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International Journal of

Biomathematics and Systems BiologyOfficial Journal of Biomathematical Society of India

Volume 1, No. 1, Year 2014

The role of space in marine plankton ecosystem inpresence of toxin producing phytoplankton

Anal Chatterjee1 and Samares Pal2

1 Department of Mathematics, Sheikhpara A.R.M. Polytechnic, Sheikhpara-742308, India.E-mail: [email protected]

2Department of Mathematics, University of Kalyani, Kalyani- 741235, India. Email:[email protected]

Received: 10 April 2014 Accepted: 2 July 2014

Abstract. In this paper, we investigate the complex dynamics of a nutrient phytoplankton zoo-plankton reaction-diffusion system with Holling type-II functional response. In this model, it isassumed that phytoplankton releases toxic chemical for self defense against their predators. Themodel system is studied analytically and the threshold values for the existence and stability of var-ious steady states are worked out. Moreover we have found out a condition for diffusive instabilityof a locally stable equilibrium. Also it is shown that the system undergoes Hopf bifurcation whenthe maximal zooplankton conversion rate crosses a certain critical value. Computer simulationshave been carried out to illustrate various analytical results.

Key words: Phytoplankton, Zooplankton, Toxic, Bloom

1 Introduction

Dynamics of food-web population models largely depends on nutrients. Tiny plankton populates in the surface waters of the oceans,rivers, and lakes. An algal bloom or marine bloom or water bloom is a rapid increase in the population of algae in an aquatic systemwhich is generally caused by high nutrient levels and favorable conditions. Plankton populations comprise a large number of differentspecies and are in the bottom of the food chain. In addition, effects of toxin producing phytoplankton play a vital role in marineplankton ecology. After the pioneering work of [1], many papers on plankton dynamics have been appeared in the literature.

2 Corresponding author: Tel.:+91-33-25666571 ; Email: [email protected]

ISSN: 2394-7772

2 Chatterjee and Pal

The role of toxin producing phytoplankton in the reduction of grazing pressure by zooplankton has been acknowledged in severalpublication [2, 3, 4]. The growth rates of both plankton populations are diminished when phytoplankton can uptake toxin in onemodel and in the other model zooplankton can uptake toxin have been discussed in [5]. While studying phytoplankton-zooplanktoninteraction, [6] found that the occurrence of toxin producing phytoplankton may not be always harmful rather help to maintain stableequilibrium in trophodynamics through coexistence of all the species. Recently role of competition in phytoplankton population forthe occurrence and control of plankton bloom in the presence of environmental fluctuations and effect of toxin on NPZ model havebeen studied in [7, 8, 9, 10, 11]. The importance of nutrient for growth of plankton in phytoplankton-herbivore interaction model havebeen discussed in [12, 13, 14, 15]. Multi nutrient model and effects of harmful algal bloom have been discussed in [16, 17] whileworking on NPZ model.

The exploration of pattern formation mechanisms in nonlinear complex systems is the one of the major scientific problem. Thedevelopment of the theory of self-organized temporal, spatial or functional structuring of nonlinear systems far from equilibriumhas been discussed in [18, 19, 20]. Recently, the role of space in stage-structured cannibalism model and spatial effects on viraldisease in plankton system have been discussed in [21, 22]. A three dimensional mathematical model of plankton dynamics withhelp of reaction-diffusion equations has been studied in [23]. The authors in [24] consider spatiotemporal pattern formation in aratio-dependent predator-prey system and show that the system can develop patterns both inside and outside of the Turing parameterdomain. The researchers in [25] represent a theoretical analysis of processes of pattern formation that involves organism distributionand their interaction of spatially distributed population with local diffusion.

In this paper we extended the model proposed in [7] by taking into account the competition between phytoplankton and zoo-plankton in reaction-diffusion system. Further we established the conditions for stability in diffusive system. It is assumed here thatzooplankton does not take nutrients directly, i.e., the predator is obligate, and all dead zooplankton and phytoplankton are recycledback into nutrient. The main aim in this present study is to see the combined effect of bottom up and top down approach on the growthof phytoplankton especially when they have the ability to release toxic chemicals in diffusive system.

