dynamical analysis of toxin generating phytoplankton ......reproduce when infected by the german...

Dynamical analysis of toxin generating Phytoplankton-Zooplankton

system with time delay and infectious Phytoplankton population

Kulbhushan Agnihotria, Harpreet Kaurb,c,∗

aDepartment of Applied Science and Humanities, Shaheed Bhagat Singh State Technical Campus, Ferozpur,Punjab, India

bResearch scholar, I.K. Gujral Punjab Technical University, Jalandhar, Punjab, IndiacDepartment of Applied Sciences, Lala Lajpat Rai Institute of Engineering and Technology, Moga, Punjab, India

Abstract

The dynamical conduct of a delayed system consisting of toxin generating phytoplankton and

zooplankton species with infectious phytoplankton species is investigated. The destabilization

of the model system due to the time lag in the toxin generation mechanism is established.

Oscillations are seen in populations due to of Hopf-bifurcation. Stability of the bifurcating

periodic orbits is investigated with the help of Central manifold arguments and Normal form

theory. Numerical simulations are used to verify the theoretical findings.

Keywords: Zooplankton; Toxin generating phytoplankton; Infection; Time delay; Normal form

theory; Center manifold theorem.

1. Introduction

Chief part of aquatic life is based on Phytoplankton, also called micro algae. Phytoplankton

are the primary producers, capable of photo-synthesis and in this manner balance out nature

by absorbing nearly half of the universe CO2 and release huge oxygen. The noteworthy feature

linked with several phytoplankton species is the quick increase in their abundance accompanied

by a sharp decline after a fixed interval of time. This kind of rapidly increased phytoplankton

concentration is termed as bloom and is classified into two parts: spring bloom and red bloom.

Spring bloom is connected to the weather fluctuations and occurs as a result of seasonal variations

in temperature and nutrient levels of water. Red bloom is a localised shoot-up related to the

Journal of University of Shanghai for Science and Technology ISSN: 1007-6735

Volume 22, Issue 11, November - 2020 Page-69

alterations in water temperature and high salt content of the water column and increased rate of

growth [1]. During the algae bloom of phytoplankton, an immense accumulation of dead organic

material begins to decompose as a result of the transient nature of the algae. The process of

decomposition absorbs oxygen dissolved in the water, resulting in hypoxiation, and thus triggering

the massive fatality of plants and animals, and negatively affecting human health, tourism,

water quality, marine environment, fishing industry, and the ecosystem. ‘Harmful Algal Blooms’

(HABs) comprise of phytoplankton species that harm other organisms, causing mass mortality

through the development of natural toxins, mechanical damage to different organisms, or through

various means. Some of the toxin-generating phytoplankton species are Gambierdiscus toxicus,

Gymnodinium catenatum, Pseudo-nitzschia spp., Alexandrium spp., Alexandrium ostenfeldii,

Amphidinium spp., Ostreopsis siamensis, Ostreopsis lenticularis, Pyrodinium bahamense and

Prorocentrium lima. For the control of such issues, a thorough analysis of the plankton system

is needed.

Viruses are particularly abundant in the marine. They assume a pivotal part in the interaction,

existence and extinction of the plankton population. Several researchers have explored the

eco-epidemiological models [2–9]. [10] researched that a stable dynamics of the system of

phytoplankton population and its predator can be de-stabilized by a tiny amount of infectious

agent. Anderson and May [11] have established that the infected population would not survive

if the rate of infection becomes less than a minimum threshold value.

Noctiluca scintillans is a marine dinoflagellate. Despite the fact it is not a toxin-producer,

it is observed that it builds up toxic scales of ammonia and hence works as a killing agent in

blooms [12]. It is demonstrated experimentally that Noctilucas scintillans becomes unable to

reproduce when infected by the German Bight disease [13]. This system has been explored from

the perspective of stability and bifurcation by S. Gakkhar, K. Negi [14]. They have studied and

analysed the dynamical conduct of a toxin generating phytoplankton population infected by a

viral disease and zooplankton system. They found conditions for the coexistence of species. They

established that the system undergoes periodic oscillations for the disease parameter in a small

region and shows quasi-periodical behaviour for higher estimates of the rate of infection. They

have considered the release of toxins as a spontaneous process. However, it involves an inevitable



time delay [15, 16], the justification for this is that the toxin generating phytoplankton species

may require some time for maturity to reduce the predation strains of zooplankton. Thus, it

becomes necessary to incorporate the time delay factor in the study of such systems. In this

study we will explore this system by incorporating time delay in the toxin liberation mechanism.

2. Model

S. Gakkhar and K. Negi [14] studied the following eco-epidemiological model incorporating

functional response of Holling type-II for the interaction terms:

dP1

dt= rP1

(1− P1 + P2

K1

)− cP1P2 − b1P1Z1

dP2

dt= cP1P2 − b2P2Z1 − δP2 (2.1)

dZ1

dt= h1P1Z1 + h2P2Z1 − dZ1 − θ

P1 + P2

γ + P1 + P2

Z1

With P (t) as the toxin generating phytoplankton (TGP) population at any time t, which is

partitioned into two categories: susceptible phytoplankton P1(t) and infected phytoplankton

P2(t). Z1(t) is the population of zooplankton species at any time t. It is inferred that susceptible

phytoplankton grow logistically and become infected under the encounter of viruses. Both of

phytoplankton and zooplankton species are assumed to follow simple Lotka–Volterra form of

interaction. It is further supposed that infected phytoplankton species do not grow. The various

parameters taken for the system (2.1) are given in table 2.1. The authors investigated the impact

of infection rate and toxins produced by toxin generating phytoplankton species on the dynamics

of the model system (2.1). In the present work, we extend the model system (2.1) by introducing

time lag in the toxin liberation mechanism and study the following system of delay differential

equations:

dP1

dt= rP1

(1− P1 + P2

K1

)− cP1P2 − b1P1Z1

dP2

dt= cP1P2 − b2P2Z1 − δP2 (2.2)

dZ1

dt= −dZ1 − θ

(P1(t− τ) + P2(t− τ)

