Noise-induced phenomena in environmental systems - Noise-induced Phenomena in the Environmental Sciences - page 136

A m

a

1

0.8

0.6

______

Stable
Unstable

0.4

0.2

s c

0

s gn

0.5 1 1.5 2 2.5

p A

b

4

3.5

______

s gn 2.00

3

s gn 0.25

2.5

2

1.5

1

0.5

A

0.2 0.4 0.6 0.8 1

Figure 4.21. (a) Stable and unstable states of the stochastic harvest process with
random reproduction rate a with Gaussian distribution, mean a 0 , and intensity
s gn ( a 0 = 1). (b) The pdf of state variable A of the stochastic

harvest process with multiplicative noise [same parameters as in (a)].

1, k = 0 . 3, and A c =

This deterministic model has at most two equilibrium points (i.e., a stable and an
unstable state), which are roots of the second-order polynomial on the right-hand side
of ( 4.32 ). With parameter a treated as a random variable (mean α 0 and intensity s gn ),

Eq. ( 4.31 ) becomes

d A
d t =

a 0 A (1 − A ) + β − A + A (1 − A ) ξ . (4.33) gn

The properties of this stochastic genetic model were investigated in detail by Hors-
themke and Lefever ( 1984 )and Lefever ( 1990 ). As in the previous case the right-hand
side of Eq. ( 3.48 ) is a polynomial, which increases its order by one as s gn increases
from zero to s gn > 0. Thus noise might be capable of increasing the number of stable

states of the system. Figure 4.22 (a) shows the equilibrium states of ( 4.33 ) as a function
of noise intensity (i.e., of s gn ). As s gn exceeds a critical value s c , the system undergoes
a noise-induced transition: The stable state of the underlying deterministic dynamics