2 The mathematical model

Let N(t) be the concentration of the nutrient at time t. Let P(t) and Z(t) be the concentration of phytoplankton and zooplanktonpopulation respectively at time t. Let N0 be the constant input of nutrient concentration, D is the dilution rate [26]. The constant 1

D hasthe physical dimension of a time and represents the average time that nutrient and waste products spend in the system [27]. D1 and D2

denote the dilution rates of the phytoplankton and zooplankton populations respectively. Let α1 and α2 be the nutrient uptake rate forthe phytoplankton population and conversion rate of nutrient for the growth of phytoplankton population respectively (α2 ≤ α1). Letµ1 be the mortality rate of the phytoplankton population and µ2 be the mortality rate of the zooplankton population. Let µ3 (µ3 ≤ µ1)

be the nutrient recycle rate after the death of phytoplankton population and µ4 (µ4 ≤ µ2) be the nutrient recycle rate after the death ofzooplankton population. Let γ1 be the maximal zooplankton ingestion rate and γ2 (γ2 ≤ γ1) be the maximal zooplankton conversionrate. We choose Holling type II functional form to describe the grazing phenomena with K1 and K2 as half saturation constant. It isassumed θ is the rate of zooplankton decay due to toxin producing phytoplankton. With these assumptions our model system is

dNdt

= D(N0 −N)− α1PNK1 +N

+µ3P+µ4Z ≡ F1(N,P,Z)

dPdt

=α2PNK1 +N

− γ1PZK2 +P

− (µ1 +D1)P ≡ F2(N,P,Z)

dZdt

=(γ2 −θ)PZ

K2 +P− (µ2 +D2)Z ≡ F3(N,P,Z). (2.1)

The system (2.1) has to be analyzed with the following initial conditions,

N(0)> 0, P(0)> 0, Z(0)> 0. (2.2)

3 Some preliminary results

3.1 Positive invariance

By setting X = (N, P, Z)T ∈ R3 and F(X) = [F1(X), F2(X), F3(X)]T , with F : R3+ → R3 and F ∈ C∞(R3), equation (2.1) becomes

International Journal of Biomathematics and Systems Biology 3

X = F(X), (3.1)

together with X(0) = X0 ∈ R+3. It is easy to check that whenever choosing X(0)∈ R+

3 with Xi = 0, for i=1, 2, 3, then Fi(x) |Xi=0≥ 0.Then any solution of equation (3.1) with X0 ∈ R+

3, say X(t) = X(t;X0), is such that X(t) ∈ R+3 for all t > 0.

3.2 Equilibria

The system (2.1) possesses the following equilibria:(i) The plankton free equilibrium E0 = (N0,0,0).(ii) The zooplankton free equilibrium E1(N1,P1,0) with

N1 =(µ1+D1)K1

α2−(µ1+D1), P1 =

Dα2 [N0α2−(N0+K1)(µ1+D1)][α2−(µ1+D1)][α1(µ1+D1)−µ3α2 ]

, which exists if

max{(µ1 +D1),

(N0 +K1)(µ1 +D1)

N0

}< α2 <

α1(µ1 +D1)

µ3.

(iii) The positive interior equilibrium E∗ = (N∗,P∗,Z∗) where, P∗ = K2(µ2+D2)(γ2−θ)−(µ2+D2)

,

Z∗ = K2(γ2−θ)[α2N∗−(µ1+D1)(K1+N∗)]γ1(K1+N∗)[(γ2−θ)−(µ2+D2)]

, N∗ =−B1+

√B2

1−4A1C12A1

with A1 = Dγ1[(γ2 −θ)− (µ2 +D2)],

B1 =−[(DN0 −DK1)[(γ2 −θ)− (µ2 +D2)]γ1 +(µ3 −α1)(µ2 +D2)K2γ1 +K2µ4(γ2 −θ)[α2 − (µ1 +D1)]],

C1 =−[DN0K1γ1[(γ2 −θ)− (µ2 +D2)]+(µ2 +D2)K1K2µ3γ1 − (µ1 +D1)K1K2(γ2 −θ)µ4)]),

where, L =−B1 +√

B21 −4A1C1. So that N∗ = L

2γ1D[(γ2−θ)−(µ2+D2)].

Thus the condition for the existence of the interior equilibrium point E∗(N∗, P∗,Z∗) is given by,

γ2 > (µ2 +D2 +θ), (3.2)

andL

2γ1D[(γ2 −θ)− (µ2 +D2)]>

K1(µ1 +D1)

α2 − (µ1 +D1). (3.3)

3.3 Criterion for the extinction of phytoplankton

Theorem 1. Let the inequality

R0 =α2N0

(µ1 +D1)(K1 +N0)< 1 (3.4)

hold. Then limt→∞(N(t),P(t),Z(t)) = E0.

