Noise-induced phenomena in zero-dimensional systems - Noise-induced Phenomena in the Environmental Sciences - page 55

k 2 ), and the superscript denotes the derivative, for example

where τ c = 1 / ( k 1 +

φ = φ m , φ = φ m .

d g ( φ ) d f ( φ )

g ( f ( φ m ) = φ m ) = (3.4)

d φ d φ

The impact of noise properties on the shape of the pdf clearly appears in Eq. ( 3.3 ).
In fact, the first term is independent of the noise parameters and remains even when
the noise term in Eq. ( 3.1 ) is turned off. In these conditions the modes and antimodes
of p ( ) coincide with the stable states of the underlying deterministic dynamics in

that they are given by the condition f (

φ

0. The second term expresses the effect
of the multiplicative nature of the noise [i.e., it is present when g (

φ m ) =

φ ) = const]; the

third term results from the asymmetry of the noise (i.e, 1 =− 2 ); and the fourth

term is due to the noise autocorrelation.
If the mechanistic interpretation is adopted, it is convenient to rewrite Eq. ( 3.3 )in
terms of the functions f 1 ( φ )and f 2 ( φ ); we obtain

f 1 ( φ m ) f 2 ( f 2 ( φ m ) f 1 ( φ m ) − φ m )

− k 1 f 2 ( φ m ) − k 2 f 1 ( φ m ) = 0 . (3.5)

f 2 ( φ m ) − f 1 ( φ m )

It is clear from Eq. ( 3.5 ) that the stable points of the dichotomic dynamics φ m can

be very different from their deterministic counterparts φ st , 1 and φ st , 2 , which are the

zeros of f 1 ( φ )and f 2 ( φ ).

3.2.1.1 Noise-induced transitions for processes driven by additive DMN
To provide an example of how noise may profoundly affect the dynamical properties
of a system through noise-induced transitions, we consider the dynamics described
in the previous chapter (Subsection 2.2.3 , Example 2.1) where DMN is used in the
mechanistic framework, i.e., to switch the dynamics between the two functions

f 1 ( φ ) = 1 − φ, f 2 ( φ ) =− φ, (3.6)

with deterministic steady states 0, respectively. Thus the stoch-

astic dynamics resulting from the random switching between these two functions are
naturally bounded between 0 and 1. In this example, Eq. ( 3.5 ) can be solved to give

φ st , 1 = 1and φ st , 2 =

1 − k 2

φ =

− k 1 − k 2 .

(3.7) m

2

Thus the mode or antimode φ m is between the boundaries of the interval ]0 , 1[ if

k 1 < 1and k 2 < 1or k 1 > 1and k 2 > 1. In the first case φ m is an antimode, whereas

in the second case φ m is a mode. It is also of interest to explore the behavior of the

pdf close to the boundaries. Using approximation ( 2.41 ) we obtain

k 2 − 1

) k 1 − 1 ; lim

φ → 0 p (

φ ) ∼ φ , lim

φ → 1 p (

φ ) ∼ (1 − φ