BioSystems 68 (2003) 5 /17
www.elsevier.com/locate/biosystems
Dynamics of nutrient phytoplankton interaction in the
presence of viral infection
/
J. Chattopadhyay a,*, R.R. Sarkar a, S. Pal b
a
b
Embryology Research Unit, Indian Statistical Institute, 203, B.T. Road, Calcutta 700 035, India
Ramakrishna Mission Vivekananda Centenary College, Rahara, North 24 Parganas, Calcutta 743 186, India
Received 27 May 2001; received in revised form 11 March 2002; accepted 21 July 2002
Abstract
The present paper deals with the problem of a nutrient /phytoplankton (N /P) populations where phytoplankton
population is divided into two groups, namely susceptible phytoplankton and infected phytoplankton. Conditions for
coexistence or extinction of populations are derived taking into account general nutrient uptake functions and Holling
type-II functional response as an example. It is observed that the three component systems persist when the infected
phytoplankton population is not able to consume nutrient.
# 2002 Elsevier Science Ireland Ltd. All rights reserved.
Keywords: Nutrient; Susceptible phytoplankton; Infected phytoplankton; Coexistence; Extinction
1. Introduction
Plankton are the basis of all aquatic food chain
and phytoplankton in particular occupies the first
trophic level. Phytoplankton transform mineral
nutrients into primitive biotic material using
external energy, provided by the sun. The dynamic
relationship between phytoplankton and nutrients
has long been of great interest in both experimental and mathematical ecology, its universal
existence and importance.
After the pioneering paper of Riley et al. (1949),
many papers on plankton dynamics have appeared
in the literature. Numerical models with increasing
* Corresponding author.
E-mail address: [email protected] (J. Chattopadhyay).
degrees of complexity have been constructed and
some of these have been discussed and compared
by Patten (1968), Platt et al. (1977), Fasham et al.
(1983). But these models were concerned mainly
with phytoplankton /herbivore interactions. The
importance of nutrients to the growth of plankton
leads to explicit incorporation of nutrients concentrations in the plankton /herbivore models and
have been considered by Evans and Parslow
(1985), Frost (1987), Taylor (1988), Wroblewski
et al. (1988). Spatial distribution of plankton have
also been studied extensively by Levin (1986) (see
also the references therein). Busenberg et al. (1990)
studied the dynamics of plankton/nutrient interaction and observed that under certain conditions
the coexistence of phytoplankton and zooplankton
occurs in an orbitally stable oscillatory mode.
0303-2647/02/$ - see front matter # 2002 Elsevier Science Ireland Ltd. All rights reserved.
PII: S 0 3 0 3 - 2 6 4 7 ( 0 2 ) 0 0 0 5 5 - 2
6
J. Chattopadhyay et al. / BioSystems 68 (2003) 5 /17
Ruan (1993) considered plankton /nutrient interaction models consisting of phytoplankton, zooplankton, and dissolved limiting nutrient with
general nutrient uptake functions and instantaneous nutrient recycling. He worked out the
conditions for boundedness, existence, stability
and persistence of the system. Further he considered the zooplankton /phytoplankton /nutrient interaction models with a fluctuating nutrient input
and with a periodic washout rate and showed that
coexistence of the zooplankton and phytoplankton
may arise due to positive bifurcating periodic
solutions. Recently, Pardo (2000) studied a phytoplankton nutrient system and observed the
global behaviour of the system. He showed that
in the case of extinction of the phytoplankton,
introduction of nutrient in the system reveals the
bloom of phytoplankton, which appears in upwelling conditions.
