Noise-induced phenomena in environmental systems - Noise-induced Phenomena in the Environmental Sciences - page 138

the interested reader to this topic and to Ludwig et al. ( 1978 ) for more details on this
process.

4.4.3 Noise-induced extinction
The effect of random fluctuations in the carrying capacity ( Levins , 1969 ; May , 1973 )
may induce interesting phenomena in the logistic process. May ( 1973 ) treated the car-
rying capacity in Eq. ( 4.27 ) as a normally distributed random variable, A c = A 0 + ξ gn ,
with mean A 0 > 0 and intensity s gn , and determined the steady-state probability

distribution of A . When the noise intensity (i.e., s gn ) is relatively small, the ran-
dom environment induces stochastic fluctuations of A about its long-term mean

is smaller than A 0 and decreases with increasing values of s gn ,
suggesting that noise has the effect of reducing the mean value of A with respect
to the carrying capacity A 0 of the underlying deterministic dynamics. As s gn ex-
ceeds a critical-threshold value, the random fluctuations become so intense that no
steady-state pdf of A exists, and the system tends to extinction. The occurrence of
extinction as the noise intensity exceeds a critical value is clearly a noise-induced
phenomenon.
When A c =

A . Moreover, A

A 0 + ξ gn ,Eq.( 4.27 ) can be rewritten as ( 4.26 ) with rescaled time

t = at ,and

f ( A ) = A ( A 0 − A ) , g ( A ) = A . (4.35)

Notice that, because for A = 0 g ( A ) = 0, the dynamics have an intrinsic boundary at

0; it can be shown ( Horsthemke and Lefever , 1984 ) that this boundary is also a
natural boundary (i.e., it cannot be accessed by A )if s gn

A =

A 0 .
Conditions conducive to noise-induced extinction were determined by May ( 1973 ),
who used Ito's interpretation of the stochastic equation (see Chapter 2), leading to the
steady-state probability distribution of A :

<

C − 1 A
s gn

A 0
s gn −

2

A
s gn

e − p ( A ) = , (4.36)

with normalization constant

A 0
s gn −

1 C = s gn (4.37)

and where ( · ) is the gamma function. Equation ( 4.37 ) is obtained from the integration

of p ( A )in[0 1. When s gn exceeds A 0

there is no finite value of C and no equilibrium solution exists. In this case, A

, +∞ ). Notice that C is finite only if A 0 / s gn >

0
is no longer a natural boundary, in that the whole probability mass is concentrated at
this intrinsic boundary; the pdf of A becomes p ( A )

=

) is the Dirac
delta function ( Horsthemke and Lefever , 1984 ). Thus when the intensity s gn of the

= δ ( A ), where δ ( ·