Theorem (1) reveals that if the ratio of maximal nutrient conversion rate for the growth of phytoplankton and the loss rate of thephytoplankton is less than to unity then the phytoplankton population will remove from the system.

The proof is obvious.

3.4 Eigenvalue analysis and Hopf-bifurcation

In this section, local stability analysis of the system around the biological feasible equilibria is performed. The central aim of thepresent analysis is to find out suitable mechanism to explain the planktonic blooms and a possible solution to control it.Lemma 1. If R0 > 1, then the plankton free steady state E0 of the system (2.1) is unstable.

Lemma 2. There exists a feasible zooplankton free steady state E1 of the system (2.1) which is unstable if

R1 =Dα2[N0α2 − (N0 +K1)(µ1 +D1)]a

(µ2 +D2)[α2 − (µ1 +D1)]bK2> 1, (3.5)

where a = [γ2 −θ−µ2 −D2] and b = [α1(µ1 +D1)−µ3α2].


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Fig. 1 (a) The equilibrium point E∗ is locally asymptotically stable for the parameter values given in this paper. (b) The figuredepicts oscillatory behavior around the positive interior equilibrium point E∗ of system (2.1) for increasing γ2, from 1 to 2 with otherparameter values are unaltered. (c) The bifurcation diagram for γ2 with all parameter values as given in this paper.

Local Asymptotic Stability (LAS) analysis of the system (2.1)Let E = (N,P,Z) be any arbitrary equilibrium. Then the variational matrix about E is given by

V =

−D− α1K1P

(K1+N)2 − α1NK1+N +µ3 µ4

K1α2P(K1+N)2 x − γ1P

K2+P

0 (γ2−θ)K2Z(K2+P)2 y

,

where x = α2NK1+N − K2γ1Z

(K2+P)2 − (µ1 +D1), y = (γ2−θ)PK2+P − (µ2 +D2).

Proof of Lemma 1. By computing the variational matrix for the equilibrium E0 of the system (2.1) we find that the eigenvalues of thevariational matrix V0 are λ1 =−D < 0, λ2 = (D1 +µ1)(R0 −1), λ3 =−(D2 +µ2)< 0. Clearly E0 is asymptotically stable if and onlyif R0 < 1. When R0 > 1, the plankton free steady state is unstable and there exist a feasible zooplankton free steady state E1.Proof of Lemma 2. The eigenvalues of the variational matrix V1 around the equilibrium E1 of the system (2.1) are λ1

′, λ2

′which are

the roots of the equation

λ2 +λ(

D+α1K1P1

(K1 +N1)2

)+

K1α2P1

(K1 +N1)2

(α1N1

K1 +N1−µ3

)= 0


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Fig. 2 (a) Biomass distribution of nutrient over time and space of the model (3.9) for parameter values as given in this paperwith diffusion coefficients dn = 0.3, dp = 0.3 and dz = 0.3. (b) Biomass distribution of phytoplankton over time and space of themodel (3.9) for same set of parameter values as given in this paper with diffusion coefficients dn = 0.3, dp = 0.3 and dz = 0.3. (c)Biomass distribution of zooplankton over time and space of the model (3.9) for parameter values as given in this paper with diffusioncoefficients dn = 0.3, dp = 0.3 and dz = 0.3.

and λ3′= (γ2−θ)P1

K2+P1− (D2 +µ2). Clearly λ1

′and λ2

′have negative real parts because for equilibrium point E1(N1,P1,0), the last term

of above quadratic equation is positive which can be easily seen from equation (2.1) by putting Z=0 . Now E1 is unstable i.e., λ3′> 0

if condition (3.5) is satisfied. Now we study the stability analysis of the positive interior equilibrium of the system (2.1).