Viruses are evidently the most abundant entities
in the sea */nearshore and offshore, tropical to
polar, sea surface to seafloor, and in sea ice and
sediment pore water. Quite a good number of
studies (Wommack and Colwell, 2000; Bergh et
al., 1989; Tarutani et al., 2000; Suttle et al., 1990)
showed the presence of pathogenic viruses in
phytoplankton communities. Fuhrman (1999)
synthesized the accumulated evidence regarding
the nature of marine viruses and their ecological as
well as biogeological effects. Suttle et al. (1990)
showed by using electron microscopy that the viral
disease could infect bacteria and even phytoplankton in coastal water. Parasites may modify the
behaviour of the infected member of the prey
population. Viral infections can also cause phytoplankton cell lysis. Virus-like particles have been
described for many eukaryotic algae (van Etten et
al., 1991; Reisser, 1993), cyanobacteria (Suttle et
al., 1993), and natural phytoplankton communities (Peduzzi and Weinbauer, 1993). Viruses
have been held responsible for the collapse of
Emiliania huxleyi blooms in mesocosms (Bratbak
et al., 1995) and in the North Sea (Brussaard et al.,
1996) and have been shown to induce lysis of
Chrysochromulina (Suttle and Chan, 1993). Because viruses are sometimes strain-specific, they
can increase genetic diversity (Nagasaki and
Yamaguchi, 1997). Nevertheless, despite the in-
creasing number of reports, the role of viral
infection in the phytoplankton population is still
far from understood.
Ecology and epidemiology are major field of
studies in their own right. The role of diseases in
ecological systems cannot be ignored. But little
attention has been paid so far in this direction. To
the best of our knowledge, except for the papers of
Hadeler and Freedman (1989), Freedman (1990),
Beltrami and Carroll (1994), Venturino (1995),
Beretta and Kuang (1998), Chattopadhyay and
Arino (1999), Chattopadhyay et al. (1999), no
work has been carried out in such eco-epidemiological system. Beltrami and Carroll (1994) proposed and analyzed a predator /prey system in
which some of the susceptible phytoplankton cells
were infected by viral contamination and formed a
new group (infected). The role of viral disease in
recurrent phytoplankton blooms was discussed.
They compared their results to the actual data by a
non-linear forecasting technique. They concluded
that only a minute amount of infectious agent can
destabilize the otherwise stable trophic configuration between a phytoplankton species and its
grazer. Chattopadhyay and Pal (2002) modified
the model equations of Beltrami and Carroll and
also the model of Venturino (1995). They observed
that there is a possibility for the coexistence of the
system when the contact rate follows the law of
mass action rate, which is similar to the results of
Venturino. But if the contact rate follows the law
of standard incidence rate then only a minute
amount of infection can destabilize the system and
the result is similar to Beltrami and Carroll.
The applicability of modelling is assessed for
key processes in ecology: physical transmission,
retention of viability, host defenses, behaviour
within the host, and ecosystem-scale effects.
Nutrient /phytoplankton (N /P), nutrient /zooplankton (N /Z), phytoplankton /zooplankton
(P/Z), and nutrient /phytoplankton /zooplankton
(N /P /Z) systems are well studied. Experimental
observations of viruses in planktonic communities
are well documented. In the present paper, the
virus infection in phytoplankton populations and
its effect in N /P model has been analyzed.
Different threshold values of spread of infection
have been calculated. However, these thresholds
J. Chattopadhyay et al. / BioSystems 68 (2003) 5 /17
will give new insight to the experimental biologists
to explain the coexistence of susceptible and
infected phytoplankton population and also the
controllability of the spread of virus infection in
the N/P system. In Section 2, we have studied the
dynamics of phytoplankton /nutrient system having a viral disease in the phytoplankton species.
We have proposed a mathematical model consisting of concentration of nutrient, susceptible phytoplankton and infected phytoplankton with
nutrient uptake rate as a general continuous
functional form. In Sections 3 and 4, we have
studied the boundedness, local stability and persistence of the system. Our analysis leads to
different thresholds, which are expressible in terms
of the model parameters and determine the
existence and stability of various equilibrium states
of the system and these thresholds can be compared with the basic reproductive ratio of epidemiology. In Section 5, we have analyzed nutrient,
susceptible phytoplankton and infected phytoplankton system with nutrient uptake rate as
Holling type-II functional form. Further in Section
6, in a special situation we have considered that the
members of infected phytoplankton population
are not able to consume nutrients and obtained the
different threshold values of the system parameters
for which the infected phytoplankton will exist or
not. Section 7 contains the general discussion of
the paper. Our analytical as well as numerical
study revealed the essential mathematical features
regarding the role of viral infection in the phytoplankton community and their dependence on
nutrient concentration.