)P1(t− τ) + P2(t− τ) + γ

Z1 + h1P1Z1 + h2P2Z1

Where τ represents the time delay that is assimilated with the assumption that the release of the

toxin is not a spontaneous process, instead it is interposed with a time delay. System-associated



Table 2.1: Various parameters taken for the system (2.1)

Parameters Biological meaning

K1 carrying capacity for the phytoplankton species

r growth rate of phytoplankton species

c infection rate

δ natural death rate of infected phytoplankton population

b1, b2 predation rates of zooplankton for susceptible and infected phytoplankton respectively

h1, h2 growth rates of zooplankton as a result of susceptible and infected phytoplankton predation respectively

d natural death rate of zooplankton population

θ toxin liberation rate of the toxin generating phytoplankton (TGP) species

initial history functions are

P1(υ) = ϑ1(υ), P2(υ) = ϑ2(υ), Z1(υ) = ϑ3(υ), −τ ≤ υ ≤ 0. Where

ϑ=

(ϑ1, ϑ2, ϑ3)T ∈ C+ | ϑ1(υ), ϑ2(υ), ϑ3(υ) ≥ 0 ∀υ ∈ [−τ, 0]

, and norm of ϑ ∈ C+ is given by

‖ϑ‖ = sup−τ≤υ≤0

∣∣ϑ1(υ)∣∣ ,∣∣ϑ2(υ)

∣∣ ,∣∣ϑ3(υ)∣∣.

C+ : [−τ, 0] −→ R3+ being a Banach space of continuous functions.

For biological feasibility, it is further assumed that P1(υ) > 0, P2(υ) ≥ 0, Z1(υ) > 0.

3. Mathematical analysis of system without delay

For the model (2.1) to have bounded solutions, we give the following lemma.

Lemma 3.1. Ω bounds the solutions of model (2.1) initiating in R3+ for 0 < Λ ≤ mind, δ and

b2h1 ≥ b1h2, where

Ω = (P1, P2, Z1) ∈ R3+ : P1 + P2 +

b1

h1

Z1 =K1

4rΛ(r + Λ)2 + ε, ε > 0.

The various steady-states for model (2.1) are:

1. The zero steady-state E0(0, 0, 0).

2. The boundary steady-state E1(K1, 0, 0).

3. The planar steady state E2(P1, P2, 0) on P1 − P2 plane, where

P1 =δ

c, P2 =

r(K1c− δ)c(r +K1c)



4. The equilibrium point E3(P′′

1 , 0, Z′′

1 ), where

P′′

1 =−(h1γ − d− θ) +

√(h1γ − d− θ)2 + 4h1dγ

2h1

,

Z′′

1 =r

b1

(1− P

′′1

K1

)

5. The non-trivial equilibrium point is E∗(P ∗1 , P

∗2 , Z

∗1), where

P ∗2 =

K1(b2r + b1δ)− (rb2 + b1cK1)P ∗1

b2(r + cK1)> 0,

Z∗1 =

cP ∗1 − δb2

> 0

whereδ

c< P ∗

1 <d

h1 − θand P ∗

1 is the root of the quadratic equation (h1b22(r+cK1)2−b2(r+

cK1)(h2 +h1)(b2r+b1cK1)+h2(b2r+b1cK1)2)P ∗1

2 +(b22(h1γ−d−θ)(r+cK1)2−2h2K1(b2r−

b1δ)(b2r + b1cK1)− b2(r + cK1)(h2γ − θ − d)(b2r + b1cK1) + b2K1(h2 + h1)(b2r + b1δ)(r +

cK1))P ∗1 +(−dγb2

2(r+cK1)2 +h2K21 (b2r+b1δ)

2 +b2K1(r+cK1)(b2r+b1δ)(h2γ−θ−d)) = 0.

The non-trivial equilibrium point exists if P ∗1 satisfies the relation:

δ

c< P ∗

1 <K1(b2r + b1δ)

(rb2 + b1cK1)

The existence and local stability conditions for the steady states of the system (2.1) are presented

in table 3.1:

Table 3.1: Existence and local stability conditions for various steady-states of the system (2.1)

Boundary Equilibria Existence condition Stability condition

E0 Always exists Stable in P2 and Z1 direction but unstable in P1 direction

E1 Always exists K1 <δ

cand d+

θK1

γ +K1

> h1K1

E2 K1 >δ

ch1P1 + h2P2 < d+

θ(P1 + P2)

γ + P1 + P2

E3 θ < (h1K1 + h1γ − d)− dγ

K1

cP′′

1 − b2Z′′

1 − δ < 0

E∗ δ

c< P ∗

1 <K1(b2r + b1δ)

(rb2 + b1cK1)

ch1K1

cK1 + r<

(h1 −

θγ

(γ + P1 + P2)2

)<b1(h2 − h1)

(b2 − b1)



4. The delay model (2.2)

4.1. Stability properties

Here, we will explore the delay model (2.2) for local stability about the endemic stationary

point E∗. Linearization of (2.2) about E∗ gives

dx1

dt= l100x1(t) + l010x2(t) + l001x3(t)

dx2

dt= m100x1(t) +m010x2(t) +m001x3(t) (4.1)

dx3

dt= n100x1(t) + n010x2(t) + n001x3(t) + n∗

100x1(t− τ) + n∗010x2(t− τ)

Where, l100 = −rP∗1

K1

, l010 = −cP ∗1 −

rP ∗1

K1

, l001 = −b1P∗1 , m100 = cP ∗

2 , m010 = 0, m001 = −b2P∗2 ,

n100 = h1Z∗1 , n010 = h2Z

∗1 , n001 = 0, n∗

100 =−γθZ∗

1

(γ + P ∗1 + P ∗

2 )2,

n∗010 =

−γθZ∗1

(γ + P ∗1 + P ∗

2 )2.

