Environmental Engineering Reference
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Harrison et al.
(
2001
) interpreted this transition to chaos as a possible mechanism
able to maintain the coexistence of two species.
Lai and Liu
(
2005
) investigated the
ability of a stochastic forcing to induce species coexistence in Holt-McPeek systems
in nonchaotic conditions. To this end, they accounted for the effect of environmental
noise by considering the carrying capacity as a random variable
K
j
, modeled as a
white-noise term with mean
K
j
and intensity
s
gn
[i.e.,
K
j
K
j
=
+
ξ
ξ
j
(
t
), where
j
is a Gaussian-white-noise term with zero mean and intensity
s
gn
]. When the noise
level (i.e.,
s
gn
) exceeds a critical value, the stochastic forcing can prevent
p
1
(
t
) from
converging to zero [see Fig.
4.30
(b)]. As
s
gn
increases, the average frequency
p
1
of species 1 increases at the expenses of species 2. Thus coexistence occurs for
intermediate noise levels.
Lai and Liu
(
2005
) interpreted this behavior as a noise-
induced effect, particularly as a stochastic resonance. However, in the classification
presented in Chapter 3 this behavior is suggestive of a coherence resonance. In fact,
Lai and Liu
(
2005
) did not use any periodic deterministic forcing. The stochastic
forcing seems to trigger abrupt increases in the size of the population of species 1
with a temporary transition to chaotic dynamics, followed by a relaxation phase back
to stable dynamics. Thus the process resembles the dynamics of excitable systems
with endogenous time scales.
4.8.4 Coherence resonance in excitable predator-prey systems
Predator-prey dynamics are known for their ability to exhibit interesting nonlinear
deterministic behaviors, including limit cycles (e.g.,
Goel et al.
,
1971
). Thus, in the
presence of a stochastic driver, these dynamics could lead to coherence resonance and
behave as excitable systems. An example of coherence resonance in predator-prey
systems can be found in the coupled dynamics of phytoplankton (
P
) and zooplankton
(
Z
). These dynamics are often investigated with the
Truscott and Brindley
(
1994
)
model:
a
2
P
2
d
P
d
t
=
rP
(1
−
P
)
−
b
2
P
2
Z
,
1
+
a
2
P
2
d
Z
d
t
=
b
2
P
2
Z
−
m
3
Z
;
(4.79)
1
+
the first equation expresses the dynamics of the phytoplankton (prey) as a harvest
process with a harvest (or
grazing
) rate proportional to the zooplankton biomass and
dependent on
P
according to a Holling type III grazing model with maximum rate
a
2
b
2
. The dynamics of
Z
are expressed by a growth-death model with the growth
rate determined by the grazing process and a mortality term proportional to
Z
with rate
m
3
. The parameter
/
determines the time scale of the response of
P
,and
r
expresses
the growth rate of the logistic term in the dynamics of
P
.
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