Stability analysis of the positive interior equilibrium of the system (2.1)The variational matrix of system (2.1) around the positive equilibrium E∗ = (N∗,P∗,Z∗) is

V ∗ =

a11 a12 a13

a21 a22 a23

0 a32 0

,

where a11 =−D− K1α1P∗

(K1+N∗)2 < 0, a12 =− α1N∗

K1+N∗ +µ3 < 0, a13 = µ4 > 0a21 =K1α2P∗

(K1+N∗)2 > 0,

a22 =γ1Z∗

K2+P∗ − K2γ1Z∗

(K2+P∗)2 > 0, a23 =− γ1P∗

K2+P∗ < 0, a32 =K2(γ2−θ)Z∗

(K2+P∗)2 > 0.The characteristic equation is

y3 +Ay2 +By+C = 0,


where A =−(a11 +a22), B = a11a22 −a12a21 −a23a32; C = a11a23a32 −a13a32a21. By the Routh-Hurwitz criteria, all roots of aboveequation have negative real parts if and only if A > 0, C > 0 and AB−C > 0.A =−(a11 +a22) = D+ K1α1P∗

(K1+N∗)2 +K2γ1Z∗

(K2+P∗)2 −γ1Z∗

K2+P∗ > 0

if D+ K1α1P∗

(K1+N∗)2 +K2γ1Z∗

(K2+P∗)2 > γ1Z∗

K2+P∗ .

Also B > 0 if a12a21 +a23a32 < a11a22 since a12a21 < 0, a23a32 < 0, a11a22 < 0.

C = a11a23a32 −a13a32a21

=

(−D− K1α1P∗

(K1 +N∗)2

)(−γ1P∗

K2 +P∗

)(K2(γ2 −θ)Z∗

(K2 +P∗)2

)− µ4K2(γ2 −θ)Z∗

(K2 +P∗)2K1α2P∗

(K1 +N∗)2

=Dγ1(γ2 −θ)K2P∗Z∗

(K2 +P∗)3 +K2(γ2 −θ)P∗Z∗

(K2 +P∗)2K1

(K1 +N∗)2

[α1γ1P∗

K2 +P∗ −α2µ4

]=

Dγ1(γ2 −θ)K2P∗Z∗

(K2 +P∗)3 +K1K2(γ2 −θ)P∗Z∗

(K2 +P∗)2(K1 +N∗)2

[α1γ1(µ2 +D2)

(γ2 −θ)−α2µ4

]=

Dγ1(γ2 −θ)K2P∗Z∗

(K2 +P∗)3 +K1K2(γ2 −θ)P∗Z∗

(K2 +P∗)2(K1 +N∗)2

[α1γ1µ2 +α1γ1D2 −α2(γ2 −θ)µ4

(γ2 −θ)

]> 0,

since α1 > α2, γ1 > γ2, µ2 > µ4, γ2 > θ. Finally, AB−C = −(a11 +a22)(a11a22 −a12a21 −a23a32)− (a11a23a32 −a13a32a21) > 0,if −(a11 +a22)(a11a22 − a12a21 − a23a32) > (a11a23a32 − a13a32a21). Therefore according to Routh-Hurwitz criterion, E∗ is locallyasymptotically stable.

Theorem 2. When the maximal zooplankton conversion rate γ2 crosses a critical value, say γ2∗, the system (2.1) enters into Hopf-

bifurcation around the positive equilibrium and that induces oscillations of the populations.Proof. Now we will study the Hopf-bifurcation of the system given by (2.1) taking γ2 as the bifurcation parameter, the necessaryand sufficient conditions for the existence of the Hopf-bifurcation for γ2 = γ2

∗ if it satisfies (i) A(γ∗2) > 0, B(γ∗2) > 0, C(γ∗2) > 0,(ii)A(γ2

∗)B(γ2∗)−C(γ2

∗) = 0 and (iii) the eigenvalues of above characteristic equation should be of the form λi = ui + ivi, andduidγ2

= 0, i = 1,2,3. We will now verify the Hopf-bifurcation condition (iii), putting λ = u+ iv in above equation, we get

(u+ iv)3 +A(u+ iv)2 +B(u+ iv)+C = 0. (3.6)

On separating the real and imaginary parts and eliminating v between real and imaginary parts, we get

8u3 +8Au2 +2u(A2 +B)+AB−C = 0. (3.7)

It is clear from the above that u(γ2∗) = 0 iff A(γ2

∗)B(γ2∗)−C(γ2

∗) = 0. Further, at γ2 = γ2∗, u(γ2

∗) is the only real root, since thediscriminate 8u2 +8Au+2(A2 +B) = 0 is 64A2 −64(A2 +B)< 0. Again, differentiating (3.7) with respect to γ2, we have24u2 du

dγ2+ 16Au du

dγ2+ 2(A2 + B) du

dγ2+ 2u[2A dA

dγ2+ dB

dγ2] + dS

dγ2= 0 where S = AB −C. Now since at γ2 = γ2

∗, u(γ2∗) = 0 we

get[

dudγ2

]γ2=γ2∗

=− dS

dγ22(A2+B) = 0. This ensures that the above system has a Hopf-bifurcation around the positive interior equilibrium

E∗. From the Theorem (2), it is observed that higher the maximal zooplankton conversion rate is the chance for bloom formation.Next we study the global asymptotic stability of the equilibria E1.

