dynamical analysis of a delay model of phytoplankton–zooplankton interaction

Applied Mathematical Modelling 36 (2012) 638–647

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Applied Mathematical Modelling

journal homepage: www.elsevier .com/locate /apm

Dynamical analysis of a delay model of phytoplankton–zooplanktoninteraction q

Mehbuba Rehim a,⇑, Mudassar Imran b

a College of Mathematics and System Sciences, Xinjiang University, Urumqi 830046, People’s Republic of Chinab Department of Mathematics, Lahore University of Management Sciences, Pakistan

a r t i c l e i n f o a b s t r a c t

Article history:Received 12 May 2010Received in revised form 5 June 2011Accepted 1 July 2011Available online 23 July 2011

Keywords:PhytoplanktonZooplanktonEquilibriumDelayToxic

0307-904X/$ - see front matter � 2011 Elsevier Incdoi:10.1016/j.apm.2011.07.018

q Supported by The National Natural Science Foun(BS080106).⇑ Corresponding author.

E-mail addresses: [email protected] (M. Rehim

The interaction of toxic-phytoplankton–zooplankton systems and their dynamical behav-ior will be considered in this paper based upon nonlinear ordinary differential equationmodel system. We induced a discrete time delay to the both of the consume response func-tion and distribution of toxic substance term to describe the delay in the conversion ofnutrient consumed to species and the fact that the time required for the phytoplanktonspecies to mature before they can produce toxic substances. We generalized the modelin [1] and explicit results are derived for globally asymptotically stability of the boundaryequilibrium. Using numerical simulation method, we determine there is a parameter rangefor the delay parameter s where more complicated dynamics occurs, and this appears to bea new result. Significant outcomes of our numerical findings and their interpretations fromecological point of view are provided in this paper.

� 2011 Elsevier Inc. All rights reserved.

1. Introduction

Plankton is the basis of all aquatic food chains, and phytoplankton in particular lies on the first trophic level of the foodchain. The animals in the plankton community are known as zooplankton. The phytoplankton are consumed by zooplankton,their animal counterparts, which considered to be most favorable food sources for fish and other aquatic animals. The mostcommon features of the phytoplankton population is rapid increase of biomass due to rapid cell proliferation and almostequally rapid decrease in populations, separated by some fixed time period. This type of rapid change in phytoplankton pop-ulation density is known as ‘‘bloom’’ [1]. Due to the accumulation of high biomass or to the presence of toxicity, some ofthese blooms, more adequately called ‘‘harmful algal blooms’’ (HABs [2]), are noxious to marine ecosystems or to humanhealth and can produce great socioeconomic damage. There has been a global increase in harmful plankton blooms in lasttwo decades [3–5] and considerable scientific attention towards harmful algal blooms has been paid in recent years [6–8].

Reduction of grazing pressure of zooplankton due to release of toxic substances by phytoplankton is one of the mostinteresting topics of research in the last few decades [3,7,9–14]. A broad classification of harmful algal blooms species dis-tinguishes two groups: the toxin producers, which can contaminate seafood or kill fish, and the high-biomass producers,which can cause anoxia and indiscriminate mortalities of marine life after reaching dense concentrations. Some toxin harm-ful algal phytoplankton species have characteristics for both group. Although the attention given to this issue, the nature ofharmful plankton and its possible control mechanism are not yet well established and require special attention [12]. Hence

. All rights reserved.

dation of P.R. China [10961022, 10901130] and The Natural Science Foundation of Xinjiang University

), [email protected] (M. Imran).

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M. Rehim, M. Imran / Applied Mathematical Modelling 36 (2012) 638–647 639

the study of marine ecology is an important consideration for the survival of our earth and the experimental as well as math-ematical modeling are necessary in this field.

