Environmental Engineering Reference
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2
I
III
p B
1.5
p B
B
B
a
2
1
p B
0.5
B
II
p B
IV
B
0.1
0.2
0.3
0.4
0.5
0.6
P
0
Figure 4.8. Shapes of the probability distributions of biomass
B
as functions of the
parameters
P
0
and
a
2
of the state-dependent dichotomous process (calculated for
a
1
=
0
.
2 and
b
=
0
.
4).
between
B
and
k
1
,
2
. Figure
4.8
shows the shape of
p
(
B
) for different values of the
parameters.
In Subsection
4.2.1
we showed how random fluctuations in the resource
q
may
generate an interesting variety of behaviors, including the emergence of bistable
dynamics, which did not exist in the deterministic counterpart of the system. These
bistable dynamics are clearly noise induced (
Horsthemke and Lefever
,
1984
;
Bena
,
2006
). As noted, bistabilitymay also emerge as an effect of positive feedbacks (see Box
4.1) between environmental drivers and the state of the system (
Wilson and Agnew
,
1992
). Comparing Figs.
4.1
and
4.8
, we observe that case V in Fig.
4.1
disappears in
the presence of the state dependency induced by the feedback, whereas the size of the
area III in the parameter space increases with the “strength”
b
of the feedback. We
interpret this effect as an enhancement of bistability induced by the positive feedback
between resources and state of the system. Thus noise induces a bistable behavior
in the dynamics even in the absence of the feedback, and the feedback enhances the
bistability of the system by broadening the interval of the parameter space in which
the probability distribution of
B
is bimodal. A similar effect is found in Subsection
4.3.2
in the case of state-dependent Poisson processes.
4.3 Environmental systems forced by white shot noise
Some environmental drivers are not always continuously active but occur as random,
episodic events of relatively short duration, which can be modeled as instantaneous
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