2. The mathematical model
Let N (t) be the concentration of the nutrient at
time t. Let S (t ) and I (t) be the concentration of
susceptible phytoplankton population and infected
phytoplankton, respectively at time t. Let N0 be
the constant input of nutrient concentration; D ,
the washout rate of the nutrient; D1, mortality rate
of susceptible population and D2, mortality rate of
infected population. Let us also consider the
maximal nutrient uptake rate for the susceptible
population, maximal nutrient uptake rate for the
7
infected population, and the infection rate as a , b
and l, respectively. The mathematical model is:
dN
dt
dS
dt
D(N 0 N)aSU(N)bIV (N);
aSU(N)
lSI
SI
D1 S;
dI
lSI
bIV (N)
D2 I:
dt
SI
(1)
System (1) has to be analyzed with the following
initial conditions:
N(0)] 0;
S(0)]0;
I(0)] 0:
(2)
Here the functions U (N ) and V (N ) describe the
nutrient uptake rates of susceptible and infected
population, respectively. We assume the following
hypotheses on the nutrient uptake functions:
i)
U (N ) and V (N ) are continuous functions
defined on [0, ).
ii) U (0) /0, dU /dN /0, and
lim U(N)1:
N0
(3)
iii) V (0) /0, dV /dN /0, and
lim V (N) 1:
N0
(4)
3. Some basic results
Let P (t) /N0/N (t)/S (t)/I(t) then the system (1) may be written equivalently in the following form:
dP
dt
5D0 P(t);
dS
lSI
aSU(N 0 S I P)
D1 S;
dt
SI
dI
lSI
bIV (N 0 S I P)
D2 I:
dt
SI
8
J. Chattopadhyay et al. / BioSystems 68 (2003) 5 /17
Here N0/S (0)/I (0)/P (0) ]/0, S (0) ]/0, I(0) ]/
0, D0 /min(D , D1, D2).
Clearly limt 0 P (t)/0 and so the omega limit
set of any solution of Eq. (1) is contained in the
set,
Now we are in a position to prove the boundedness of the system (1).
Theorem 2.1.
bounded .
All the solutions of Eq. (1) are
V3 f(N; S; I)½ N ]0; S ] 0; I ]0; P0g: (5)
The limiting system obtained by restricting the
initial conditions to the set V3, is
Proof 2.1.
wN S I:
dw
dt
0
bIV (N S I)
lSI
SI
D2 I:
(6)
These equations, of course, are restricted to the
region,
0
V f(S; I)½S ] 0; I ]0; S I 5N g:
The boundary of V satisfies the following properties (S/I)t /N0, for some t ]/0, imply
dS
dt
(t)S
l(N 0 S)
N0
dt
5D0 (N S I)DN 0 :
(10)
It is clear that the right-hand side of Eq. (10) is
bounded.
Then we can find a constant m /0 such that:
dw
D0 w5 m:
dt
Using the variation of constant formula, this
inequality is transformed into,
w(N(t); S(t); I(t))
5
D 50;
m
(1eD0 t )w(N(0); S(0); I(0)) eD0 t ;
D0
from which we get two properties,
m
(N S I)(t)5max (N(0)S(0)I(0));
;
D0
and
dI
lI
(t)I
Dl 50;
dt
N0
(9)
The time derivative of Eq. (9) along the solutions
of Eq. (1) is,
dS
lSI
aSU (N 0 S I)
D1 S;
dt
SI
dI
We define a function,
(7)
t]0;
and for t 0/, we have
m
if D/l /0 then S (t )/0 and I (t)/0 for some
t ]/0, imply that dS /dt(t )/0 and
Sup(N S I)(t)5
dI
(t)0;
dt
Hence all the solutions of Eq. (1) which initiate in
R3 are eventually confined in the region,
(8)
respectively.
Therefore, V is a positively invariant region
(where uniqueness of initial value problem is
applied in case (8)). Similar arguments show that
V3 defined in Eq. (5) is positively invariant.
D0
:
(11)
B
m
3
(N; S; I) R : N S I o; o 0 :
D0
Hence the theorem follows.I
J. Chattopadhyay et al. / BioSystems 68 (2003) 5 /17
9
4. Local stability analysis
50;
Before going to eigenvalue analysis of the
system, we establish some results for which the
susceptible phytoplankton or infected phytoplankton or both populations will eventually go extinct.
4.1. Criterion for the extinction of susceptible
phytoplankton population
Theorem 2.2.
Let the inequality
if the condition Eq. (14) holds.