The characteristic equation for the system (4.1) at the non-zero steady state E∗ is given by

the following transcendental equation:

λ3 +B1λ2 +B2λ+B3 = [B4λ+B5]e−λτ (4.2)

where

B1 =rP ∗

1

K1

B2 = b2h2P∗2Z

∗1 + b1h1P

∗1Z

∗1 +

(c2 +

rc

K1

)P ∗

1P∗2

B3 =

[rb2(h2 − h1)

K1

+ c(b1h2 − b2h1)

]P ∗

1P∗2Z

∗1 (4.3)

B4 =θγ

(γ + P ∗1 + P ∗

2 )2(b2P

∗2Z

∗1 + b1P

∗1Z

∗1)

B5 =−cθγ

(γ + P ∗1 + P ∗

2 )2(b2 − b1)P ∗

1P∗2Z

∗1

The interior stationary point E∗ will be locally asymptotically stable if every eigenvalue corre-

sponding to the characteristic equation (4.2) for the model (2.2) has negative real part. But

(4.2) has infinite many roots, it being a transcendental equation. Thus the steady-state stability



can not be examined by using Routh-Hurwitz criterion. If λ(τ) = ρ(τ) + iσ(τ) is the latent root

of the characteristic equation (4.2), then we have

ρ3 − 3ρσ2 +B1(ρ2 − σ2) +B2ρ+B3 = [(B4ρ+B5) cosστ +B4σ sinστ ]e−ρτ (4.4)

and

−σ3 + 3ρ2σ + 2B1ρσ +B2σ = [−(B4ρ+B5) sinστ +B4σ cosστ ]e−ρτ (4.5)

Set ρ = 0 in (4.4) and (4.5), which is the necessary condition for the stability changes of E∗, we

get

B3 −B1σ2 = B5 cosστ +B4σ sinστ (4.6)

B2σ − σ3 = B4σ cosστ −B5 sinστ (4.7)

On eliminating τ , equations (4.6) and (4.7) give

σ6 + (B21 − 2B2)σ4 + (B2

2 − 2B1B3 −B24)σ2 + (B2

3 −B25) = 0 (4.8)

Taking σ2 = η, (4.8) becomes

K1(η) = η3 + σ1η2 + σ2η + σ3 = 0 (4.9)

where σ1 = B21 − 2B2, σ2 = B2

2 − 2B1B3 −B24 , σ3 = B2

3 −B25 .

There will be least one positive root of the equation (4.9) if σ3 < 0. Now we present the theorem

given below:

Theorem 4.1. Let E∗ be the locally asymptotically stable (LAS) non-zero steady-state for the

model (2.2) with τ = 0 and let η0 = σ20 be the positive root of (4.9). Then

• there exists τ = τ ∗ such that for τ ∈ [0, τ ∗], E∗ will be LAS.

• E∗ will be unstable for τ > τ ∗.

• For H(σ)F (σ)−G(σ)J(σ) > 0, Hopf-bifurcation will arise around E∗ at τ = τ ∗.



Proof. (4.2) admits a pair of imaginary roots: ±iσ0 as η0 is a positive root of the equation (4.9),

Substituting λ = iσ0 in (4.2), separating real and imaginary parts, we get

B3 −B1σ20 −B5 cosσ0τ = B4σ0 sinσ0τ (4.10)

B2σ0 − σ30 −B4σ0 cosσ0τ = −B5 sinσ0τ (4.11)

on solving (4.10) and (4.11), we get τp∗, a function of σ0 for p = 0, 1, 2, 3, ..., which is given by

τ ∗p =2πp

σ0

+1

σ0

arccos

[−B4σ

40 + (B2B4 −B1B5)σ2

0 +B3B5

(B4σ0)2 + (B5)2

](4.12)

According to Butler’s lemma, the endemic steady-state E∗ of the system (2.2) will be stable for

τ < τ ∗, τ ∗ = minp≥0

τp∗ provided it is stable for τ = 0.

For the occurrence Hopf-bifurcation, the following transversality condition must hold,[dρ(τ)

dτ

]τ=τ∗

> 0. (4.13)

Taking derivative of (4.4) and (4.5) w.r.t. τ and considering ρ = 0,

F (σ)dρ

dτ+G(σ)

dσ

dτ= H(σ)

−G(σ)dρ

dτ+ F (σ)

dσ

dτ= J(σ)

(4.14)

where

F (σ) = −3σ2 +B2 −B4 cosστ + τ [B5 cosστ + σB4 sinστ ]

G(σ) = −2σB1 +B5τ sinστ −B4στ cosστ −B4 sinστ

H(σ) = −B5σ sinστ +B4σ2 cosστ

J(σ) = −B4σ2 sinστ −B5σ cosστ

(4.15)

On solving (4.14), we get [dρ(τ)

dτ

]τ=τ∗

=H(σ)F (σ)−G(σ)J(σ)

F 2(σ) +G2(σ)

using (4.13), we have

H(σ)F (σ)−G(σ)J(σ) > 0. (4.16)

Hence, the system undergoes Hopf-bifurcation around E∗ for τ = τ ∗p , p ∈W.



5. Stability, direction and period of bifurcating periodic solutions

In the preceding section, the occurrence of Hopf-bifurcation around the non-zero steady-state

E∗ at τ = τ ∗ for the model (2.2) under the condition given by (4.16) is established. Now, the

stability, direction and period for the bifurcating periodic solutions will be examined for the

non–zero steady-state E∗ at τ = τ ∗p by means of the central manifold theorem and normal form

theory [17].

For this, we use the transformation

x1(t) = P1 − P ∗1 , x2(t) = P2 − P ∗

2 , x3(t) = Z1 − Z∗1 ,

The model system (2.2) reduces to

dx1

dt=l100x1(t) + l010x2(t) + l001x3(t) +

∑`≥2

lijkxi1(t)xj2(t)xk3(t)

= K1(x1, x2, x3)

dx2

dt=m100x1(t) +m010x2(t) +m001x3(t) +

∑`≥2

mijkxi1(t)xj2(t)xk3(t)

= K2(x1, x2, x3)

dx3

dt=n100x1(t) + n010x2(t) + n001x3(t) + n∗

100x1(t− τ) + n∗010x2(t− τ)

+∑`≥2

n1ijkx

i1(t)xj1(t− τ)xk2(t) +

∑`≥2

n2ijkx

i1(t)xj1(t− τ)xk3(t)

+∑`≥2

n3ijkx

i1(t)xj2(t)xk2(t− τ) +

∑`≥2

n4ijkx

i2(t)xj2(t− τ)xk3(t)

+∑`≥2

n5ijkx

i1(t)xj2(t− τ)xk3(t) +

∑`≥2

n6ijkx

i1(t− τ)xj2(t)xk3(t)

+∑`≥2

n7ijkx

i1(t− τ)xj2(t− τ)xk3(t)

+∑`≥2

n8ijkx

i1(t)xj1(t− τ)xk2(t− τ)

+∑`≥2

n9ijkx

i1(t− τ)xj2(t)xk2(t− τ) +

∑`≥2

n10ijkx

i1(t)xj2(t)xk3(t)

= K3(x1, x2, x3)

(5.1)



where

` = i+ j + k, lijk =1

i!j!k!