Theorem: 3. If the non negative equilibrium E1 exists, then (N1, P1) is globally asymptotically stable in the N −P plane.

Proof. Let us define a Lyapunov function [28]

W (N,P) =∫

N1

N x−N1

xdx+

α1N1 −µ3(K1 +N1)

α2N1

∫P1

P x−P1

xdx.

Then W (N,P) = 0 if and only if N = N1, P = P1 and from the existence condition of E1, W (N,P)≥ 0 in the N −P plane.The time derivative of W along the trajectories of the subsystem is


dWdt

=dNdt

[N −N1

N

]+

α1N1 −µ3(K1 +N1)

α2N1

[dPdt

][P−P1

P

]

= [N −N1]

[(N0 −N)D− α1PN

K1+N +µ3P

N

]+

α1N1 −µ3(K1 +N1)

α2N1(P−P1)

(α2N

K1 +N− (µ1 +D1)

)

= [N −N1]

[D(N0 −N)

N−P(

α1

K1 +N− µ3

N)

]+

[α1 −

µ3(K1 +N1)

N1

][N

K1 +N− µ1 +D1

α2

](P−P1)

=

[D(N0 −N)

N−P

(α1

K1 +N− µ3

N

)−D

(N0 −N1

N1

)+P1

(α1

K1 +N1− µ3

N1

)][N −N1]

+

[α1 −

µ3(K1 +N1)

N1

][N

K1 +N− N1

K1 +N1

](P−P1)

= [N −N1][D(N0 −N)

N− D(N0 −N1)

N1−P(

α1

K1 +N− µ3

N)

+P1

(α1

K1 +N1− µ3

N1

)+

(α1

K1 +N1− µ3

N1

)(K1

K1 +N

)(P−P1)]

≤ [N −N1][D(N0 −N)

N−P(

α1

K1 +N− µ3

N)− D(N0 −N1)

N1+P(

α1

K1 +N1− µ3

N1)]

≤ [N −N1][D(N0 −N)

N− D(N0 −N1)

N1+

D(N0 −N1)

N− D(N0 −N1)

N+P(

µ3

N− µ3

N1)]

≤ [N −N1]

[−D(N −N1)

N+

N1 −NNN1

(Pµ3 +D(N0 −N1)

)]≤ −(N −N1)

2 DN

− (N −N1)2

NN1[Pµ3 +D(N0 −N1)]. (3.8)

If N1 < N0, the second term is negative. The first term is obviously negative. Thus dWdt ≤ 0 and dW

dt = 0 if and only if N = N1. Thelargest invariant subset of the set of the point where dW

dt = 0 is (N1,P1). Therefore by LaSalle’s theorem [29] (N1,P1) is globallyasymptotically stable in the N −P plane.

3.5 Stability analysis in the presence of diffusion

We have also incorporated the spatial effects on the system (2.1) via simple diffusions. If dn, dp, dz are the self diffusion coefficientsof nutrient, phytoplankton and zooplankton population respectively then the model system (2.1) in the presence of one-dimensionaldiffusion has the following form:

∂N∂t

= D(N0 −N)− α1PNK1 +N

+µ3P+µ4Z +dn∇2N

∂P∂t

=α2PNK1 +N

− γ1PZK2 +P

− (µ1 +D1)+dp∇2P

∂Z∂t

=(γ2 −θ)PZ

K2 +P− (µ2 +D2)Z +dz∇2Z. (3.9)

To study the effect of diffusion on the model system, we have considered the linearized form of the system (3.9) about (N, P, Z)is:

∂U∂t

= a11U + a12V + a13W +dn∇2N

∂V∂t

= a21U + ˆa22V + a23W +dp∇2P

∂W∂t

= a31U + a32V + a33W +dz∇2Z, (3.10)

where N = N+U,P= P+V,Z = Z+W and a11 =−D− K1α1P(K1+N)2 , a12 =− α1N

K1+N+µ3, a13 = µ4, a21 =

K1α2P(K1+N)2 , a22 =

γ1 ZK2+P

− K2γ1 Z(K2+P)2 ,

a23 =−γ1PK2+P

, a32 =K2(γ2−θ)Z(K2+P)2 , a33 =

(γ2−θ)PK2+P

− (µ2 +D2).