This results shows that the infected phytoplankton will be eliminated from the dynamical system
if the ratio of the maximal nutrient uptake rate by
the infected population and the difference of
mortality rate and infection rate is less than
unity.I
Consequently, we arrive at the following situation.
0
R0 aU(N )
B1
D1
(12)
Corollary 2.4.
hold. Then limt 0 S (t)/0.
If Eqs. (12) and (14) hold , then
lim (N(t); S(t); I(t)) E0 :
t0
From system (1), we have
Proof 2.2.
dS
lSI
aSU (N)
D1 S 5S[aU(N 0 )D1 ]
dt
SI
(13)
50:
Since there is no invariant set such that S /0 is
constant, the theorem follows.I
Theorem 2.2 demonstrates that regardless of
infection if the ratio of maximal nutrient uptake
rate for the susceptible phytoplankton and the
mortality rate of the susceptible phytoplankton is
less than unity then the susceptible phytoplankton
will be eliminated from the dynamical system.
4.2. Criterion for the extinction of infected
phytoplankton population
Theorem 2.3.
Let the inequality
0
R?0 bV (N )
D2 l
B1
hold. Then limt 0 I (t )/0.
Proof 2.3.
From system (1), we have that
dI
lSI
bIV (N)
D2 I
dt
SI
I
[S(bV (N 0 ) l D2 ) I(bV (N 0 ) D2 )]
SI
(14)
By corollary 2.4 if Eqs. (12) and (14) hold then
both of the susceptible phytoplankton and infected
phytoplankton become extinct and hence the
feasibility for persistence of system (1) does not
arise.
4.3. Eigenvalue analysis to establish local stability
Let us first consider the plankton-free steady
state of the system (1) E0 /(N0, 0, 0). The variational matrix of system (1) at E0 is,
2
3
D
aU(N 0 )
bV (N 0 )
5:
V0 4 0 aU(N 0 )D1
0
0
0
0
bV (N )D2
The eigenvalues of the variational matrix V0
are m1 //D B/0, m2 /aU (N0)/D1 /D1(R0/1),
m3 bV (N 0 )D2 R?0 (D2 l)D2 :/
Clearly, this steady state is asymptotically stable
if and only if R0 B/1 and R?0 B1BD2 =(D2 l); in
this case it attracts all the feasible solutions. When
R0 /1 and R?0 1; the plankton free steady state is
unstable (saddle) and there exist a feasible infected
phytoplankton free steady state E1(N1, S1, 0) with
N1 /U 1(D1/a),
S1 /(D /D1)[N0/U1(D1/a )]
and a feasible susceptible phytoplankton free
steady state E2(N2, 0, I2) with N2 /V1(D2/b ),
I2 /(D /D2)[N0/V1(D2/b )].
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J. Chattopadhyay et al. / BioSystems 68 (2003) 5 /17
Existence conditions of E1 and E2 lead to the
following results.
Theorem 2.5.
D1
B1
a
If the inequalities
(15)
and
R0 1
(16)
hold , then the system (1) has a non -negative
equilibrium E1 /(N1, S1, 0) where N1 and S1 are
defined above.
Theorem 2.6.
D2
B1
b
mƒ1 ; mƒ2 ; which are the roots of the equation,
m2 m(DbI2 V ?(N2 ))D2 bI2 V ?(N2 ) 0;
and mƒ3 aU(N2 )lD1 :/
Clearly, mƒ1 and mƒ2 have negative real parts. Now
if mƒ3 0; i.e. R?1 [aU(N2 )=(D1 l)] 1 then, E2
is a saddle point and hence E2 is unstable in the
direction orthogonal to N /I coordinate plane.
Next we study the global asymptotic stability of
the equilibria E1 and E2.
Theorem 2.7. If the non -negative equilibrium E1
and E2 exist , then (N1, S1) and (N2, I2) are globally
asymptotically stable in the N /S plane and N /I
plane , respectively .
If the inequalities
(17)
Let us define a Liapunov function,
Proof 2.7.
and
N
R?0 1
(18)
W (N; S)
then the system (1) has a non -negative equilibrium
E2 /(N2, 0, I2) where N2 and I2 are defined above.