∂`K1

∂P i1(t)∂P j

2 (t)∂Zk1 (t)

, mijk =1

i!j!k!

∂`K2

∂P i1(t)∂P j

2 (t)∂Zk1 (t)

,

n1ijk =

1

i!j!k!

∂`K3

∂P i1(t)∂P j

1 (t− τ)∂P k2 (t)

, n2ijk =

1

i!j!k!

∂`K3

∂P i1(t)∂P j

1 (t− τ)∂Zk1 (t)

,

n3ijk =

1

i!j!k!

∂`K3

∂P i1(t)∂P j

2 (t)∂P k2 (t− τ)

, n4ijk =

1

i!j!k!

∂`K3

∂P i2(t)∂P j

2 (t− τ)∂Zk1 (t)

,

n5ijk =

1

i!j!k!

∂`K3

∂P i1(t)∂P j

2 (t− τ)∂Zk1 (t)

, n6ijk =

1

i!j!k!

∂`K3

∂P i1(t− τ)∂P j

2 (t)∂Zk1 (t)

,

n7ijk =

1

i!j!k!

∂`K3


2 (t− τ)∂Zk1 (t)

n8ijk =

1

i!j!k!

∂`K3

∂P i1(t)∂P j

1 (t− τ)∂P k2 (t− τ)

,

n9ijk =

1

i!j!k!

∂`K3


2 (t)∂P k2 (t− τ)

, n10ijk =

1

i!j!k!

∂`K3

∂P i1(t)∂P j

2 (t)∂Zk1 (t)

The linear term coefficients are

l100 = −rP∗1

K1

, l010 = −cP ∗1 −

rP ∗1

K1

, l001 = −b1P∗1 , m100 = cP ∗

2 , m010 = 0, m001 = −b2P∗2 ,

n100 = h1Z∗1 , n010 = h2Z

∗1 , n001 = 0, n∗

100 = − γθZ∗1

(γ + P ∗1 + P ∗

2 )2, n∗

010 = − γθZ∗1

(γ + P ∗1 + P ∗

2 )2.

The coefficients of non-linear terms are

l200 = − r

K1

, l020 = 0, l002 = 0, l110 = − r

K1

− c, l101 = −b1, m110 = c, m011 = −b2, m101 = 0,

n10110 = 0, n10

101 = h1, n10110 = h2, n2

110 = 0, n2011 = − γθ

(γ + P ∗1 + P ∗

2 )2, n2

021 = − γθ

(γ + P ∗1 + P ∗

2 )2.

Let τ = τ ∗p + κ, νi(t) = νi(τt) and νt(µ) = ν(t + µ) for µ ∈ [−1, 0], neglecting the bars for

simplification, model (5.1) takes the form of functional differential equation in C([−1, 0],R3

)as

below:

ν(t) = zκ(νt) + f(κ, νt) (5.2)

with ν(t) = (x1, x2, x3)T ∈ R3 and the functions zκ : C→ R3, f : R× C→ R3 are:

zκ(ζ) = (κ+ τ ∗p )[A11ζ(0) + A22ζ(−1)] (5.3)

where

A11 =

l100 l010 l001

m100 0 m001

n100 n010 0

, A22 =

0 0 0

0 0 0

n∗100 n∗

010 0



f(κ, ζ) =

(κ+ τ∗p )

(l200ζ21 (0) + l110ζ1(0)ζ2(0) + l101ζ1(0)ζ3(0))

m110ζ1(0)ζ2(0) +m011ζ2(0)ζ3(0)

(∑`≥2

n1ijkζ

i1(0)ζj1(−1)ζk2 (0) +

∑`≥2

n2ijkζ

i1(0)ζj1(−1)ζk3 (0)+∑

`≥2

n3ijkζ

i1(0)ζj2(0)ζk2 (−1) +

∑`≥2

n4ijkζ

i2(0)ζj2(−1)ζk3 (0)+∑

`≥2

n5ijkζ

i1(0)ζj2(−1)ζk3 (0) +

∑`≥2

n6ijkζ

i1(−1)ζj2(0)ζk3 (0)+∑

`≥2

n7ijkζ

i1(−1)ζj2(−1)ζk3 (0) +

∑`≥2

n8ijkζ

i1(0)ζj1(−1)ζk2 (−1)

+∑`≥2

n9ijkζ

i1(−1)ζj2(0)ζk2 (−1) +

∑`≥2

n10ijkζ

i1(0)ζj2(0)ζk3 (0))

(5.4)

According to Riesz representation theorem, ∃ a bounded variation function χ(µ, κ) for µ ∈ [−1, 0]

with

zκ(ζ) =

0∫−1

dχ(µ, κ)ζ(µ) (5.5)

where

χ(µ, κ) = (κ+ τ ∗p )[A11δ(µ)− A22δ(µ+ 1)] (5.6)

δ being Dirac delta function. It is clear from theorem 4.1 that the model (5.2) experiences

Hopf-bifurcation about the interior steady state and ±iσ0τ∗p are the latent roots of the associated

characteristic equation for κ = 0.

For ζ ∈ C([−1, 0],R3

), define the function

B(κ)ζ =

dζ(µ)

dµµ ∈ [−1, 0)

0∫−1

dχ(µ, κ)ζ(µ) µ = 0

S(κ)ζ =

0 µ ∈ [−1, 0)

f(κ, ζ) µ = 0(5.7)



Thus, we have

ν(t) = B(κ)ut + S(κ)ut (5.8)

For ϕ ∈ C1([0, 1], (R3)∗

), define

B∗ϕ(s) =

−dϕ(s)

dss ∈ (0, 1]

0∫−1

ϕ(−t)dχ(t, 0) s = 0

and a bilinear inner product

< ϕ(s), ζ(µ) >= ϕ(0)ζ(0)−0∫

−1

µ∫ρ=0

ϕ(ρ− µ)dχ(µ)ζ(ρ)dρ (5.9)

where χ(µ) = χ(µ, 0). Clearly, the operators B(0) and B∗ are adjoint operators. Hence ±iσ0τ∗p

are eigenvalues of B∗, as these are the eigenvalues of B(0).