Buskey and Stockwell [10], Nielsen et al. [15], Ives [16] and Nejstgaard and Solberg [17] observed in their field as well aslaboratory studies that the toxic substance plays one of the important role on the groth of the zooplankton population andhave a great impact on phytoplankton-zooplankton interactions. In order to establish an alternative approach (the effect oftoxic chemicals on zooplankton on the contrary to viral infection on phytoplankton) to explain the suitable mechanism forthe occurrence of planktonic blooms and its possible control, they observed both in experimentally and modeling point ofview. They put their emphasis to observe the effects of toxin producing phytoplankton on planktonic blooms and succes-sions. Their tested toxin producing phytoplankton species is Noctiluca scintillans belonging to the group Dinoflagellates ofthe Division Dinophyta. It is a very common heterotrophic dinoflagellate and is known to feed on bacteria, diatoms, otherflagellates and ciliate protozoans [12]. Amang zooplankton species they choose Paracalanus belonging to the group Copep-oda which dominate the zooplankton community in all over the world oceans, and are the major herbivore which determinethe form of the phytoplankton curve. Based on their these field observations, Chattopadhyay et al. formulated the followingordinary differential equations:

P0ðtÞ ¼ rPðtÞ 1� PðtÞk

� �� af ðPÞZðtÞ;

Z0ðtÞ ¼ bf ðPÞZðtÞ � lZðtÞ � hgðPÞZðtÞ:ð1:1Þ

Here, P(t) and Z(t) denote the concentration of the toxin producing Phytoplankton population and the zooplankton popula-tion at time t, respectively. f(p) represents the predational response function and g(p) represents the distribution of toxicsubstances. The authors [12] analyzed the model system (1.1) by taking various combinations of functional response f(p)and g(p). Their analysis includes the existence and local stability of various steady states as well as existence of excitablebehavior of the system.

Biological delay systems of one type or another have been considered by a number of authors [18–21]. These system gov-erned by integro-differential equations exhibit much more rich dynamics than ordinary differential systems. In the phyto-plankton–zooplankton system considered above, the liberation of toxic substances by the phytoplankton species is not aninstantaneous process but is mediated by some time delay. This phenomenon is supported by the observation that zooplank-ton mortality due to the toxic phytoplankton bloom occurs after some time lapse. The field study conducted by Chattopad-hyay et al. [22] suggests that the abundance of Paracalanus (zooplankton) population reduces after some time lapse of thebloom of toxic phytoplankton Noctiluea Scintillans. This time delay can be interpreted biologically as the time required forthe phytoplankton species to mature before they can produce toxic substances. Therefore, in [22], the following type of mod-el system was presented


� �� aPðtÞZðtÞ;

Z0ðtÞ ¼ bPðtÞZðtÞ � lZðtÞ � qPðt � sÞcþ Pðt � sÞ ZðtÞ:

ð1:2Þ

For model (1.2), the authors [22] studied the oscillation of phytoplankton and zooplankton populations along with the sta-bility condition for oscillatory behavior. The above model is based on the model (1.2) with delay response function in zoo-plankton population concentration equation. In this model the author take functional response f(P) as a linear function anddistribution of toxic substances g(P) as a delay function.

Saha and Bandyopadhyay [1] extended the model (1.2) to the following form


� �� bPðtÞ

cþ PðtÞ ZðtÞ;

Z0ðtÞ ¼ b1PðtÞcþ PðtÞ ZðtÞ � dZðtÞ � qPðt � sÞ

cþ Pðt � sÞ ZðtÞ:ð1:3Þ

Their extension includes non-linear Holling’s type II function for f(P). Based upon the model (1.3) the authors [1] studied theperiodic oscillatory nature of phytoplankton and zooplankton populations by considering a discrete time delay as bifurcationparameter. They also obtained the conditions for the global existence of periodic solutions for the model system (1.3).

In a real ecological context, the interaction between phytoplankton and zooplankton will not be essentially instanta-neous. Instead, the response of zooplankton to contacts with phytoplankton is likely to be delayed due to a gestation period.Another fact, during the interaction between phytoplankton and zooplankton, is that the liberation of toxic substances byphytoplankton must be mediated by some time lag which is required for the maturity of toxic-phytoplankton. Based onthe above fact, we extend the system (1.3) by introducing these time delay.


