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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