The variational matrix of system (1) at
E1(N1, S1, 0) is,
V 21
DaS1 U?(N1 )
4 aS1 U?(N1 )
0
aU(N1 )
aU(N1 )D1
0
3
bV(N1 )
5:
l
bV(N1 )D2 l
Further, the eigenvalues of the variational matrix
V1 are m?1 ; m?2 ; which are the roots of the equation,
g
U(x) U(N1 )
U(x)
V2 2
3
DbI2 V ?(N2 )
aU(N2 )
bV(N2 )
4
5:
0
0
aU(N2 )D1 l
bI2 V ?(N2 )
l
bV (N2 )D2
The eigenvalues of the variational matrix V2 are
dx
N1
g
S1
x S1
dx:
x
(19)
Then W (N , S )/0 if and only if N /N1, S/S1
and W(N , S )]/0 in the N /S plane.
The time derivative of W along the trajectories
of the subsystem is,
dW dN U(N) U(N1 )
dS S S1
dt
dt
U(N)
dt
S
m2 m(DaS1 U?(N1 ))D1 aS1 U?(N1 )0;
and m?3 bV (N1 )lD2 :/
Clearly, m?1 and m?2 have negative real parts. Now
if m?3 0; i.e. R1 /[bV (N1)/(D2/l)] /1 then E1 is
a saddle point and hence E1 is unstable in the
direction orthogonal to N /S coordinate plane.
The variational matrix of system (1) at
E2(N2, 0, I2) is,
S
[U(N)U(N1 )]
(N 0 N)D
U(N)
aS a(S S1 )
(U(N)U(N1 )):
Since
aS1 D
(N 0 N1 )
U(N1 )
[U(N)U(N1 )]
0
D(N 0 N)
U(N)
D(N 0 N1 )
U(N1 )
J. Chattopadhyay et al. / BioSystems 68 (2003) 5 /17
[U(N)U(N1 )]
D(N 0 N) D(N 0 N1 ) D(N 0 N1 )
U(N)
U(N1 )
U(N)
0
D(N N1 )
U(N)
D(N N1 )
[U(N)U(N1 )]
U(N)
0
2 D(N N1 )
[U(N)U(N1 )]
:
U(N)U(N1 )
Since N1 B/N0, the second term is negative. The
first term is negative because U (N ) is an increasing
function. Thus dW /dt 5/0 and dW /dt /0 if and
only if N /N1. The largest invariant subset of the
set of the point where dW /dt/0 is (N1, S1).
Therefore, by LaSalle’s theorem (see Khalil,
1992), (N1, S1) is globally asymptotically stable
in the N /S plane.I
Similarly, we can prove the global asymptotically stability of (N2, I2) in the N /I plane.I
From the above observations we observe that if
the inequalities R1 B/1 and R?1 B1 hold then both
the planar equilibrium E1 and E2 are stable. It is
interesting to note that R1 /1 and R?1 1 imply
the non-existence of the positive equilibrium.
Now we shall perform the local stability analysis
of the system (1) around the positive equilibrium.
The variational matrix of system (1) around the
positive equilibrium E * /(N *, S *, I*) is,
2
3
a11 a12 a13
V 4a21 a22 a23 5;
a31 a32 a33
where a11 //D/aS *U ?(N *)/bI*V ?(N *), a12 /
/aU (N *), a13 //bV (N *), a21 /aS *U ?(N *),
a22 /lS*I */(S */I *)2, a23 //lS *2/(S */I*)2,
a31 /bI*V ?(N *), a32 /lI *2/(S */I*)2, a33 //
lS *I*/(S */I*)2.
The above matrix is equivalent to,
2
3
a11 a12 a13
V 4 0 a?22 a?23 5;
0 a?32 a?33
11
where a?22 a11 a22 a21 a12 ; a?23 a11 a23 a21 a13 ;
a?32 a11 a32 a31 a12 ; a?33 a11 a33 a31 a13 :/
Again the matrix can be written equivalently as:
2
3
a11 a12 a13
V 4 0 aƒ22 0 5;
0 a?32 a?33
where aƒ22 a?33 a?22 a?23 a?32 :/
The eigenvalues of the variational matrix are
m1 /a11, m2 a?33 ; m3 aƒ22 :/
It is clear that m2 /0. Therefore, we claim that
the system (1) around E * is locally unstable.
Hence, we may conclude finally that if R1 /1
and R?1 1; the persistence of the system can never
be attained.