Let, for the eigenvalue iσ0τ∗p of B(0), q(µ) = (1, u1, v1)T eiσ0µτ

∗p be the corresponding eigenvector

of B(0). Thus, we have

B(0)q(µ) = iσ0τ∗p q(µ)

From the definitions of B(0), zκ(ζ) and χ(µ, κ), above expression reduces to

[(A11 + A22e−iσ0τ∗p )− iσ0I]q(0) = 0 (5.10)

Here I is a 3× 3 unit matrix. Substituting various values, (5.10) givesl100 − iσ0 l010 l001

m100 −iσ0 m001

C1 C2 −iσ0

1

u1

v1

= 0

where C1 = n100 + n∗100e

−iσ0τ∗p , C2 = n010 + n∗010e

−iσ0τ∗p .

Which further implies that

q(0) =

1

u1

v1

=

1

l001m100 + (iσ0 − l100)m001

iσ0l001 +m001l010iσ0(iσ0 − l100)− l010m100

iσ0l001 +m001l010



From the definition of B∗, we have

B∗ϕ(s) =

0∫−1

dχ(t, 0)ϕ(−t)

= τ ∗p [AT11ϕ(0) + AT22ϕ(−1)]

As for the eigenvalue −iσ0τ∗p of B∗, q∗(s) is the corresponding eigenvector, therefore we have

B∗q∗(0) = −iσ0τ∗p q

∗(s)

or

τ ∗p [(AT11 + AT22eiσ0τ∗p ) + iσ0I](q∗(0))T = 0 (5.11)

Here I is a 3× 3 unit matrix. Substituting the values, (5.11) gives tol100 + iσ0 m100 C3

l010 iσ0 C4

l001 m001 iσ0

1

u∗1

v∗1

= 0

where C3 = n100 + n∗100e

iσ0τ∗p , C4 = n010 + n∗010e

iσ0τ∗p .

Which further implies that

q∗(0) =

1

u∗1

v∗1

=

1

l010C3 − (l100 + iσ0)C4

m100C4 − iσ0C3iσ0(l100 + iσ0)− l010m100

m100C4 − iσ0C3

From (5.9), we have

< q∗(s), q(µ) >=U(1, u∗1, v∗1)(1, u1, v1)T−

0∫−1

µ∫ρ=0

U(1, u∗1, v∗1)e−i(ρ−µ)σ0τ∗p dχ(µ)(1, u1, v1)T eiρσ0τ

∗p dρ

or

< q∗(s), q(µ) >= U [1 + u1u∗1 + v1v∗1 + v∗1(n∗100 + u1n

∗010)τ ∗p e

−iσ0τ∗p ]

Thus

U =1

eiσ0τ∗p1 + u1

∗u1 + v1∗v1 + v1

∗(n∗100 + u1n∗

010)τp∗



Now, we shall evaluate the coordinates to explain the centre manifold C0 at κ = 0. At κ = 0,

let the solution of (5.8) be νt and define

y(t) = < q∗, νt >

W (t, µ) = νt(µ)− 2<y(µ)q(µ) (5.12)

On the centre manifold C0,

W (t, µ) = W (y(t), y(t), µ)

= W20(µ)y2

2+W11(µ)yy +W02(µ)

y2

2+W30(µ)

y3

6+ . . .

(5.13)

where y, y represent local coordinates for the centre manifold C0 in the direction of q and q∗

respectively. Here, we have considered only real solutions. For the solution νt ∈ C0 of (5.8), as

κ = 0, we have

y(t) = iσ0τ∗p y + q∗(0)f(0,W (y, y, 0) + 2<(yq(µ)))

= iσ0τ∗p y + q∗(0)f0(y, y) (5.14)

or

y(t) = iσ0τ∗p y + g(y, y) (5.15)

where

g(y, y) =q∗(0)f0(y, y)

=g20(µ)y2

2+ g11(µ)yy + g02(µ)

y2

2+ g21(µ)

y2y

2+ . . .

(5.16)

From (5.12) and (5.13), we get

νt(µ) =W20(µ)y2

2+W11(µ)yy +W02(µ)

y2

2+W30(µ)

y3

6

+ (1, u1, v1)T eiσ0τ∗pµy + (1, u1, v1)T e−iσ0τ

∗pµy + . . .

(5.17)



From (5.4) and (5.16), we have

g(y, y) = D1

(l200ν21t(0) + l110ν1t(0)ν2t(0) + l101ν1t(0)ν3t(0))

(m110ν1t(0)ν2t(0) +m011ν2t(0)ν3t(0))

(n10101ν1t(0)ν3t(0) + n10

011ν2t(0)ν3t(0)

+n2011ν1t(−1)ν3t(0) + n5

011ν2t(−1)ν3t(0)

+n2021ν

21t(−1)ν3t(0) + n4

021ν22t(−1)ν3t(0)

+n7111ν1t(−1)ν2t(−1)ν3t(0) + n1

020ν21t(−1)

+n3002ν

22t(−1) + n7

110ν1t(−1)ν2t(−1)

+n1030ν

31t(−1) + n7

210ν21t(−1)ν2t(−1)

+n7120ν1t(−1)ν2

2t(−1) + n7020ν

32t(−1) + h.o.t.)

