Noise-induced phenomena in environmental systems - Noise-induced Phenomena in the Environmental Sciences - page 121

We notice that this bistable behavior does not emerge in the deterministic coun-
terpart of the system. In fact, as noted, the deterministic harvest process [Eq. ( 4.10 )]
has only one stable state. We can obtain the equilibrium points A ∗ of the underlying
deterministic dynamics from ( 4.9 ), eliminating the random fluctuations by replacing
the noise term with its average value,

. Thus the equilibria A ∗ are given by the roots

λα

of

aA ∗ (1 − A ∗ ) − λα A ∗ =

0 . (4.13)

The solution of ( 4.13 ) leads to A ∗ = 0 (unstable state) and A ∗ = 1 − λα/ a (stable

state). Thus, for a given value of α , the stable state of the system increases from 0

to 1 as to 0, consistent with the behavior of the modes of

the stochastic dynamics through zones III, II, and I of Fig. 4.9 .For

λ/ a decreases from 1 /α

the
fire-induced disturbances are too frequent and intense to allow the process to leave the
state A

λ/ a > 1 /α

0, which is stable both in the deterministic and in the stochastic (zone V in
Fig. 4.9 ; see caption) dynamics. Thus the behaviors found in zones I, II, III, and V exist
also in the deterministic system and these transitions are not noise induced. However,
the bimodal behavior in zone IV is completely noise induced. In fact, because f ( A )is
a quadratic function, the deterministic system is not bistable. For the bounds to induce
bistability, an unstable deterministic state should exist between 0 and 1, as in the case
of the bounded Takacs process ( Porporato and D'Odorico , 2004 ). In the present case,
such a state does not exist. Thus we conclude that multiplicative noise is able to create
new states that did not exist in the underlying deterministic dynamics.
Within a certain portion of the parameter space, the dynamics become bimodal, i.e.,
they exhibit two preferential states, whereas the deterministic system [i.e., Eq. ( 4.10 )]
has only one stable state. The emergence of bimodality indicates that, in this part
of the parameter space (zone IV), conditions favorable to tree-grass coexistence are
unstable, whereas woodland and grassland are alternative statistically stable states of
the system ( D'Odorico et al. , 2006b ). The emergence of this noise-induced bistability
causes a discontinuous response of the system to changes in parameter values, as
shown by the existence of threshold effects and hysteresis in the bifurcation diagram
in Fig. 4.10 . Subsection 4.3.2 shows how feedbacks between fires and vegetation may
enhance this noise-induced behavior.

=

4.3.2 Poisson harvest process with state-dependent harvest rate
In Subsection 4.3.1 we investigated the properties of a stochastic harvest model in
relation to the case of fire-vegetation dynamics in mesic savannas. Here we show
how a state-dependent rate λ ( A ) of the Poisson process may induce other interesting

noise-induced behaviors in the dynamics. The dependence of the probability of fire
occurrence λ on A has been well documented (e.g., van Wilgen et al. , 2003 )and

is due to the fact that in savannas the fuel load is contributed by grass biomass.