Noise-induced phenomena in environmental systems - Noise-induced Phenomena in the Environmental Sciences - page 135

a 0 and intensity s gn ; i.e., a = a 0 + ξ gn . Thus the stochastic dynamics can be expressed

as ( 4.26 ) with

f ( A ) = a 0 A ( A c − A ) − kA , g ( A ) = A ( A c − A ) . (4.29)

This model is well suited to investigate the possible emergence of noise-induced
transitions as an effect of random environmental fluctuations in the growth rate
of the dynamics. In what follows we normalize A with respect to A c and use the
state variable A = A / A c . Moreover, to simplify the notation we drop the prime

superscript.
The steady-state pdf of A can be calculated with Eq. ( 2.83 ) from Chapter 2:

1
s gn

k 1

s gn ( a 0

1
s gn ( a 0

Ce −

− k ) − 1

A ) −

− k ) − 1

p ( A ) = A (1 − , (4.30)

1 − A

where C is the normalization constant. It can be shown that when k = 0 the normal-

ization constant C is not finite and p ( A ) 0),

as the noise intensity (i.e., s gn ) increases, the dynamics undergo a noise-induced tran-
sition from a system with only one stable state (for the deterministic system or for
low values of s gn ) to bistable dynamics. Figure 4.21 (a) shows the modes of A calcu-
lated with Eq. ( 3.48 ): As s gn exceeds a critical value s c =

= δ ( A − 1). In all the other cases ( k =

k ), a noise-induced
transition occurs, in that multiple equilibria emerge and the probability distribution
of A becomes bimodal [Fig. 4.21 (b)]. Thus the multiplicative noise is able to in-
duce a new stable state A

( a 0 −

= 0 that did not exist in the underlying deterministic

system.

4.4.2 Stochastic genetic model
Logistic harvest model ( 4.29 ) can be generalized to account for possible inputs of A
occurring at constant rate

β

.Knownasa genetic model ( Horsthemke and Lefever ,
1984 ), this process finds a number of applications to population biology and dy-
namic ecology. For example, the genetic model could describe the dynamics of a
population affected by logistic growth, state-dependent emigration or harvest, and
state-independent immigration ( ). If, for the sake of simplicity, A is normalized with

respect to A c in a way that it ranges between 0 and 1, these dynamics are expressed
by

β

d A
d t =

aA (1 − A ) + β − kA . (4.31)

Dividing both sides of ( 4.31 )by k , normalizing the model's parameters a =

a / k ,

β = β/ k , rescaling the time variable ( t =

kt ), and dropping the prime superscripts,

we obtain

d A
d t =

aA (1 − A ) + β − A . (4.32)