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Fig. 3 (a) Biomass distribution of nutrient over time and space of the model (3.9) for γ2 = 2 and other parameter values as givenin this paper with diffusion coefficients dn = 0.3, dp = 0.3 and dz = 0.3, the system is oscillatory about E∗. (b) Biomass distributionof phytoplankton over time and space of the model (3.9) for γ2 = 2 and other parameter values as given in this paper with diffusioncoefficients dn = 0.3, dp = 0.3 and dz = 0.3. (c) Biomass distribution of zooplankton over time and space of the model (3.9) for γ2 = 2and other parameter values as given in this paper with diffusion coefficients dn = 0.3, dp = 0.3 and dz = 0.3.

It may be noted that (U,V,W ) are small perturbations of (N,P,Z) about the equilibrium point (N, P, Z). We start by assuming

solutions of the form

UVW

=

v1

v2

v3

eλt+i(kxx+kyy) where λ > 0 is the frequency, vi > 0 represents the amplitude (i = 1,2,3) and

kx,ky > 0 are wave number of the perturbations in time t. Substituting k2 = k2x + k2

y , the system (3.10) becomes

∂U∂t

= (a11 −dnk2)U + a12V + a13W

∂V∂t

= a21U +( ˆa22 −dpk2)V + a23W

∂W∂t

= a31U + a32V +(a33 −dzK2)W, (3.11)

The eigenvalues for the plankton-free steady state E0, obtained from the characteristic equation of the linearized system (3.11)are −D− dnk2,−(µ2 +D2)− dzk2 and α2N0

K1+N0 − (µ1 +D1)− dpk2 = (µ1 +D1)[R0 − 1]− dpk2. In absence of diffusion, E0 is locallyasymptotically stable for R0 < 1. Under the same conditions E0 (in presence of diffusion) is also spatially stable.


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Fig. 4 (a) Biomass distribution of nutrient over time and space of the model (3.9) for γ2 = 2 and θ = 1 with other parameter valuesas given in this paper with diffusion coefficients dn = 0.3, dp = 0.3 and dz = 0.3. (b) Biomass distribution of phytoplankton overtime and space of the model (3.9) for γ2 = 2 and θ = 1 with other parameter values as given in this paper with diffusion coefficientsdn = 0.3, dp = 0.3 and dz = 0.3 . (c) Biomass distribution of zooplankton over time and space of the model (3.9) for γ2 = 2 and θ = 1with other parameter values as given in this paper with diffusion coefficients dn = 0.3, dp = 0.3 and dz = 0.3.

Again the eigenvalues of the variational matrix for the steady state E1 = (N1,P1,0) in presence of diffusion are λ1′′, λ2

′′which

are the roots of the equation

λ2 +λT +T dpk2 +K1α2P1

(K1 +N1)2

(α1N1

K1 +N1−µ3

)= 0

where T =(

D+ α1K1P1(K1+N1)2 + k2dn

)and λ3

′′= (γ2−θ)P1

K2+P1−(D2+µ2)−dzk2. Clearly λ1

′′and λ2

′′have negative real parts for equilibrium

point E1(N1,P1,0). Now E1 is unstable i.e., λ3′′> 0 if condition (3.5) is satisfied. Thus under the same conditions E1 (in presence of

diffusion) is also spatially stable.

At E∗, the characteristic equation of the linearized system (3.11) is λ3 +A∗λ2 +B∗λ+C∗ = 0, whereA∗ = A+(dn +dp +dz)k2,B∗ = B− [a22dn +a11dp +(a11 +a22)dz]k2 +(dndp +dpdz +dzdn)k4 andC∗ =C+[−a23a32dn +(a11a22 −a12a21)dz]k2 − (a11dpdz +a22dndz)k4 +dndpdzk6.

Here a11 = −D− K1α1P∗

(K1+N∗)2 < 0, a12 = − α1N∗

K1+N∗ + µ3 < 0, a13 = µ4 > 0; a21 = K1α2P∗

(K1+N∗)2 > 0, a22 = γ1Z∗

K2+P∗ − K2γ1Z∗

(K2+P∗)2 > 0, a23 =

− γ1P∗

K2+P∗ < 0, a31 = 0, a32 =K2(γ2−θ)Z∗

(K2+P∗)2 > 0, a33 = 0.


