Environmental Engineering Reference
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induce DO transitions, its combined effect with noise synchronizes the transitions,
inducing order and regularity in the intervals between events.
Depending on whether DO events result from coherence resonance or stochastic
resonance, the underlying dynamics would exhibit different properties: (i) in the case
of coherence resonance the 1500-yr periodicity would be inherent to the dynamics
of the system ( Timmermann et al. , 2003 ); (ii) in the case of stochastic resonance this
periodicity would result from an externally imposed periodic fluctuation in freshwater
inputs ( Ganopolski and Rahmstorf , 2002 ). Thus a stochastic-resonance theory of DO
events requires an explanation of the physical processes determining a 1500-yr peri-
odicity in freshwater inputs. These inputs are presumably associated with fluctuations
in hydrologic conditions. It is not clear what could induce 1500-yr periodicity in these
fluctuations ( Ganopolski and Rahmstorf , 2001 ), though some more recent analyses
have shown ( Braun et al. , 2005a ) that the 1500-yr cycle could emerge when the peri-
odic forcing is obtained as a suitable superposition of two harmonics corresponding to
the 87-yr and 210-yr solar cycles. Thus nonlinear excitable or bistable systems forced
by a periodic driver with more than one frequency may show resonance at a frequency
different from those of its driver. This effect is also known as ghost resonance .
4.8.3 A coherence-resonance mechanism of biodiversity
In Subsection 4.2.3 we showed how noise can promote species diversity through the
beneficial effect of environmental fluctuations of intermediate frequency. Lai and Liu
( 2005 ) demonstrated how species coexistence (and hence diversity) can also result
from a coherence-resonance mechanism.
Recent studies have shown how environmental variability may favor the coexis-
tence of species with different dispersal rates (e.g., Holt andMcPeek , 1996 ). Temporal
variability may either be induced by externally imposed drivers or result from en-
dogenous chaotic dynamics ( Harrison et al. , 2001 ). A simplified framework recently
used to investigate these dynamics is based on the Holt-McPeek model ( Holt and
McPeek , 1996 ), i.e., a two-species and two-patch system with four state variables,
N i , j ( i
2), representing the density of species i in patch j . Species compete
within patches and disperse between patches. Each species behaves identically within
each patch (i.e., with the same competitive abilities) but disperses differently between
patches. Thus, in each patch, the density dependence is a function of the total species
density, N t , j =
N 1 , j +
N 2 , j ( j
2). The local population growth W j in patch j
( j
2) is species independent:
exp r j 1
N t , j ( t )
K j
W j =
where K j is the carrying capacity in patch j and r j is the a parameter determining the
growth rate in patch j . Dispersal is modeled assuming that a fraction e i of species i
migrates at each time step (generation) from its patch to the other one. This migratory
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