings may prevent the emergence of multiple preferential states in
the regional climate.
The role of colored noise in multivariate environmental dynamics was recently
investigated in the context of trophic interactions in population dynamics ( Petchey
et al. , 1997 ; Heino , 1998 ; Heino et al. , 2000 ; Laakso et al. , 2001 ; Greenman and
Benton , 2003 ; Xu and Li , 2003 ; Vasseur , 2007 ). Most of these studies concentrated
on a particular case of autocorrelated noise, which is characterized by power spectra
with power-law behavior, 1 f β . The exponent

/ β of the spectrum is zero in the case

of white noise, and values of β greater than zero correspond to autocorrelated (or

f β noise. Numerical simulations have shown that populations that are
either more sensitive to environmental fluctuations or more prone to extreme events
are less exposed to extinction risk in a fluctuating environment if the environmental
noise is autocorrelated ( Schwager et al. , 2006 ). Moreover, it was found (e.g., Vasseur
and Fox , 2007 ) that in consumer-producer populations resource variability decreases,

colored) 1 /