� �� bPðtÞZðtÞ;

Z0ðtÞ ¼ e�dsb1Pðt � sÞZðt � sÞ � dZðtÞ � e�dsqPðt � sÞcþ Pðt � sÞ Zðt � sÞ:

ð1:4Þ

640 M. Rehim, M. Imran / Applied Mathematical Modelling 36 (2012) 638–647

In model (1.4), P(t) denotes the concentration of the toxin producing phytoplankton population at time t and Z(t) de-notes the density of the zooplankton population at time t subject to the non-negative initial value P(0) = P0 P 0,Z(0) = Z0 P 0. In this model parameters r is the intrinsic growth rate ‘‘k’’ is the environmental caring capacity of the toxinproducing phytoplankton population, b(>0) is the maximum uptake rate for zooplankton species, b1(>0) denotes the ratioof biomass conversion (satisfying the obvious restriction 0 < b1 < b) and d(>0) is the natural death rate of zooplankton.q(>0) denotes the rate of toxic substances produced by per unit biomass of phytoplankton,and c is the half saturationconstant.

The delay constant s in the first and last term of the second equation is the gestation delay of Zooplankton and the dis-crete time period required for the maturity of toxic-phytoplankton, respectively. In general case, the gestation delay of Zoo-plankton different from time lag which is required for the maturity of toxic-phytoplankton, but in this paper we study thespecial case, that is the gestation delay of zooplankton the same as the discrete time period required for the maturity oftoxic-phytoplankton.

Model (1.4) extended the phytoplankton–zooplankton interaction model to the more general case. We induced a discretetime delay to the both of the consume response function and distribution of toxic substance term. Due to the outflow (whichequal to natural death of zooplankton and poisoned zooplankton), only e�ds Z(t � s), not Z(t), of zooplankton that consumednutrient (phytoplankton) s unit of time previously survive the s units of time assumed necessary to complete the process ofconverting the nutrient to new cells.

This paper is organized as follows. In Section 2, we establish some preliminary results about model (1.4) that used inlater sections. In Section 3, we consider the special case of model (1.4) without delay and discuss the local and globallyasymptotically stability of the equilibria. In Section 4, under the aids of numerical simulation method, we analyse themodel (1.4) and determine there is a parameter range for the delay parameter s where more complicated dynamics oc-curs. Significant outcomes of our numerical findings and their interpretations from ecological point of view are provided inSection 5.

2. Preliminary results

For any integer n P 1, define Rnþ ¼ ðx1; x2; . . . ; xnÞ 2 Rn; xi P 0; 1 6 i 6 n

� �and its interior is denoted by

IntRnþ ¼ ðx1; x2; . . . ; xnÞ 2 Rn; xi > 0; 1 6 i 6 n

� �. For s > 0, let C([�s,0],R2) denotes the space of continuous functions map-

ping the interval [�s,0] into R2 with the uniform norm, i.e., if / 2 C([�s,0]), R2, k/k = suph2[�s, 0]j/(h)j, where j�j is any normin R2. Let C2

þ ¼ Cð½�s;0�;R2þÞ and IntC2

þ ¼ Cð½�s;0�;IntR2þÞ. If r > 0 and / : [�s,r) ? IntR2

þ, define /t(h) = /(t + h) for t 2 [0,r]and h 2 [�s,0]. Then if / is continuous on [�s,r), then /t 2 C2

þ. For system (1.4), we consider any initial data in IntC2þ.

There are advantages in analyzing dimensionless equations. We take variable changes for model (1.4) as following:

t� ¼ rt; P�ðt�Þ ¼ PðtÞk; Z�ðtÞ ¼ bZðtÞ

r;

s� ¼ sr; b�1 ¼kb1

r; d� ¼ d

r; q� ¼ q

r; c� ¼ c

k:

ð2:1Þ

A direct calculation, using (1.1) gives:

dP�dt� ¼

1kr

dPdt¼ 1

krrPðtÞ 1� PðtÞ

k

� �� bPðtÞZðtÞ

� �¼ PðtÞ

k1� PðtÞ

k

� �� PðtÞ

kbZðtÞ

r¼ P�ðtÞ 1� P�ðtÞð Þ � P�ðtÞZ�ðtÞ;

dZ�

dt�¼ b

r2 e�dsb1Pðt � sÞZðt � sÞ � dZðtÞ � qe�dsPðt � sÞcþ Pðt � sÞ Zðt � sÞ

� �

¼ e�drsr kb1

rPðt � sÞ

kbZðt � sÞ

r� d

rbZðtÞ

r� e�

drsr

qr

Pðt�sÞk

ckþ

Pðt�sÞk

bZðt � sÞr

¼ b�1e�d�s�P�ðt� � s�ÞZ�ðt� � s�Þ � d�Z�ðt�Þ � q�e�d�s�P�ðt� � s�Þc� þ P�ðt� � s�Þ Z�ðt� � s�Þ:

Omitting the stars, in order to simplify the notation, the nondimensional version of model (1.1) can be written:

P0ðtÞ ¼ PðtÞ 1� PðtÞð Þ � PðtÞZðtÞ;

Z0ðtÞ ¼ b1e�dsPðt � sÞZðt � sÞ � dZðtÞ � qe�dsPðt � sÞcþ Pðt � sÞ Zðt � sÞ:

ð2:2Þ

For system (2.2), we make an assumption that b1 >qc, that is, the ratio of biomass consumed by zooplankton is greater

than the 1c times the rate of toxic substance liberation by phytoplankton species. For the existence, uniqueness and positivity

of the solutions of the system (2.2), We have the following results.

Lemma 2.1. Let P(h), Z(h) P 0 on �s 6 h < 0, and P(0), Z(0) > 0. Then all solutions of (2.2) with initial data / 2 IntC2þ exist on

[0,r), for some r > 0, and are unique and positive for 0 < t < r.


Proof. By Theorem 2.1 and 2.3 in Hale and Lunel [23], solutions of (2.2) with initial data / 2 IntC2þ exist on 0 < t < r for

some r > 0, and are unique. Suppose (P(t), Z(t)) is a solution of (2.2) for t 2 [0,r). Without loss of generality, assume thatt 2 [0,r) is the maximum internal of the solution and r =1 if the solution exists for any t > 0. Integrating the first equa-tion of (2.2) gives

PðtÞ ¼ /1ð0Þ expZ t

0ð1� PðuÞÞ � ZðuÞð Þdu

� > 0; t 2 ½0;rÞ:

To prove the Z(t) > 0 for any t 2 [0,r), use the method of contradiction. Suppose there exists t⁄ 2 [0,r) such that

Zðt�Þ ¼ 0; Z0ðt�Þ 6 0; and ZðtÞ > 0 for any t 2 ½�s; t�Þ:

From the second equation of the system (2.2), we have

Z0ðt�Þ ¼ b1e�dsPðt� � sÞZðt� � sÞ � dZðt�Þ � qe�dsPðt� � sÞcþ Pðt� � sÞ Zðt� � sÞP e�ds b1 �

qc

� �Pðt� � sÞZðt� � sÞ > 0:

This contradicts to Z0(t⁄) 6 0. Hence Z(t) > 0 for all t 2 [0,r). h

Lemma 2.2. The solutions of (2.2) are bounded for all t P 0. In addition, limsupt?1P(t) 6 1.

Proof. Let P(h), Z(h) be a given solution of (2.2). From the first equation of (2.1), we have

P0ðtÞ 6 PðtÞð1� PðtÞÞ:

It is easy to obtain

lim supt!1

PðtÞ 6 1:

Therefore, there exists a T > 0 such that

PðtÞ < 1þ �0 for all t P T;

where constant �0 > 0 is sufficient small. Define

WðtÞ ¼ ZðtÞ þ b1e�dsPðt � sÞ ð2:3Þ

for all t P 0. Then it follows from (2.2) that

W 0ðtÞ ¼ b1e�dsPðt � sÞZðt � sÞ � dZðtÞ � qe�dsPðt � sÞcþ Pðt � sÞ Zðt � sÞ þ b1e�dsPðt � sÞð1� Pðt � sÞÞ � b1e�dsPðt � sÞZðt � sÞ

