Environmental Engineering Reference
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capacity was modeled as white Gaussian noise. Thus it was implicitly accepted that,
although the mean is larger than zero, the random fluctuations may lead to negative
values of the carrying capacity. These negative fluctuations could be responsible for
the effects of noise-induced extinction previously discussed. These negative values
have limited physical meaning when the state variable is defined as a positive quantity
(e.g., population density, vegetation cover, or biomass). To assess whether noise-
induced extinction can emerge even when the carrying capacity A c is modeled as
a positive random variable, we express A c as the exponential of a white Gaussian
noise. Analytical solutions are difficult to obtain for the stochastic logistic process;
however, an analytical expression for the steady-state probability distribution of the
state variable can be determined in the case of the Gompertz process (e.g., Goel et al. ,
1971 ):
d A
d t =−
A c .
aA log
This process resembles to some extent Verhulst (logistic) process ( 4.27 )inthatthey
both model the growth of a population with carrying capacity A c . 4
To solve stochastic equation ( 4.42 ) we consider the auxiliary variable B =
log( A )
and define the parameter B c =
log( A c ). Equation ( 4.42 ) can be rewritten as
d B
d t = a ( B c B )
Thus we can solve Eq. ( 4.42 ) by finding a solution for Eq. ( 4.45 ) using the ana-
lytical solution for Eq. ( 4.26 ), driven by Gaussian white noise. Using Stratonovich's
interpretation, we obtain the probability distribution of B [see Eq. ( 2.83 )]. The distri-
bution of A is then determined as a derived distribution of p ( B ). It is found that the
mode of A decreases with increasing noise intensities. However, this mode always
remains positive and different from zero. Thus the stochastic Gompertz process does
not exhibit the noise-induced transitions that were reported for the case of the logistic
In both processes the growth rate is proportional to A and to a function G ( A , A c ), which accounts for a decrease in
the growth rate ( saturation effect ) as the population approaches its carrying capacity:
d A
d t = aAG ( A , A c ) ,
A c 1 q
1 q
G ( A , A c ) =
The parameter q is zero in the case of the logistic process, whereas it tends to 1 in the Gompertz process (e.g., Goel
et al. , 1971 ).
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