010

2030

40

020

4060

80

1000

0.5

1

1.5

2

Space (a) Time

Nu

trie

nt

010

2030

40

020

4060

80

1000

0.5

1

1.5

2

Space (b) Time

Ph

yto

pla

nk

ton

010

2030

40

020

4060

80100

−2

0

2

4

6

8

10

x 10−4

Space (c) Time

Zo

op

lan

kto

n

Fig. 5 (a) Biomass distribution of nutrient over time and space of the model (3.9) for γ2 = 2 and θ = 2.1 with other parameter valuesas given in this paper with diffusion coefficients dn = 0.3, dp = 0.3 and dz = 0.3. (b) Biomass distribution of phytoplankton over timeand space of the model (3.9) for γ2 = 2 and θ = 2.1 with other parameter values as given in this paper with diffusion coefficientsdn = 0.3, dp = 0.3 and dz = 0.3. (c) Biomass distribution of zooplankton over time and space of the model (3.9) for γ2 = 2 and θ = 2.1with other parameter values as given in this paper with diffusion coefficients dn = 0.3, dp = 0.3 and dz = 0.3.

Lemma 3. If (3B+a211 +a2

22)d >−(3a11 +a23a32 +a12a21)M, the stability of the non-spatial system (2.1) at E∗ implies the stabilityof the diffusive system (3.9) at E∗.Proof. If the non-spatial system (2.1) is locally asymptotically stable at E∗, then A > 0, B > 0, C > 0 and AB−C > 0.Here, A > 0 implies A∗ > 0.Now A∗B∗−C∗ = AB−C+ k6H1 − k2H2 + k2H3, whereH1 = (dn +dz)(dndp +dpdz +dzdn)+d2

p(dn +dz)> 0.H2 = a11dn(dp +dz)+a22dp(dn +dz)+(dn +dp +dz){a22dn +a11dp +(a11 +a22)dz} andH3 = B(dn +dp +dz)+a2

11(dp +dz)+a222(dn +dz)+a11a22(dn +dp +dz)+a23a32dn +a12a21dz.

Since A > 0, it follows thatH2 ≤ 2M2(a11 +a22)+3M.2D(a11 +a22) = 8M2(a11 +a22) =−8M2A < 0, where M = max{dn,dp,dz}.Also, B > 0 implies H3 > 0 if (3B+a2

11 +a222)d >−{3a11 +a23a32 +a12a21}M where d = min{dn,dp,dz}. Therefore k6H1 −k4H2 +

k2H3 > 0 and consequently A∗B∗ >C∗ (since AB>C). Hence, it follows that whenever (3B+a211+a2

22)d >−(3a11 +a23a32 +a12a21)M,the stability of the non-spatial system (2.1) at E∗ implies the stability of the system (3.9) at E∗.


Lemma 4. The condition for diffusive instability of the system at the positive equilibrium E∗ is given by K6H1 −K4H2 +K2H3 >

C−AB and H2 ≥ 2√

H3H1.Proof. Diffusive instability occurs if A∗B∗ ≤ C∗. This happens if K4H1 −K2H2 +H3 ≤ 0. Since H1 > 0 and H3 > 0, it follows thatK4H1 −K2H2 +H3 ≤ 0 if H2 ≥ K4H1+H3

K2 > 0. Let H(η) = η2H1 −ηH2 +H3, η = K2. Since H ′′ = 2H1 = 2(dn +dz)(dndp +dpdz +

dzdn)+d2p(dn+dz)> 0. it follows that H has a minimum at η=K2

c , where K2c = H2

2H1. Thus, Hmin =K4

c H1−K2c H2+H3 =H3−

H22

4H1≤ 0

if H2 ≥ 2√

H1H3.

4 Numerical simulations

In this section, we investigate the effect of various parameters on the qualitative behavior of the system by using numerical simulation.In order to illustrate the analytical results we have used the MATLABs in-build solver pdepe which adopts finite difference schemefor solving partial differential equations. The performed numerical simulations are carried out in a one-dimensional spatial domain.