¼ �dWðtÞ þ b1e�dsPðt � sÞðdþ 1� Pðt � sÞÞ � qe�dsPðt � sÞcþ Pðt � sÞ Zðt � sÞ

6 �dWðtÞ þ b1e�dsPðt � sÞðdþ 1� Pðt � sÞÞ 6 �dWðtÞ þ 14

b1e�dsðdþ 1Þ2

and since Pðt � sÞ � 12 ðdþ 1Þ

�2 P 0; Pðt � sÞðdþ 1� Pðt � sÞÞ 6 ðdþ1Þ24 . Therefore, by the comparison theory [24, Theo-

rem 1.4], W(t) 6 U(t), where UðtÞ ¼ Uð0Þe�dt þ 14d b1e�dsðdþ 1Þ2ð1� e�dtÞ is the solution of initial value problem

U0ðtÞ ¼ �dUðtÞ þ 14

b1e�dsðdþ 1Þ2;Uð0Þ ¼Wð0Þ:

Consequently, WðtÞ 6Wð0Þ þ 14d b1e�dsðdþ 1Þ2. By (2.3),

ZðtÞ þ b1e�dsPðt � sÞ ¼WðtÞ 6Wð0Þ þ 14d

b1e�dsðdþ 1Þ2 6 Zð0Þ þ b1e�dsPð�sÞ þ 14d

b1e�dsðdþ 1Þ2:

Thus all nonnegative solutions of (2.2) are bounded for t > 0.We will need the following simple and well known result, which is a direct application of Theorem 4.9.1 in [13, p.

159]. h

Lemma 2.3. If a < b, then the solution of the equation

y0ðtÞ ¼ ayðt � sÞ � byðtÞ;

where a, b, s > 0, and y(t) > 0 for �s 6 t 6 0, satisfies limt?1y(t) = 0.Let �Eð�P; �ZÞ denotes any one of the equilibrium point of system (2.2). Linearizing (2.2) about E we obtain


P01ðtÞ ¼ ð1� 2�P � �ZÞP1 � �PZ1;

Z01ðtÞ ¼ e�ds�Z b1 �qc

ðcþ �PÞ2

!P1ðt � sÞ � dZ1ðtÞ þ e�ds�P b1 �

qcþ �P

!Z1ðt � sÞ:

ð2:4Þ

The characteristic equation of (2.4) is A(k,s) = 0, where

Aðk; sÞ ¼1� 2�P � �Z � k ��P

e�ðdþkÞs�Z b1 � qcðcþ�PÞ2

� e�ðdþkÞs�P b1 � q

cþ�P

� � d� k

��: ð2:5Þ

In what follows, we will study the equilibria of model (2.2) without and with delay.

3. The stability analysis of model (2.2) without delay

Let s = 0 in model (2.2). Then the model has three equilibria E0(0,0), E1(1,0) and E+(P⁄(0), Z⁄(0)), where

P�ð0Þ ¼qþ d� cb1 þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðqþ d� cb1Þ

2 þ 4b1dcq

2b1; Z�ð0Þ ¼ 1� P�ð0Þ: ð2:5Þ

The positive equilibrium E+(P⁄(0), Z⁄(0)) exists and distinct from E1(1,0) if and only if

b1 > dþ q1þ c

: ð2:6Þ

From Aðk;0ÞjE0¼ 0 and Aðk;0ÞjE1

¼ 0 we get that E0(0,0) is always a saddle point. E1(1,0) is locally stable if b1 < dþ qcþ1 and it

is unstable if the inequality reserved. As for global stability of E1(1,0), we have the following theorem.

Theorem 3.1. Suppose that b1 < dþ q1þc. Then E1(1,0) is globally asymptotically stable.