5. Phytoplankton /nutrient model with Holling
type-II uptake rate
Here, we consider the nutrient uptake as Holling
type-II functional form and rewrite the system
equation as:
dN
dt
dS
dt
D(N 0 N)
aSN
K1 N
aSN
K1 N
lSI
SI
bIN
K2 N
;
D1 S;
dI
bIN
lSI
D2 I:
dt K2 N S I
(20)
Here U (N )/N /(K1/N ), V (N )/N /(K2/N ) and
K1, K2 are the half-saturation constants.
The dynamics of the system (20) are depicted in
the following theorem. There are different thresholds R0, R?0 ; R1, and R?1 ; which are defined in
Section 4.3 for the general uptake function and
these thresholds govern the existence and stability
of different states of the system.
5.1. Main theorem
The system (20) always has a plankton-free
steady state E0(N0, 0, 0). This steady state is stable
if and only if,
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J. Chattopadhyay et al. / BioSystems 68 (2003) 5 /17
R0 aN 0
B 1;
D1 (K1 N 0 )
and
R?0 bN 0
(D2 l)(K2 N 0)
(21)
B1B
D2
D2 l
;
(22)
in which case it attracts all feasible solutions.
When R0 /1 and R?0 D2 =(D2 l); the plankton
free steady state is unstable and there exists a
feasible infected phytoplankton free steady state
E1(N1, S1, 0) where,
N1 S1 D1 K1
;
a D1
D [N 0 (a D1 ) K1 D1 ]
D1
a D1
(R0 1);
and a feasible susceptible phytoplankton free
steady state E2(N2, 0, I2) where,
N2 b D2
V
2
aSK
bIK
6D (K N)2 (K N)2
6
6
6
aSK
6
6
(K N)2
6
6
bIK
4
(K N)2
;
D [N 0 (b D2 ) K2 D2 ]
D2
[b D2 ]
D(K2 N 0 )
(R?0 (D2 l)D2 ):
D2 (b D2 )
The steady state E1 is stable if and only if R0 /
1, a /D1 /1 and
R1 bN1
B1;
(D2 l)(K2 N1 )
(23)
in which case it attracts all feasible solutions.
When R1 /1, the infected phytoplankton free
steady state is unstable and there exists a coexistence steady state E *(N *, S *, I *).
The steady state E2 is stable if and only if R?0 D2 =(D2 l); b /D2 /1 and
R?1 3
bN
(K N) 7
7
7
lS2 7
7:
(S I)2 7
7
7
lSI 5
2
(S I)
By applying the same technique as we did in
Section 4.3, we see that one of the eigenvalues of
the characteristic equation of V* is always positive
and given by,
D
aSK
bIK
lSI
2
2
(K N)
(K N) (S I)2
I2 aN
(K N)
lSI
(S I)2
lI2
(S I)2
[a D1 ]
D(K1 N 0 )
D2 K2
in which case it attracts all feasible solutions.
When R?1 1; the susceptible phytoplankton free
steady state is unstable and the coexistence steady
state exists.
It is easy to observe that the variational matrix
of system (20) around this coexisting steady state
E*
is,
aN2
B1;
(D1 l)(K1 N2 )
(24)
b2 IKN
:
(K N)3
Hence the system (20) remains always unstable
around the steady state E *.
The above analysis shows if both the susceptible
and infected phytoplankton consume nutrient then
the coexistence of both susceptible and infected
population is not feasible. But any one of the
populations can exist with proper consumption of
nutrient provided the thresholds R0, R?0 ; R1 and R?1
follow certain parametric relations.
The study of Uhlig and Sahling (1992) on the
dinolagellate Noctiluca scintillans in the German
Bight showed that infected phytoplankton cells do
not feed any more and also not in a position to
reproduce. Motivated by this study, we assume
that the infected phytoplankton population is
incapable of consuming nutrient. We modify the
system (1) accordingly and study the dynamics of
the system around the possible steady states.
J. Chattopadhyay et al. / BioSystems 68 (2003) 5 /17
6. Phytoplankton /nutrient model when infected
population is incapable of nutrient consumption
I
In this case, system (1) takes the following form:
dN
D(N 0 N)aSU (N);
dt
dS
dt
dI
dt
aSU (N)
lSI
SI
lSI
SI
D1 S;
D2 I:
(25)
(26)
lD2 0;
holds, then E1 is a saddle point and is unstable in
the direction orthogonal to the N /S plane.
