Environmental Engineering Reference
In-Depth Information
The dynamics expressed by Eqs. (
4.79
) may exhibit interesting cyclic behaviors
(
Fernandez et al.
,
2002
), similar to those of other grazing systems (
Noy-Meir
,
1975
).
Sieber et al.
(
2007
) used this model to investigate the viral infection of the phytoplank-
ton population. In particular, they concentrated on the effect of lysogenic infection,
i.e., on the case of viruses integrated in the genome of the host cells. Thus the re-
production of the host genome occurs concurrently to the reproduction of the viral
genome.
Sieber et al.
(
2007
) modified Eqs. (
4.79
) to account for the fact that the
phytoplankton may be either susceptible to viruses (
Z
s
) or infected (
Z
i
). Indicating
the zooplankton population by
Z
z
, we can express the dynamics of the state variable,
Z
(
t
)
={
Z
s
(
t
)
,
Z
i
(
t
)
,
Z
z
(
t
)
}
,as
=
−
1
rZ
s
(1
a
2
Z
s
(
Z
s
+
d
Z
s
(
t
)
d
t
Z
i
)
Z
s
Z
i
Z
s
+
−
Z
s
−
Z
i
)
−
Z
i
)
2
Z
z
−
λ
,
1
+
b
2
(
Z
s
+
Z
i
=
−
1
rZ
i
(1
Z
i
−
m
2
Z
i
a
2
Z
i
(
Z
s
d
Z
i
(
t
)
d
t
+
Z
i
)
Z
s
Z
i
Z
s
+
−
Z
s
−
Z
i
)
−
Z
i
)
2
Z
z
+
λ
,
1
+
b
2
(
Z
s
+
a
2
(
Z
s
+
Z
i
)
2
d
Z
z
(
t
)
d
t
=
Z
i
)
2
Z
z
−
m
3
Z
z
,
(4.80)
1
+
b
2
(
Z
s
+
where
represents the virus transmission rate. The fraction of infected phytoplankton
(
Z
i
) increases as an effect of virus transmission, while the susceptible population
(
Z
s
) decreases. The infected population is then prone to virus-induced mortality at
rate
m
2
. The zooplankton dynamics are the same as in (
4.79
). It can be shown that
in the dynamics described by (
4.80
) the ratio
i
=
Z
i
/
λ
(
Z
s
+
Z
i
) remains constant for
λ
=
m
2
.
Sieber et al.
(
2007
) used this property to investigate (
4.80
) as a 2D system.
Depending on the initial value of
i
, the system may exhibit a variety of dynamical
behaviors, including the convergence to a stable limit cycle or to a stable focus. For
relatively low values of
m
3
the dynamics may exhibit excitation followed by relaxation
to stable conditions.
Interesting properties emerge when this system is forced with a multiplicative noise.
To this end,
Sieber et al.
(
2007
) added to the right-hand side of Eqs. (
4.80
) a term,
Z
j
ξ
j
(
t
)(
j
ξ
i
(
t
) is a zero-mean Gaussian white noise with intensity
s
gn
. The effect of this stochastic forcing is to induce noise-sustained oscillations.
With adequate noise intensity this system may exhibit coherence resonance, i.e., a
high degree of regularity (or
coherence
) in noise-induced oscillations. An optimal
value of
s
gn
exists at which these fluctuations are most coherent. This effect is shown
in Fig.
4.31
for three different noise levels (with
r
=
s
,
i
,
z
), where
=
1,
a
=
4,
b
=
12,
λ
=
m
2
=
10
−
3
): With relatively high or low noise levels (Fig.
4.31
) we observe a random occurrence of excitation episodes followed by relaxation.
With intermediate values of
s
gn
the dynamics exhibit coherence resonance. This
effect can be assessed by calculation of the CV of the time between two consecutive
0
.
01,
m
3
=
0
.
0525, and
=
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