(5.18)

where U1 = U(1, u∗1, v∗1)τ ∗p . Further simplification gives

g(y, y) =Uτ ∗p [(p1 + u∗1p2 + v∗1p3)y2 + (p4 + u∗1p5 + v∗1p6)yy

+ (p7 + u∗1p8 + v∗1p9)y2 + (p10 + u∗1p11 + v∗1p12)y2y + . . .](5.19)

Where p1 = l200 + u1l110 + v1l101, p2 = u1m110 + u1v1m011

p3 = v1n9101 + u1v1n

9011 + v1e

−iσ0τ∗p (n2011 + u1n

5011) + e−2iσ0τ∗p (n1

020 + u21n

3002 + u1n

7110)

p4 = (2l200 + (u1 + u1)l110 + (v1 + v1)l101)

p5 = ((u1 + u1)m110 + (u1v1 + v1u1)m011)u∗1

p6 = ((v1 + v1)n9101 +(u1v1 +v1u1)n9

011 +(v1e−iσ0τ∗p +v1e

iσ0τ∗p )n2011 +(u1v1e

−iσ0τ∗p +v1u1eiσ0τ∗p )n5

011 +

2n1020 + 2u1u1n

3002 + (u1 + u1)n7

110)v∗1

p7 = l200 + l110u1 + l101v1, p8 = u1m110 + u1v1m011

p9 = v1n9101 + u1v1n

9011 + (v1n

2011 + u1v1n

5011)eiσ0τ

∗p + (n1

020 + u21n

3002 + u1n

7110)e2iσ0τ∗p

p10 =(W

(1)20 (0) + 2W

(1)11 (0)

)l200 +

(W

(2)11 (0) +

1

2W

(2)20 (0) +

u1

2W

(1)20 (0) + u1W

(1)11 (0)

)l110

+

(1

2W

(3)20 (0) +

1

2v1W

(1)20 (0) + v1W

(1)11 (0) +W

(3)11 (0)

)l101

p11 =

(1

2W

(2)20 (0) +

u1

2W

(1)20 (0) + u1W

(1)11 (0) +W

(2)11 (0)

)m110

+

(u1

2W

(3)20 (0) +

v1

2W

(2)20 (0) + u1W

(3)11 (0) + v1W

(2)11 (0)

)m011



p12 =

(1

2W

(3)20 (0) +

v1

2W

(1)20 (0) +W

(3)11 (0) + v1W

(1)11 (0)

)n9

101

+

(u1

2W

(3)20 (0) +

v1

2W

(2)20 (0) + u1W

(3)11 (0) + v1W

(2)11 (0)

)n9

011

+

(1

2eiσ0τ

∗pW

(3)20 (0) +

v1

2W

(1)20 (−1) + e−iσ0τ

∗pW

(3)11 (0) + v1W

(1)11 (−1)

)n2

011

+

(u1

2eiσ0τ

∗pW

(3)20 (0) +

v1

2W

(2)20 (−1) + u1e

−iσ0τ∗pW(3)11 (0) + v1W

(2)11 (−1)

)n5

011

+(v1e

−2iσ0τ∗p + 2v1

)n2

021 +(u2

1v1e−2iσ0τ∗p + 2u1u1v1

)n4

021

+(u1v1e

−2iσ0τ∗p + u1v1 + u1v1

)n7

111

+(eiσ0τ

∗pW

(1)20 (−1) + 2e−iσ0τ

∗pW

(1)11 (−1)

)n1

020

+(u1e

iσ0τ∗pW(2)20 (−1) + 2u1e

−iσ0τ∗pW(2)11 (−1)

)n3

002

+

(1

2eiσ0τ

∗pW

(2)20 (−1) +

u1

2eiσ0τ

∗pW

(1)20 (−1) + u1e

−iσ0τ∗pW(1)11 (−1)

)n7

110

+ 3e−iσ0τ∗pn1

030 + (u1 + 2u1) e−iσ0τ∗pn7

210

+(u2

1 + 2u1u1

)e−iσ0τ

∗pn7

120 + 3u21u1e

−iσ0τ∗pn7020 +O(|y, y|3)

From (5.16) and (5.19), we get

g20 = 2Uτ ∗p [p1 + u∗1p2 + v∗1p3]

g11 = Uτ ∗p [p4 + u∗1p5 + v∗1p6]

g02 = 2Uτ ∗p [p7 + u∗1p8 + v∗1p9]

g21 = 2Uτ ∗p [p10 + u∗1p11 + v∗1p12]

(5.20)

To determine g21, we need to compute W20(µ) and W11(µ). (5.8) and (5.12) give

W = νt − 2<y(t)q(µ)

Hence

W =

B(0)W − 2<q∗(0)f0q(µ) µ ∈ [−1, 0)

B(0)W − 2<q∗(0)f0q(0)+ f0 µ = 0

Now

W = B(0)W +H(y, y, µ) (5.21)

where

H(y, y, µ) = H20(µ)y2

2+H11(µ)yy +H02

y2

2+ . . . (5.22)

On the centre manifold C0, near origin

W = Wyy +Wy¯y



From (5.21), we get

Wyy +Wy¯y −B(0)W = H(y, y, µ)

which implies that

−H20(µ) = (B − 2iσ0τ∗p I)W20(µ) (5.23)

BW11(µ) = −H11(µ) (5.24)

Taking µ ∈ [−1, 0) in (5.21) gives

H(y, y, µ) = −g(y, y)q(µ)− g(y, y)q(µ) (5.25)

(5.16), (5.22) and (5.25) imply that

H20(µ) = −g20q(µ)− g02q(µ) (5.26)

H11(µ) = −g11q(µ)− g11q(µ) (5.27)

Definitions of B(0), (5.23) and (5.26) give

W20(µ)− 2iσ0τ∗pW20(µ) = g20q(µ) + g02q(µ) (5.28)

substituting q(µ) = (1, u1, v1)T eiσ0τ∗pµ, we find that

W20(µ) =ig20

σ0τ ∗pq(0)eiσ0τ

∗pµ +

ig02

3σ0τ ∗pq(0)e−iσ0τ

∗pµ + K1e

2iσ0τ∗pµ (5.29)

And, from (5.24), (5.27), we have

W11(µ) =−ig11

σ0τ ∗pq(0)eiσ0τ

∗pµ +

ig11

σ0τ ∗pq(0)e−iσ0τ

∗pµ +K2 (5.30)

where the vectors K1 = (K(1)1 , K

(2)1 , K

(3)1 ) ∈ R and K2 = (K

(1)2 , K

(2)2 , K

(3)2 ) ∈ R can be obtained

by substituting µ = 0 in H(y, y, µ).

g(y, y) = q∗(0)f0, gives

U(1, u∗1, v∗1)f0 = 2Uτ∗p (1, u∗1, v

∗1)

p1

p2

p3

y2

2+ Uτ∗p (1, u∗1, v

∗1)

p4

p5

p6

yy + . . .



or

f0 = 2τ ∗p

p1

p2

p3

y2

2+ τ ∗p

p4

p5

p6

yy + . . . (5.31)

Taking µ = 0 in (5.21) gives

H(y, y, 0) = −q∗(0)f0q(0)− q∗(0)f0q(0) + f0

or

H20(0)y2

2+H11(0)yy +H02(0)

y2

2+ . . .