Proof. Choosing e1 > 0 small enough such that b1 < dþ qcþ1þe1

. Also, by positive of solutions, P0(t) 6 P(t)(1 � P(t)). This implies

that limsupt?1P(t) 6 1. Choosing e2 ¼ 12

db1�

qcþ1þe1

� 1� �

> 0 and e = min (e1,e2). Then there exists T > 0 such that P(t) < 1 + e for

all t P T. Consequently, for t P T,

Z0ðtÞ ¼ �dZðtÞ þ b1 �q

cþ PðtÞ

� �PðtÞZðtÞ 6 �dZðtÞ þ b1 �

qcþ 1þ e

� �ð1þ eÞZðtÞ

6 �dZðtÞ þ b1 �q

cþ 1þ e1

� �ð1þ e2ÞZðtÞ 6 �dZðtÞ þ 1

2ðb1 �

qcþ 1þ e1

Þ þ d2

� �ZðtÞ:

By comparison, Z(t) is bounded above by the solution y(t) of

y0ðtÞ ¼ 12

b1 �q

cþ 1þ e1

� �� d

2

� �yðtÞ; t > T:

Since b1 � qcþ1þe1

� < d, which yields that y(t) ? 0. Hence Z(t) ? 0.

Let g 2 (0,1). Then there exists Tg > 0 such that, for t P Tg,

P0ðtÞP PðtÞð1� g� PðtÞÞ:

By another comparison argument,

lim inft!1

PðtÞP 1� g:

Since g 2 (0,1) is arbitrary, liminft?1P(t) P 1. We already have limsupt?1P(t) 6 1. Hence limt?1P(t) = 1. The proof of the-orem is complete. h

Theorem 3.2. If b1 > dþ q1þc, then interior equilibrium E+(P⁄(0), Z⁄(0)) is globally asymptotically stable.

Proof. First of all, let RðP; ZÞ ¼ 1PZ. Obviously R(P,Z) > 0 if P > 0, Z > 0. We define

g1ðP; ZÞ ¼ Pð1� PÞ � PZ;

g2ðP; ZÞ ¼ b1PZ � dZ � qPcþ P

Z:


A direct computation gives this quantity as

@ðg1RÞ@P

þ @ðg2RÞ@Z

¼ �1Z< 0:

The Bendixon–Dulac criterion then excludes any periodic orbits in the first positive quadrant.Next, condition b1 > dþ q

1þc implies the existence of the interior positive equilibrium E+(P⁄(0),Z⁄(0)) and E1(0,0) isunstable (a saddle point). E0(0,0) is also a saddle point. Butler–McGehee Theorem [25, p. 12] shows that neither E0

nor E1 can be an Omega limit point on a trajectory with initial conditions in first positive quadrat. Poincare–Bendixon Theory shows that all orbits tends to E+(P⁄(0),Z⁄(0)) as t tends to infinity. This completes the proof ofTheorem 3.2. h

4. The stability analysis of model (2.2) with delay

Let us turn to the case s P 0. System (2.2) has three equilibria E0(0,0),E1(1,0) and E+(P⁄(s),Z⁄(s)), where

P�ðsÞ ¼qþ deds � cb1 þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðqþ deds � cb1Þ

2 þ 4b1dcedsq

2b1; Z�ðsÞ ¼ 1� P�ðsÞ: ð2:7Þ

It is clear that E+(P⁄(s),Z⁄(s)) exists and distinct from E1(1,0) if and only if b1 > dþ q1þc and 0 6 s < sc, where

sc ¼1d

ln1d

b1 �q

cþ 1

� �� :

Examining the characteristic equation A(k,s) = 0, one can easily know that E0 is always a saddle point. As for E1(1,0), we havethe following result.

Theorem 4.1. Suppose that b1 > dþ q1þc , then E1(1,0) is unstable if 0 6 s < sc and globally asymptotically stable if s > sc.

Proof. The characteristic equation evaluated atE1(1,0) is given by

Aðk; sÞjE1¼ ðkþ 1Þ kþ d� e�ðdþkÞs b1 �

qcþ 1

� �� ¼ 0:

One of the roots of the characteristic equation is k = �1. The other roots given by solution of

ðkþ dÞeðdþkÞs ¼ b1 �q

cþ 1: ð2:8Þ

For any fixed 0 6 s < sc, assume k is a real root. The left hand side of (2.8) is a monotone increasing function in k for any fixeds. It takes the value deds at k = 0 and takes to positive infinity as k ? +1. Since 0 6 s < sc; deds < b1 � q

cþ1. By the intermediatevalue Theorem, there exists a unique k(s) > 0 such that Eq. (2.8) holds and so Aðk; sÞjE1

¼ 0 has at least one positive root k(s).Hence E1 is unstable for 0 6 s < sc.

