In this paper, we formulate two chemostat type models of phytoplankton and zooplankton population dynamics with instant nutrient recycling to study the role of viral infection on phytoplankton. The infection is transmitted only among phytoplankton population and it makes them more vulnerable to predation by zooplankton. It is observe that the chemostat system is very stable in the absence of viral infection but the presence of viral infection make the chemostat system sensitive with respect to the grazing rate of infectedphytoplankton by zooplankton. Further, if the grazing rate is less than certain threshold the system remain stable and exhibits Hopfbifurcation after crossing it.
AMS Mathematics Subject Classification : 92D30, 92D40, 37G15.
1. Introduction
Phytoplankton and micro zooplankton are one celled organisms those drift with the current on the surface of open ocean. They are the staple item for the food web and are producer and recycler of most of the energy that flow through the oceanic ecosystem. Plankton play important role not only in maintaining the fish stock but stabilize environment by consuming half of the universe Carbon dioxide and release huge Oxygen. Pollution of fresh water in marine system by anthropogenic sources has become a concern over the last decades. Researchers have found out that each tea spoon of ocean water contains 10 million to 100 million of viruses. Again, viral infection is known to cause a cell lysis in phytoplankton and biomass is the common criterion for nontoxic species but the measurable level of toxin due to harmful species is responsible for the bloom dynamics. In coastal area viral disease can infect bacteria and phytoplankton
[1]
. Virus like particle is found in many eukastic algae
[23]
and natural phytoplankton community
[20]
. These virus like particle have also been described in many eukaryotic algae
[7
,
8]
. The parasite modifying behavior has also been exhibited by the infected individuals of host population. This may happen by reducing stamina, disorientation and altering responses
[9]
. Kill fish(fundulus parvipinnes) tend to come closer to the surface of the sea on contracting disease, which make them more vulnerable to predation by birds
[21]
. Viruses have been held responsible for the collapse of Emiliaxia Huxleye bloom in mesocosms
[13
,
23]
. Since virus have major role in shaping the dynamics of plankton. Many researchers have developed a epidemiological model and concluded that infected pollution does not persist if infection rate is below some threshold value
[3
,
4
,
5
,
14
,
15
,
16
,
17
,
21
,
22]
. Recently, the role of toxic producing phytoplankton and delay on planktonic ecosystem in the presence of a toxic producing phytoplankton are studied
[24
,
25]
.
These kind of transmissible disease cause death and behavior change in aquatic species. As a result of this many planktonic species show bloom in their population (i.e., the rapid growth and decay in their population). The blooms are of two type, one is spring bloom and other is red bloom. The spring bloom is seasonal. It is due to change in temperature and nutrient level associated with season. On the other hand Red bloom is localized out break associated with change in water temperature and with greater stability of water column
[10]
. Different criteria have been used in the literature to define and classify bloom
[11
,
22]
. Recently, pattern formation have been studied for instant nutrient replenishment on plankton dynamics with diffusion in a closed system
[6]