= −q(0)

(g20

y2

2+ g11yy + g02

y2

2+ . . .

)

− q(0)

(g20

y2

2+ g11yy + g02

y2

2+ . . .

)+ f0 (5.32)

Using (5.31), we obtain

H20(0) = −g20q(0)− g20q(0) + 2τ ∗p

p1

p2

p3

(5.33)

H11(0) = −g11q(0)− g11q(0) + τ ∗p

p4

p5

p6

(5.34)

(5.23) gives

B(0)W20(0) = −H20(0) + 2iσ0τ∗pW20(0)

Thus0∫

−1

dχ(µ)W20(µ) = 2iσ0τ∗pW20(0)−H20(0) (5.35)

and similarly0∫

−1

dχ(µ)W11(µ) = −H11(0) (5.36)



and iσ0τ∗p I −

0∫−1

eiσ0τ∗pµdχ(µ)

q(0) = 0

−iσ0τ∗p I −

0∫−1

e−iσ0τ∗pµdχ(µ)

q(0) = 0

(5.37)

Substituting (5.29) and (5.33) in (5.35), we get2iσ0 − l100 −l010 −l001

−m100 2iσ0 −m001

C11 C22 2iσ0

K1 = 2

p1

p2

p3

(5.38)

where C11 = n∗100e

−2iσ0τ∗p − n100, C22 = n∗010e

−2iσ0τ∗p − n010. Thus

K1 = 2

2iσ0 − l100 −l010 −l001

−m100 2iσ0 −m001

C11 C22 2iσ0

−1

p1

p2

p3

(5.39)

Similarly substituting (5.30) and (5.34)in (5.36), we get

K2 = −τ ∗p

l100 l010 l001

m100 0 m001

n100 + n∗100 n010 + n∗

010 0

−1

p4

p5

p6

(5.40)

Hence, we can evaluate g21 from the equations (5.20), (5.29), (5.30), (5.39) and (5.40). Further,

we can evaluate the following quantities

c1(0) =i

2σ0τ ∗p

[g20g11 − 2|g11|2 −

|g02|2

3

]+g21

2

κ2 = − <c1(0)

<

dλ(τ∗p )

dτ

β2 = 2<c1(0)

T2 = −=c1(0)+ κ2=

dλ(τ∗p )

dτ

σ0τ ∗p

, p = 1, 2, 3, . . .

(5.41)



The Hopf bifurcation direction at the critical point τ ∗p is given by κ2. It will be subcritical

(supercritical) if κ2 < 0(> 0). Stability of the bifurcating periodic solutions in the centre

manifold is decided by β2. These will be stable (unstable) for β2 < 0(> 0). For T2 > 0(< 0), the

periodicity of bifurcating periodic solution increases (decreases) [17].

6. Numerical Simulations

In this section, we will use numerical simulations to validate the results found analytically.

We take the following parametric values for the model (2.2): r = 4.852h−1; K1 = 935l−1;

b1 = 0.11lh−1; b2 = 0.19lh−1; c = 0.1721lh−1; δ = 0.6lh−1; d = 0.39lh−1; h1 = 0.07lh−1;

h2 = 0.17lh−1; θ = 3.272h−1; γ = 1.48l−1;

Thus, we consider the system

dP1

dt= 4.852P1

(1− P1 + P2

935

)− 0.1721P1P2 − 0.11P1Z1

dP2

dt= 0.1721P1P2 − 0.19P2Z1 − 0.6P2 (6.1)

dZ1

dt= −0.39Z1 − θ

(P1(t− τ) + P2(t− τ)

)1.48 + P1(t− τ) + P2(t− τ)

Z1 + 0.07P1Z1 + 0.17P2Z1

For investigating the delay impact on the model (2.2) around non-zero steady-state E∗ we linearize

6.1 about E∗ and take x1(t) = P1(t)− 46.5577, x2(t) = P2(t)− 1.7988, x3(t) = Z1(t)− 39.0136.

the linearized system is given by

dx1

dt= −0.2416x1(t)− 8.2542x2(t)− 5.1213x3(t)

dx2

dt= 0.3096x1(t) + 0x2(t)− 0.3418x3(t)

dx3

dt= 2.7310x1(t) + 6.6323x2(t) + 0x3(t)− 0.0761x1(t− τ)− 0.0761x2(t− τ)

As σ3 = 11.2721 > 0, hence (4.8) has a purely imaginary root iσ0 and value of σ0 comes out to

be 4.3713 and the corresponding value of the delay parameter is τ ∗p = 0.5329. For this value, we

have [dρ(τ)

dτ

]τ=τ∗

=H(σ)F (σ)−G(σ)J(σ)

F 2(σ) +G2(σ)

= 0.1303 > 0.



Consequently, for the above taken parametric values, the transversality condition is satisfied

and Hopf-bifurcation occurs at τ ∗p = 0.5329. Furthermore, the endemic steady-state E∗ is

stable for the range 0 < τ < 0.5329 and becomes unstable for τ ≥ 0.5329. Figure 1 shows the

phase diagram about the endemic steady-state E∗ for τ = 0 and reveals the stable dynamics

of the model (2.2). Figure 2 exhibits the phase diagram about E∗ at τ ∗p = 0.4 indicating the

stable dynamics of the model (2.2) and figure 3 reveals the associated time series solution. The

occurrence of Hopf-bifurcation about the interior steady-state E∗ at τ ∗p = 0.998 is shown in

figure 4. Figure 5 shows the corresponding time series solution for the three populations.

