Noise-driven dynamical systems - Noise-induced Phenomena in the Environmental Sciences - page 11

assumed in one of the two states is a random variable; complicated DMN ( Li , 2007 )
in which both states are (Gaussian) random variables; the gamma and McFadden
dichotomous noise ( Pawula et al. , 1993 ), in which the distribution of the permanence
times in the two states follows a gamma or a McFadden probability distribution rather
than an exponential distribution, as in the classical DMN.

2.2.2 Dichotomous noise in the environmental sciences
Dichotomous noise can be encountered in a wide variety of physical and mathematical
models for two main reasons. First, dichotomous noise is a simple and analytically
tractable form of colored noise; in fact, it is possible to obtain exact analytical solu-
tions for a stochastic differential equation driven by DMN in steady-state conditions.
Thus DMN can be used to investigate the effect of an autocorrelated random driver
on a dynamical system. We define this approach as the functional usage of the DMN
because of its function as a tool to conveniently represent a correlated (i.e., colored)
random forcing. In this case (functional usage) the starting point is a given determin-
istic system, say d ), and DMN is typically used to investigate the effect

of a zero-mean correlated random driver in this system. There are several examples of
processes in which the autocorrelation is one of the key characteristics of the external
forcing. For example, consider the variety of biogeochemical processes that are af-
fected by (random) daily temperature or the case of fluvial processes forced by river
flow. In these processes the autocorrelation of the random forcing is relevant, and it
cannot be neglected. Dichotomous noise is one of the two main mathematical tools
available for the study of the effects of colored noise on dynamical systems. Colored
Gaussian noise, described in Section 2.5 , is another type of autocorrelated noise,
which is often used in dynamical models with analytical solutions. The functional
usage of DMN can be also motivated by the fact that both white Poisson noise and
white Gaussian noise can be recovered from the dichotomous noise by taking suitable
limits, as shown in Subsections 2.3.2 and 2.4.2 .
Dichotomous noise is commonly used also for its ability to model a broad class of
systems that randomly switch between two dynamical states. This approach is called
the mechanistic usage of DMN, in which DMN is used to represent a dynamical
behavior, i.e., the mechanism of random switching between two states.
The distinction between the functional and the mechanistic use of DMN is crucial
in the stochastic modeling of a process. The mechanistic approach is frequently used
for a class of processes characterized by the following three components: (i) the
dynamical system, whose state is expressed by one state variable,

φ/ d t = f ( φ

φ ( t ); (ii) a random

driver q ( t ); (iii) a threshold value of q ( t ), marking the transition between conditions

favorable to growth or to decay of

θ

could represent
vegetation biomass in semiarid environments ( D'Odorico et al. , 2005 ) or riparian
vegetation along a river ( Camporeale and Ridolfi , 2006 ); correspondingly, q could

φ . For example, the variable φ