Noise-induced phenomena in environmental systems - Noise-induced Phenomena in the Environmental Sciences - page 99

may play a more fundamental role than causing random fluctuations in the temporal
variability of the state variables. In addition to modifying the number of possible
steady states, noise can affect the way the system approaches these states, its sensitiv-
ity to disturbances, or the ability to recover after disturbances (i.e., the resilience of
the system). Noise can also enhance the regularity of temporal fluctuations and induce
a variety of other counterintuitive dynamical behaviors (see Chapter 3). This chapter
concentrates on this nontrivial effect of noise in biogeophysical systems, and it in-
vestigates conditions leading to the emergence of noise-induced transitions, coherent
fluctuations, and other phenomena in the environmental sciences (e.g., May , 1973 ;
Benzi et al. , 1982a ; Horsthemke and Lefever , 1984 ; Spagnolo et al. , 2004 ; D'Odorico
et al. , 2005 ). In particular, we focus on the temporal dynamics of spatially lumped
(i.e., zero-dimensional) univariate systems; the case of noise in spatial dynamics is
discussed in Chapter 5. Thus, consistent with the general framework of this topic (see
Chapters 2 and 3), we consider systems whose dynamics can be expressed as

d d t =

f ( φ ) + g ( φ ) ξ ( t ) , (4.1)

ξ

where ( t ) is a noise term. We consider noise-induced behaviors induced by different

types of noise, including the case of dichotomous, shot, and Gaussian noise. Some
remarks on the properties of bistable systems, reported in Box 4.1, may help in
understanding some of the results obtained in this chapter.

4.2 Dichotomous Markov noise in ecosystem dynamics
In Section 2.2 we showed how the effect of environmental variability on ecosystem
dynamics may result in the random alternation between stressed and unstressed con-
ditions. The ecosystem's total biomass (or other suitable state variables indicative of
ecosystem health or productivity) decreases or increases, depending on whether the
system is stressed or unstressed, respectively. The random alternation of these two
states is determined by fluctuations of the environmental variables (e.g., available
resources, disturbance pressure) about a tolerance threshold marking the transition
between favorable and unfavorable conditions for the ecosystem. These dynamics
are typically modeled (e.g., D'Odorico et al. , 2005 ; Camporeale and Ridolfi , 2006 ;
Borgogno et al. , 2007 ) as stochastic process

φ ( t )drivenbyDMN.

4.2.1 Noise-induced transitions due to random alternations between
stressed and unstressed conditions in ecosystems
We consider the exemplifying case of an ecosystem in which the biomass B (e.g.,
plant biomass) randomly switches between a growth and a decay state, depending on
whether the level q ( t ) of fluctuating resources (e.g., soil water) is above or below a