Figure 1: stability of interior equilibrium around E∗ at τ ∗p = 0

Figure 2: stability of interior equilibrium around E∗ at τ ∗p = 0.4



Figure 3: Time Series depicting local stability of interior equilibrium around E∗ at τ ∗p = 0.4

Figure 4: Limit cycle solution around E∗ at τ ∗p = 0.998

Figure 5: Time series revealing limit cycle around E∗ at τ ∗p = 0.998

The eigenvectors as defined in section 5 are given by



q =

1

0.0564− i0.0816

−0.1382− i0.7220

and q∗ =

1

2.0499− i2.0804

−0.1146− i1.3414

Moreover, the values of the variables which decide the behaviour of bifurcating periodic

solutions at τ = τ ∗p , are numerically found to be

c1(0) = 1.8914× 10−5 − i5.2724× 10−4, κ2 = −1.4513× 10−4

β2 = 3.7828× 10−05, T2 = 2.2634× 10−04

Following the pattern of [17], it is clear that that the nature of Hopf-bifurcation is supercritical

and bifurcating periodic solutions exist for τ > τ ∗p . In the centre manifold, there are stable

bifurcating periodic solutions as β2 < 0. As T2 > 0 the period of bifurcating periodic solutions is

increasing in nature.

7. Discussion and conclusion

In this manuscript, our objective is to study the effect of time taken by toxin generating

phytoplankton species for their maturity (time delay) on the dynamics of a toxin generating

phytoplankton species infected by a viral disease and zooplankton species. The base model of

[14] is studied by introducing this time delay factor. It is ascertained that the system remains

locally asymptotically stable when no time lag is taken for a given parametric values i.e. the

system had no limit (Figure 1). But the inclusion of time lag τ changes the dynamics of the

system and gives rise to Hopf-bifurcation. 0 < τ < 0.5329 is the threshold limit for τ beyond

which the dynamics of the system admits stability exchange exhibits periodic oscillations about

E∗ for τp∗ = 0.998 (Figure 4).

In addition, with the help of the center manifold theorem and normal form theory by Hassard

et al. [17], the stability and direction of bifurcating periodic solutions is driven by computing the

values of c1(0), κ2, β2, T2. It has been observed that there is supercritical hopf bifurcation with

orbitally asymptotically stable bifurcating periodic solutions. Along these lines, it is established

that the time delay required for the adulthood of toxin generating phytoplankton assumes an

key part in the dynamics of the system.



Acknowledgments

The authors are thankful to the anonymous referee for his/her suggestions to improve the

readability of the paper. We are also thankful to the editor for his/her helpful comments.

Further authors acknowledge the I.K. Gujral Punjab Technical University, Jalandhar, Punjab

for providing research support.

References

[1] J. Truscott, J. Brindley, Ocean plankton populations as excitable media, Bulletin of

Mathematical Biology 56 (5) (1994) 981–998.

[2] J. Chattopadhyay, N. Bairagi, Pelicans at risk in salton sea-an eco-epidemiological model,

Ecological Modelling 136 (2) (2001) 103–112.

[3] J. Chattopadhyay, R. Sarkar, G. Ghosal, Removal of infected prey prevent limit cycle

oscillations in an infected prey-predator system-a mathematical study, Ecological Modelling

156 (2) (2002) 113–121.

[4] H. Hethcote, W. Wang, L. Han, Z. Ma, A predator-prey model with infected prey, Theoretical

Population Biology 66 (3) (2004) 259–268.

[5] P. Auger, R. Mchich, T. Chowdhury, G. Sallet, M. Tchuente, J. Chattopadhyay, Effects

of a disease affecting a predator on the dynamics of a predator-prey system, Journal of

Theoretical Biology 258 (3) (2009) 344–351.

[6] C. Tannoia, E. Torre, E. Venturino, An incubating diseased-predator ecoepidemic model,

Journal of biological physics 38 (4) (2012) 705–720.

[7] K. pada Das, K. Kundu, J. Chattopadhyay, A predator-prey mathematical model with both

the populations affected by diseases, Ecological Complexity 8 (1) (2011) 68–80.

[8] K. pada Das, J. Chattopadhyay, A mathematical study of a predator-prey model with

disease circulating in the both populations, International Journal of Biomathematics 8 (02)

(2015) 1550015.



[9] K. Agnihotri, H. Kaur, The dynamics of viral infection in toxin producing phytoplankton

and zooplankton system with time delay, Chaos, Solitons & Fractals 118 (2019) 122–133.

[10] E. Beltrami, T. Carroll, Modeling the role of viral disease in recurrent phytoplankton

blooms, Journal of Mathematical Biology 32 (8) (1994) 857–863.

[11] R. M. Anderson, R. M. May, B. Anderson, Infectious diseases of humans: dynamics and

control, Vol. 28, Wiley Online Library, 1992.

[12] T. Okaichi, S. Nishio, Identification of ammonia as the toxic principle of red tide of noctiluca

miliaris, Bulletin of Plankton Society of Japan (Japan).

[13] G. Uhlig, G. Sahling, Long-term studies on noctiluca scintillans in the german bight

population dynamics and red tide phenomena 1968–1988, Netherlands Journal of Sea

Research 25 (1-2) (1990) 101–112.

[14] S. Gakkhar, K. Negi, A mathematical model for viral infection in toxin producing phyto-

plankton and zooplankton system, Applied mathematics and computation 179 (1) (2006)

301–313.

[15] A. K. Sharma, A. Sharma, K. Agnihotri, Bifurcation analysis of a plankton model with

discrete delay, World Academy of Science, Engineering and Technology, International

Journal of Mathematical, Computational, Physical, Electrical and Computer Engineering

8 (1) (2014) 133–142.

[16] J. Chattopadhyay, R. Sarkar, A. El Abdllaoui, A delay differential equation model on

harmful algal blooms in the presence of toxic substances, Mathematical Medicine and

Biology 19 (2) (2002) 137–161.

[17] B. D. Hassard, N. D. Kazarinoff, Y.-H. Wan, Theory and applications of Hopf bifurcation,

Vol. 41, CUP Archive, 1981.



dynamical analysis of toxin generating phytoplankton ......reproduce when infected by the german...

Documents