Hence by Butler /McGhee Lemma (see Freedman and Waltman, 1984), inequality (26) implies
the persistence of the system (25). Thus the system
possesses
an
interior
equilibrium
E */
(N *, S *, I*) with,
1 l D2 D1
;
N U
a
D(N0 U 1 )((l D2 D1 )=a)
l D2 D1
(l D2 )
D2
D(N0 U 1 ((l D2 D1 )=a))
:
(l D2 D1 )
(27)
The variational matrix of system (23) around
the positive equilibrium E * /(N *, S *, I*) is,
2
3
b11 b12 b13
V 4b21 b22 b23 5;
b31 b32 b33
If 05/(D2/l ), then the infected phytoplankton
cannot exist; if R0 B/1 then susceptible phytoplankton and hence infected phytoplankton cannot exist.
For the plankton-free steady state of the system
(25) as F0 /(N0, 0, 0), it is easy to see that if R0 B/
1 then the steady state is locally asymptotically
stable.
When R0 /1 the plankton-free steady state is
unstable (saddle) and there exist a feasible infected
phytoplankton-free steady state F1 /(N1, S1, 0)
with N1 /U 1(D1/a ), S1 /(D /D1)[N0/U 1(D1/
a )].
If the conditions (15) and (16) hold then the
system (25) has a non-negative equilibrium F1 /
(N1, S1, 0) where N1 and S1 are defined above.
It is easy to see that if the inequality,
S
13
;
where bij (i, j/1, 2, 3) can be found from aij in
Section 4, by putting b/0.
The above matrix is equivalent to:
2
3
b11 b12 0
V 4 0 b?22 b?23 5;
0 b32 b33
where
b?22 b11 b22 b21 b12 ;
Again the matrix
2
b11 b12
V 4 0 bƒ22
0 b32
b?23 b11 b23 :
can be written equivalently as:
3
0
0 5;
b33
where bƒ22 b33 b?22 b?23 b32 :/
The eigenvalues of the variational matrix are m?1 ;
m?2 ; m?3 which are given by,
m?1 DaSU?(N);
m?2 m?3 lSI
(S I)2
;
la2 S2 IU(N)U?(N)
(S I)2
:
It is clear that all the eigenvalues are negative.
Therefore, we claim that the system (25) around
E * is locally asymptotically stable. We perform the
stability analysis of system (25) by considering
Holling type-II functional form (as we did in
Section 5) and observe that the system (25) around
the positive interior equilibrium is locally asymptotically stable.
J. Chattopadhyay et al. / BioSystems 68 (2003) 5 /17
14
Table 1
Thresholds and stability of steady states
Thresholds (R0, /R?0 ;/ R1, /R?1 )/
Steady state properties
(N0, 0, 0)
(N1, S1, 0)
For general nutrient uptake function and also for Holling type-II uptake function
Asymptotically stable
Not feasible
R0 B/1, R?0 B1/
and global attractor
R0 /1, R1 B/1, (a /D1)/1
Not feasible
Asymptotically stable
and global attractor
/R?
Not feasible
Not feasible
0 1; R?1 B1; (b /D2)/1
R1 1; R?1 1/
Not feasible
/
Not feasible
When infected phytoplankton incapable of nutrient consumption
R0 B/1
Asymptotically stable
Not feasible
and global attractor
R0 /1, R1 B/1
Not feasible
Asymptotically stable
and global attractor
R1 /1
Not feasible
Not feasible
7. Discussion
The results, which we have established for three
component models consisting of nutrient concentration, susceptible phytoplankton and infected
phytoplankton, are given in Table 1. There are two
distinct categories in this table. We have studied
the stability behaviour of the system around the
feasible steady states with general nutrient uptake
function both for susceptible and infected phytoplankton and with Holling type-II uptake function
as an example. We have also studied the behaviour
of the system with a special consideration that the
infected phytoplankton are not able to consume
nutrient any more. Our analysis have leaded to
four distinct thresholds R0, R?0 ; R1 and R?1 :
Existence and stability of various states of the
system have been determined in terms of these
thresholds.