05

1015

2025

3035

40

0

20

40

60

80

1000

0.5

1

1.5

2

2.5

3

Space (a) Time

Nu

trie

nt

010

2030

40

020

4060

80

1000

0.5

1

1.5

2

Space (b) Time

Ph

yto

pla

nk

ton

05

1015

2025

3035

40

0

20

40

60

80

1000

0.5

1

1.5

2

Space (c) Time

Zo

op

lan

kto

n

Fig. 6 (a) Biomass distribution of nutrient over time and space of the model (3.9) for N0 = 3 and other parameter values as given inthis paper with diffusion coefficients dn = 0.3, dp = 0.3 and dz = 0.3. (b) Biomass distribution of phytoplankton over time and spaceof the model (3.9) for N0 = 3 and other parameter values as given in this paper with diffusion coefficients dn = 0.3, dp = 0.3 anddz = 0.3. (c) Biomass distribution of zooplankton over time and space of the model (3.9) for N0 = 3 and other parameter values asgiven in this paper with diffusion coefficients dn = 0.3, dp = 0.3 and dz = 0.3.


For the set of parameters, N0 = 2, D = 0.7, D1 = .08, D2 = .01, α1 = 3.2, α2 = 2.4, γ1 = 2.1, γ2 = 1, µ1 = 0.6, µ2 = 0.2, µ3 =

.06, µ4 = .06, K1 = 0.6, K2 = 1, θ = 0 [reference [7]], the coexistence equilibrium point E∗ = (1.2503,0.2658,0.5677) is locallyasymptotically stable in the form of a stable focus with eigenvalues −0.6266,−0.0123±0.4558i (cf. Fig. 1(a)).

Keeping the other parameters fixed and increasing the value of γ2 = 2, the system shows oscillation (cf. Fig. 1(b)) around thepositive interior equilibrium E∗ with eigenvalues −0.6721,0.0210±0.4716i , satisfying our analytical result where it is observed thatwhen γ2 crosses certain critical value γ2

∗, Hopf-bifurcation and periodic oscillation occur (cf. Fig. 1(c)). Thus increase in maximalconversion rate of zooplankton due to the predation on phytoplankton may cause planktonic bloom depicted through oscillation.

Now keeping all parameters fixed and with diffusion coefficients dn = .3, dp = .3 and dz = .3 we have plotted the Fig. 2(a-c)which depict that all the species show stable biomass distribution and the interior positive equilibrium point E∗ is spatially stable. Theoscillatory behavior of the diffusive system is observed for high value of maximal zooplankton conversion rate γ2 = 2. (cf. Fig. 3(a-c).The above numerical simulation is done with θ = 0. With θ = 1, the diffusive system shows spatially stable at E∗ (cf. Fig. 4(a-c) )while the system was exhibiting oscillation when θ = 0 for the same set of parameter values. It is observed that in diffusive system,θ = 2.1 and γ2 = 2, the coexistence steady state E∗ approaches the zooplankton free steady state at E1 (cf. Fig. 5(a-c)). Further it isobserved that the diffusive system shows oscillatory behavior at E∗ for N0 = 3 for the same set of parametric values with diffusivecoefficients dn = .3, dp = .3 and dz = .3 (cf. Fig. 6(a-c)).

5 Discussion

In this paper we have analyzed a spatio-temporal plankton-nutrient interaction model based on the Holling type II functional response.Firstly, we have analyzed the non-spatial system and threshold conditions for the existence of various steady states. Next, we havestudied stability of various steady states of the non spatial system. Also, it is observed that the maximal zooplankton conversionrate γ2 crosses a certain critical value, the system enters into Hopf bifurcation that induces oscillation around the positive equilibrium.Finally, we investigate the spatio-temporal dynamics with constant rate of diffusion for all the species. We have also provided numericalsimulations to substantiate our analytic results. Further numerical analysis demonstrates the following conclusions: (i) if the maximalzooplankton conversion rate is increased, the system becomes oscillatory around the equilibrium of coexistence. In this case, rate ofzooplankton decay due to toxin producing phytoplankton is increased, the system stabilizes at the equilibrium of coexistence. (ii)Further increasing the value of rate of zooplankton decay due to toxin producing phytoplankton (θ) is increased, zooplankton willnot sustain in the system. (iii) If the constant nutrient input is increased, the system becomes oscillatory around the equilibrium ofcoexistence.

Throughout the article for deterministic model (analytically and numerically) an attempt has been made to search for a suitablemechanism to control and maintain a stable coexistence between all the species. It is observed that rate of zooplankton decay due totoxin producing phytoplankton plays a major role in controlling the dynamic instability of the system.

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