. A paper in American Museum of Nature History (2000) reveals that when a virus injects its DNA into a cell, it hijacks that cells replication machinery and produce hundreds of new particles. These rupture the host and are released into the environment to find new victims. Hence, the infected particle itself becomes the source of infection and this rate of infection is directly proportional to the number of infected phytoplankton and the number of susceptible phytoplankton. Therefore, in this paper, we have assumed that infected phytoplankton themselves are the source of infection.
Keeping in view the above discussion, in this paper, we shell study the role of virus in phytoplankton, zooplankton interaction by developing two chemostat model of phytoplankton and zooplankton species one without infected phytoplankton and other with infected phytoplankton class. In section 2 we shell be developing and analyzing chemostat model of phytoplankton and zooplankton species without infected phytoplankton and in section 3 will have the development and analysis of model of phytoplankton and zooplankton with infected phytoplankton class in instantaneous nutrient recycling environment. In section 4 the bifurcation analysis of nutrient plankton system with infected phytoplankton class will be carried out and section 5 will be devoted to the conclusion of analysis carried out in the previous sections.
2. NutrientPlankton System without Infected Phytoplankton Class
Here we purpose a preypredator model of phytoplankton in the absence of infected phytoplankton with instantaneous nutrient recycling. Let
N
(
t
),
P_{s}
(
t
) and
Z
(
t
) are nutrient, susceptible phytoplankton and zooplankton population densities at any time t respectively. Again,
d
_{1}
and
d
_{3}
, are the per capita death rate of susceptible phytoplankton and zooplankton respectively. In a natural plankton system, water flowing into the system brings in nutrient and out going water carries away nutrient from the system. Further, it is assumed that water is carrying away nutrient, susceptible phytoplankton along with flow in the same rate. Here
N
_{0}
is the constant nutrient input concentration at any time and
D
is the water influx rate or washout rate along with nutrient, susceptible phytoplankton and zooplankton. The nutrient uptake and grazing rate of phytoplankton by zooplankton is assumed to follow law of mass action. Let
a
_{1}
is the nutrient uptake rate by susceptible phytoplankton and
c
^{1}
is the grazing rate of susceptible phytoplankton by zooplankton. The conversional rate of the nutrient into susceptible phytoplankton is given by
a
^{1}
and
c
^{11}
is the conversional rate of susceptible phytoplankton into zooplankton. Model equation for the above model are given by
Here all the conversion rates lies between (0,1) and rest of the parameters are positive. Again, the positive feed back term’s (1 −
a
^{1}
)
aNP_{s}
, (1 −
c
^{11}
)
c
^{1}
P_{s}Z
,
d
_{1}
P_{s}
and
d
_{3}
Z
will be recycled into nutrients, i.e., all the losses are being are being replenished into nutrients.
Suppose
W
=
N
+
P_{s}
+
Z
, then
Using (1)(3), we get
Clearly,
W
→
N
_{0}
as
t
→ ∞. Hence, the solutions of the system (1)(3) exist and bounded for all time to come.
Again,
at
N
= 0, which shows that
N
(
t
) > 0, ∀
t
> 0. Similarly for
P_{s}
and
Z
. Therefore, we can state the following lemma:
Lemma 2.1.
Solutions of equations (1)(3) are nonnegative and bounded
.
Now we shell study the asymptotic dynamic of the above system (1)(3), using
N
+
P_{s}
+
Z
=
N
_{0}
, our model reduces to
Putting
P_{s}
=
x
,
Z
(
t
) =
z
,
a
^{1}
a
_{1}
=
a
,
c
^{1}
=
c
,
d
_{1}
+
D
=
δ
_{1}
,
d
^{1}
=
d
,
c
^{11}
c
^{1}
=
α
and
d
_{3}
+
D
=
δ
_{3}
, then system (4)(5) become
All possible equilibrium of the system are, (i)
E
_{0}
(0, 0),
exist if
aN
_{0}
>
δ
and
exist if
aαN
_{0}
>
aδ
_{3}
+
αδ
_{1}
.
Clearly the dynamic of the above system (6)(7) can be stated as follows:

(1)E0(0, 0) always exist and stable ifaN0<δ1, then all solution will converges toE0.

(2) The instability ofE0(0, 0), i.e.,aN0>δ1leads to the existence ofE1and all flow converges toE1ifaαN0

(3) The instability ofE1, i.e.,aαN0>aδ3+αδ1leads to the existence ofE2.
Hence the system persists uniformly. Again, the characteristic equation of above system around
E
^{*}
is
where
C
_{1}
=
x
^{*}
a
> 0 and
C
_{2}
=
α
(
a
+
c
)
x
^{*}
z
^{*}
> 0. Hence, both the eigen values of characteristic equation having negative real parts. So, interior equilibrium
E
^{*}
if exists, then it is always stable.
Remark 2.1.
From the above discussion it is clear that as the nutrient supply (i.e.,
aN
_{0}
) is less than removal rate of nutrient (i.e.,
δ
_{1}
) then the system stays in trivial equilibrium. Again, when the nutrient supply is greater than removal rate of nutrient
δ
_{1}
but less than
δ
_{1}
+
aδ
_{3}
/
α
, then system converges to the zooplankton free equilibrium. On the other hand if the nutrient supply is greater than
δ
_{1}
+
aδ
_{3}
/
α
, then the interior equilibrium exits and stable.
3. Nutrient Plankton System with Infected Phytoplankton Class
In this model we purpose a preypredator model for phytoplankton and zooplankton in the presence of viruses with instantaneous nutrient recycling. Let
N
(
t
),
P_{s}
(
t
),
P_{i}
(
t
) and
Z
(
t
) are nutrient concentration, susceptible phytoplankton, infected phytoplankton and zooplankton population densities at any time t respectively. Let
d
_{1}
,
d
_{2}
and
d
_{3}
, are the per capita death rate of susceptible phytoplankton, infected phytoplankton and zooplankton respectively. Here,
N
_{0}
is the constant nutrient input concentration and it is assumed that water is carrying away nutrient, susceptible phytoplankton, infected phytoplankton along with flow in the same rate ′
D
′. The nutrientphytoplankton and phytoplanktonzooplankton interactions are assumed to follow law of mass action. Here
a
_{1}
is the nutrient uptake rate by susceptible phytoplankton and
c
^{1}
,
d
^{1}
are the grazing rate of susceptible and infected phytoplankton by zooplankton respectively
The conversional rate of the nutrient into susceptible phytoplankton is given by
a
^{1}
, whereas
c
^{11}
and
d
^{11}
are the conversional rate of susceptible phytoplankton and infected phytoplankton into zooplankton respectively. Here
b
^{1}
is the disease contact rate. Then the model equations for the above system are given as:
Here all the conversional rates lies between (0, 1) and all other parameters are positive. Moreover, feed back term’s (1−
a
^{1}
)
aNP_{s}
, (1−
c
^{11}
)
c
^{1}
P_{s}Z
, (1−
d
^{11}
)
d
′
P_{Z}
,
d
_{1}
P_{s}
,
d
_{2}
P_{i}
and
d
_{3}
Z
are being recycled into nutrients, i.e, all the losses are being replenished into nutrients. Suppose
W
=
N
+
P_{s}
+
P_{i}
+
Z
, then
Using (9)(11), we get
. Clearly,
W
=
N
_{0}
as
t
⇒ ∞. Hence, the solution of the system exists and bounded for all time to come. Again,
Which shows that
N
(
t
) > 0, ∀
t
> 0. Similarly
P_{s}
,
P_{i}
and
Z
. Therefore, we can state the following lemma:
Lemma 3.1.
Solution of system (9)(11) are nonnegative and bounded.
Now we shell discus the asymptotic dynamic of the above system, using,
N
+
P_{s}
+
P_{i}
+
Z
=
N
_{0}
in equations (9)(11), our model reduces to
Putting
P_{s}
=
x
,
P_{i}
(
t
) =
y
,
Z
(
t
) =
z
,
a
^{1}
a
_{1}
=
a
,
b
^{1}
=
b
,
c
^{1}
=
c
,
d
_{1}
+
D
=
δ
_{1}
,
d
^{1}
=
d
,
d
_{2}
+
D
=
δ
_{2}
,
c
^{11}
c
^{1}
=
α
,
d
^{11}
d
^{1}
=
β
and
d
_{3}
+
D
=
δ
_{3}
.
Then equations (12)(14) become
All the feasible equilibrium of the system are (i)
E
_{0}
(0, 0, 0),
exist if
aN
_{0}
>
δ
_{1}
,
exist if
abN
_{0}
>
aδ
_{2}
+
bδ
_{1}
,
exist if
aαN
_{0}
>
aδ
_{3}
+
αδ
_{1}
, and (v)
E
_{4}
(
x
^{+}
,
y
^{+}
,
z
^{+}
), where
exist if
and
Now the dynamic of the above system (15)(17) can be stated as follow:

(1) The trivial steady stateE0(0, 0, 0) always exist and stable if ,aN0<δ1,i.e, the total nutrient supply is less than the wash out rate. Then all solution will converges toE0.