The thresholds R0, R?0 ; R1, and R?1 have relevant
biological interpretations. Firstly,
R0 aU(N 0 )
D1
;
is the ratio of the maximal nutrient uptake of the
susceptible phytoplankton to its mortality rate. It
measures the ability of the nutrient environment to
(N2, 0, I2)
(N *, S *, I *)
Not feasible
Not feasible
Not feasible
Not feasible
Asymptotically stable
and global attractor
Not feasible
Not feasible
Unstable
Not feasible
Not feasible
Not feasible
Not feasible
Not feasible
Asymptotically stable
and global attractor
support a susceptible phytoplankton population
and can be understood through the following
reasoning. If in a susceptible phytoplankton free
layer with nutrient level N0 one introduces a single
phytoplankton at time t/0, then it will have
probability eD1 t of surviving to time t /0, and
the expecting number of its offspring, over its
lifetime would be:
g
0
eD1 t aU(N 0 ) dt
aU(N 0 )
D1
R0 :
Thus R0 B/1, there would be, on average, less than
one offspring over the lifetime of each susceptible
phytoplankton and the nutrient level would be
insufficient to support a stable phytoplankton
population. If R0 /1, then the average reproduction of each susceptible phytoplankton is more
than one and a stable population can be established. Note that, as would be expected, R0 is an
increasing function of the total nutrient level N0
and the uptake rate a, and a decreasing function of
the mortality rate D1. Similar arguments can be
given exactly in the same way for the threshold of
infected phytoplankton population R?0 bV (N 0 )=/
/(D l):/
2
J. Chattopadhyay et al. / BioSystems 68 (2003) 5 /17
15
Fig. 1. Phase portraits of nutrient /susceptible phytoplankton-infected phytoplankton system with Holling type-II uptake rate
function depicting different behaviours for threshold conditions in Table 1. All the phase trajectories are starting from the initial point
P (N (0), S (0), I (0)). (a) For R0 B/1 and R?0 B1; the trajectories are asymptotically stable and attracting towards the equilibrium point
E0. (b) For R0 /1, R1 B/1 and (a /D1) /1, the trajectories are asymptotically stable and attracting towards the equilibrium point E1. (c)
For R?0 1; R?1 B1 and (b /D2) /1, the trajectories are asymptotically stable and attracting towards E2. (d) For R1 /1 and R?1 1; the
trajectories are unstable and no fixed positive interior equilibrium point is observed.
The second threshold:
R1 bV (N1 )
D2 l
(where N1 /U1(D1/a ), S1 /(D /D1)[N0/U 1
(D1/a )]), is the ratio of the maximal uptake rate
of the infected phytoplankton to the difference of
its mortality rate and infection rate, given that a
stable susceptible phytoplankton population (S1)
has been established. It can be interpreted exactly
in the same way as R0 was on the previous
paragraph by considering the introduction of a
single infected phytoplankton in an established
susceptible phytoplankton population which has
settled to its equilibrium S1. R1 is clearly an
increasing function of the uptake rate b and the
total nutrient level N1. As expected, it is a
decreasing function of the difference of the mortality rate and infection rate of infected phyto-
16
J. Chattopadhyay et al. / BioSystems 68 (2003) 5 /17
plankton population. Similarly we can interpret
the threshold R?1 aU(N2 )=(D1 l) (where N2 /
V 1(D2/b ), I2 /(D /D2)[N0/V 1(D2/b )]). It is
interesting to note that R?1 is an increasing function
of the uptake rate a and the total nutrient level N2
but it is decreasing function of the sum of the
mortality rate of susceptible phytoplankton population and the infection rate of the infected
phytoplankton population.
Phase portraits of nutrient/susceptible phytoplankton-infected phytoplankton system with Holling type-II uptake rate function depicting
different behaviours for threshold conditions in
Table 1 have been studied numerically and presented in Fig. 1. We have observed that coexistence of nutrient, susceptible and infected
phytoplankton population is not possible for
general uptake and Holling type-II functional
form. However, it is possible when infected
phytoplankton population is not in a state of
consuming nutrient.
Before ending the article we would like to
mention that the process of infection in planktonic
system is not yet well established. Hence the model
can be improved by considering different processes
of infection.
Acknowledgements
The authors are very much grateful to the
anonymous reviewers for their helpful comments
and suggestions.
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