Noise-induced phenomena in zero-dimensional systems - Noise-induced Phenomena in the Environmental Sciences - page 68

The deterministic states are then φ 1 if q ∗ >θ 1 ,and φ 2 if q ∗ <θ 2 .Themost st , st ,

interesting dynamics are found when θ < q ∗ <θ 1 : If the system is initially in the 2

growth state [i.e., θ ( φ 0 ) < q ∗ ], φ increases [hence θ ( φ ) increases] until φ reaches the

value φ ∗ , with θ ( φ ∗ ) = q ∗ . In these conditions the system is stable: In fact, if φ exceeds

φ ∗ , the state variable φ decreases with rate f 2 ( φ ) because θ ( φ ∗ ) > q ∗ . Conversely, if

θ φ > φ

the system is initially in the stressed (or decay) state [i.e., ( 0 ) q ∗ ], decreases

φ ∗

until it reaches (from above). The stable state of the deterministic system is

then φ ∗ .

Once the deterministic counterpart of the dynamics has been identified, it is possible
to investigate how the interaction of noise with state dependency modifies the stable
states of the system. To this end, we analyze the modes and antimodes φ m of the pdf of

the process, ( t ), forced by state-dependent dichotomous noise. We can obtain these

modes by setting equal to zero the first-order derivative of ( 2.44 )or( 2.45 ), depending
on the interpretation adopted for the DMN. In the functional interpretation, the modes
and antimodes are found from

φ

φ m ) g ( 1 + 2 ) f (

f ( φ m ) + τ c 1 2 g ( φ m ) + τ c ( φ m ) g ( φ m ) (3.37)

+ τ c 2 f (

f 2 ( m ) g ( φ φ m )

φ m ) f (

φ m ) − + g ( φ m ) τ c [ 1 k 2 ( φ m ) + 2 k 1 ( φ m )] = 0 .

g ( φ m )

The impact of the state dependency on the shape of the pdf clearly appears from
Eq. ( 3.37 ). The first four terms in Eq. ( 3.37 ) exist also when the dichotomous noise
is state independent [see Eq. ( 3.3 )]. When the noise is state dependent, the fifth term
in ( 3.37 ) appears. This term contributes to the emergence of differences in the stable
states (modes) between the stochastic and deterministic dynamics. Notice that when
the transition rates are constant (i.e., the noise parameters do not depend on )the

fifth term is zero if the noise is taken, as it is usually done, with a null average value,
1 k 2 + 2 k 1 =

φ

0. Instead, in the presence of state dependency the mean value of
noise is in general a function of .

If the mechanistic interpretation is adopted, it is convenient to write Eq. ( 3.37 )in
terms of the functions f 1 (

φ

φ )and f 2 ( φ ):

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

− k 1 ( φ m ) f 2 ( φ m ) − k 2 ( φ m ) f 1 ( φ m ) = 0 . (3.38)

f 2 ( φ m ) − f 1 ( φ m )

It is clear from Eq. ( 3.38 ) that the role of the state dependency in modifying the stable
states is due to the presence in Eq. ( 3.38 )oftheterms k 1 ( m ).

To demonstrate the possible impact of the feedback between noise and the dynam-
ical system, we consider the example described in the previous subsections, in which
the alternating processes of growth and decay are expressed by two linear functions,

φ m )and k 2 ( φ

f 1 ( φ ) = α (1 − φ ) , f 2 ( φ ) =− αφ, (3.39)