(2) The instability ofE0(0, 0, 0),i.e,aN0>δ1leads to the existence of boundary steady stateE1and all flow converges to it ifabN0

(3) The instability ofE1, forabN0>aδ2+bδ1, leads to the existence ofE2,i.e, zooplankton free equilibrium and all flow will converge toward it if

(4) The instability ofE1, foraαN0>aδ3+αδ1leads to the existence infection free steady stateE3and all solution converges to it if

(5) Endemic equilibriumE*= (x*,y*,z*), whereandexit and stable only if the equilibria pointsE0,E1,E2, andE3are unstable andNow, the endemic equilibrium, i.e.,E*exist IfandSincedd11=βand henceis the range of values ofdfor whichE*exist.
Now, we will examine the local behavior of the system around the endemic equilibrium
E
^{*}
= (
x
^{*}
,
y
^{*}
,
z
^{*}
) with respect to the change in parameter
d
, i.e., the grazing rate of infectedphytoplankton by zooplankton. The possibility of bifurcation of the solution of dynamical system will be explored by taking, the grazing rate of infectedphytoplankton by zooplankton as a control parameter. The characteristic equation of system around
E
^{*}
is as follow:
where

A1=ax*,

A2=βdy*z*+ (a+b)by*x*+ (a+c)αx*z*,

A3= (adβ+ (a+c)bβ− (a+b)dα)x*y*z*.
On substitution the values of
x
^{*}
and
z
^{*}
, it can be easily verified that
A_{i}
> 0, for
i
= 1, 2, 3. Now, from RouthHurwitz criterion a set of necessary and sufficient conditions for all the roots of the above equation (18) having negative real part are
A_{i}
> 0,
i
= 1, 2, 3 and
A
_{1}
A
_{2}
>
A
_{3}
. Now solving the above inequality , we get a sufficient condition for the stability.
Hence, we can state the following theorem:
Theorem 3.2.
If the endemic equilibrium exist, then the sufficient condition for the system (15)(17) to be locally stable around the endemic equilibrium point E^{*} is
4. Hopfbifurcation Analysis
Further, we will study the Hopfbifurcation of above system (15)(18), taking
d
(i.e., the grazing rate of infectedphytoplankton by zooplankton) as a bifurcation parameter. Now, the necessary and sufficient conditions for the existence of the Hopfbifurcation, if there exists
d
=
d
_{0}
such that (i)
A_{i}
(
d
_{0}
) > 0,
i
= 1, 2, 3, (ii)
A
_{1}
(
d
_{0}
)
A
_{2}
(
d
_{0}
) −
A
_{3}
(
d
_{0}
) = 0 and (iii) if we consider the eigen values of the characteristic equation (18) of the form
λ_{i}
=
u_{i}
+
iv_{i}
, then
i
= 1, 2, 3. After substitution of the values, the condition
A
_{1}
A
_{2}
−
A
_{3}
= 0 becomes
where

B1= (a+b)αm1l2− (a+b)abk1l2,

B2= (a+b)ab(k1l1−K2l2) + (a+c)αak1m1+ (a+c)βbl2m1− (a+b)αm1l1,

B3= (a+b)abk2l1+ (a+c)αak2m1− (a+c)βbl1m1.
For example, taking
a
= 0.4,
n
0 = 50,
b
= 0.8,
c
= 1,
δ
_{1}
= 0.4,
δ
_{2}
= 0.2,
δ
_{3}
= 0.5,
α
= 0.05,
β
= 0.5, we get a positive root
d
= 0.367 of the quadratic equation (19). Therefore, the eigen values of the characteristic equation (19) at
d
= 0.367 are of the form
λ
_{1,2}
= ±
iv
and
λ
_{3}
= −
w
, where
v
and
w
are positive real number.
Now, we will verify the condition (iii) of hopfbifurcation. Put
λ
=
u
+
iv
in (19), we get
On separating the real and imaginary part and eliminating
v
between real and imaginary part, we get
It is clear from the above that
u
(
d
_{0}
) = 0 iff
A
_{1}
(
d
_{0}
)
A
_{2}
(
d
_{0}
)−
A
_{3}
(
d
_{0}
) = 0. Further, at
d
=
d
_{0}
,
u
(
d
_{0}
) is the only root, since the discriminant
Again, differentiating (17) with respect to
d
, we have
Now, since at
d
=
d
_{0}
,
u
(
d
_{0}
) = 0, we get
which will ensure that the above system has a hopfbifurcation and it is shown graphically in
figures 1

