Environmental Engineering Reference
In-Depth Information
A major difference can be observed in the way these systems respond to disturbances.
Assume that a stable configuration of the system is disturbed: In the case of dynamics
with only one stable state, once the disturbance ceases, the system returns back to its
stable state, regardless of the amplitude of the perturbation. In bistable dynamics the
system recovers its initial stable state, [
s
1
in Fig. B4.1-1(a)] only if the magnitude of the
perturbation is relatively small, i.e., if the perturbation is unable to induce a transition to
the basin of attraction of the other stable state (
s
2
). In fact, if such a transition occurs, the
state of the system diverges toward the alternative state
s
2
. Because of the stability of the
state
s
2
, the dynamics remain locked in this state even once the disturbance is removed.
Thus bistable dynamics are prone to highly
irreversible
transitions. Moreover, these
dynamics are
highly nonlinear
in the sense that in the proximity of the unstable state
u
,
even small perturbations may lead the system far away from its initial state. These
dynamics are clearly characterized by the presence of a
threshold effect
associated with
the unstable state,
u
.
s
u
s
e
c
e
a
e
b
e
Figure B4.1-2. An example of bifurcation diagram.
e
is an environmental pa-
rameter governing the state of the system.
Ecologists define as
resilience
(
Holling
,
1973
;
Gunderson
,
2000
) of a stable state, say
s
1
, the ability of the dynamics to recover that state after a disturbance. In Fig. B4.1-1(b)
the resilience of state
s
1
is measured by the maximum magnitude of disturbances that
the system may withstand without escaping from the basin of attraction of
s
1
.Onthe
basis of the previous discussion, the states of bistable ecosystems have only limited
resilience.
In most cases, bistable behavior exists within only a certain interval of the parameter
space. Outside this interval one of the stable states disappears. The stable states are often
plotted as a function of parameters representative of environmental drivers. Figure
B4.1-2 shows a typical example of the dependence of stable (solid curve) and unstable
(dashed curve) states as a function of an environmental parameter
e
. We observe that the
number of stable states varies as
e
crosses the boundaries
e
a
and
e
b
of the bistability
range. Known as
bifurcation diagrams
, these plots show how nonlinearities and
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