4
.
The phase plane representation of three species around the endemic equilibrium, taking a = 0.4, n_{0} = 50, b = 0.8, c = 1, δ_{1} = 0.4, δ_{2} = 0.2, δ_{3} = 0.5, α = 0.05, β = 0.5, and d = 0.35
The phase plane representation of three species around the endemic equilibrium, taking a = 0.4, n_{0} = 50, b = 0.8, c = 1, δ_{1} = 0.4, δ_{2} = 0.2, δ_{3} = 0.5, α = 0.05, β = 0.5, and d = 0.35
The phase plane representation of three species around the endemic equilibrium, taking a = 0.4, n_{0} = 50, b = 0.8, c = 1, δ_{1} = 0.4, δ_{2} = 0.2, δ_{3} = 0.5, α = 0.05, β = 0.5, and d = 0.4
The phase plane representation of three species around the endemic equilibrium, taking a = 0.4, n_{0} = 50, b = 0.8, c = 1, δ_{1} = 0.4, δ_{2} = 0.2, δ_{3} = 0.5, α = 0.05, β = 0.5, and d = 0.5
5. Conclusion
In this paper, we have studied the effect of a viral infection which was spreading only among phytoplankton and rendered them more vulnerable to predation by zooplankton. It was observed that the dynamical system was more stable in the absence of viral infection. We established that in this dynamical system disease free equilibrium exist and stable only when the grazing rate of infectedphytoplankton by zooplankton, (i.e., ”d”) was greater than the threshold value
In other words we found that if the vulnerability of infected phytoplankton to the predation by the zooplankton was more than the certain threshold value
then the infected phytoplankton would extinct provided the boundary equilibrium (i.e.,
E
_{0}
and
E
_{1}
) were unstable. We have also analyzed that the above chemostat system would go without zooplankton population as long as the net conversion rate of infected phytoplankton into zooplankton, i.e,
β
was below the threshold
provided the boundary equilibrium (i.e.,
E
_{0}
and
E
_{1}
) were unstable. Further it has been established that endemic equilibrium
E
^{*}
= (
x
^{*}
,
y
^{*}
,
z
^{*}
) became stable only in some range of values of
d
. In section 4, the possibility of bifurcation with respect to the grazing rate of infectedphytoplankton by zooplankton (i.e.,
d
) was explored both analytically and numerically. It is found that viral infection has important role in shaping the dynamics of plankton and this kind of viral infection could give reason for the outbreak of bloom in phytoplankton species depending upon the rate of grazing of infectedphytoplankton by zooplankton.
BIO
Dr. Joydip Dhar received M.Sc. from VisvaBharati University, Santenekaten, India; and Ph.D from Indian Institute of Technology Kanpur, India. He is currently an Associate Professor at ABVIndian Institute of Information Technology and Management, Gwalior, India since 2006. His research interests are computational mathematics, mathematical modelling and simulation of biological, environmental, managerial and engineering systems.
Department of Applied Sciences, Applied Mathematics Section, ABVIndian Institute of Information Technology and Management, Gwalior474 015, M.P, India.
email: jdhar@iiitm.ac.in
Dr. Anuj Kumar Sharma received M.Sc. from Guru Nanak Dev University, Amritsar, India and Ph.D. from Punjan Technical University, Jalandhar, India. He is currently an Associate Professor at L.R.D.A.V. College, Jagraon. His research interests are Mathematical biology and computational mathematics.
Department of Mathematics, L.R.D.A.V. College, Jagraon142026, Ludhiana, Punjab, India.
email: anujsng@radiffmail.com
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