Now we prove E1(1,0) is globally asymptotically stable if s > sc. It follows from s > sc that deds > b1 � qcþ1

� . Then there

exists e1 > 0 such that deds > b1 � qcþ1þe1

� . Choosing e2 ¼ 1

2edsd

b1�q

cþ1þe1

� 1� �

> 0, and let e = min{e1,e2}. Then, there exists a T > 0

such that P(t) < 1 + e for all t P T, and for t P T + s,

Z0ðtÞ ¼ �dZðtÞ þ e�ds b1 �q

cþ Pðt � sÞ

� �Pðt � sÞZðt � sÞ 6 �dZðtÞ þ e�ds b1 �

qcþ 1þ e

� �ð1þ eÞZðt � sÞ

6 �dZðtÞ þ e�ds b1 �q

cþ 1þ e1

� �ð1þ e2ÞZðt � sÞ 6 �dZðtÞ þ 1

2e�ds b1 �

qcþ 1þ e1

� �þ d

2

� �Zðt � sÞ:

By comparison, Z(t) is bounded above by the solution y(t) of

y0ðtÞ ¼ �dyðtÞ þ 12

e�ds b1 �q

cþ 1þ e1

� �þ d

2

� �yðt � sÞ; t > T þ s

satisfying y(t) = Z(t) for t 2 [T,T + s]. Since 12 e�ds b1 � q

cþ1þe1

� þ d

2

� < d, Lemma 2.3 yields that y(t) ? 0. Hence Z(t) ? 0.

Using similar way of proof of Theorem 3.1, we can get that limt?1P(t) = 1. This completes the proof of Theorem 4.1 h

Remark. The biological meaning of the Theorem 4.1 is obvious: if the outflow (which equal to natural death of zooplanktonand poisoned zooplankton) rate is more than recruitment rate of the zooplankton, then predators face extinction.


Thus we have proved that if s > sc, there are only two equilibria E0, a saddle, and E1, a globally asymptotically stableequilibrium. However if 0 6 s < sc, both E0 and E1 are unstable and the interior equilibrium E+ exists.

Examining the characteristic equation Aðk; sÞjEþ ¼ 0, one notices that s does not appears only in the eks terms, but also innumerous other places since P⁄ and Z⁄ depend on s. So, one cannot compute exactly the values of s at stability switchesoccur. In the remainder of this section we therefore investigate the complexity of the behavior of the system around interiorrest point (P⁄(s),Z⁄(s)) using simulation technic.

The eigenvalue equation Aðk; sÞjEþ ¼ 0 can be written as follows

k2 þ aðsÞkþ ðbðsÞkþ cðsÞÞeks þ dðsÞ ¼ 0;

where

cðsÞ ¼ edsP� b1 �q

ðcþ P�Þ2

!ð1� P�Þ � P� b1 �

qcþ P�

� �; aðsÞ ¼ dþ P�;

bðsÞ ¼ �edsP� b1 �q

cþ P�

� �; dðsÞ ¼ dP�:

Suppose that k = ix(x > 0) is a root of Aðk; sÞjEþ ¼ 0. Then we have

�x2 þ dðsÞ þ bðsÞx sinðxsÞ þ cðsÞ cosðxsÞ þ i aðsÞxþ bðsÞx cosðxsÞ � cðsÞ sinðxsÞð Þ ¼ 0:

Separating the real and imaginary parts,

cðsÞcosðxsÞ þ bðsÞx sinðxsÞ ¼ x2 � dðsÞ;

cðsÞsinðxsÞ � bðsÞx cosðxsÞ ¼ aðsÞx2:

Solving for cos(xs) and sin(xs) gives

sinðxðsÞsÞ ¼ aðsÞcðsÞxþ bðsÞxðx2 � dðsÞÞcðsÞ2 þ b2ðsÞx2

;

cosðxðsÞsÞ ¼ cðsÞðx2 � dðsÞÞ � aðsÞbðsÞx2

cðsÞ2 þ b2ðsÞx2

ð2:9Þ

squaring both sides of Eq. (2.9), adding them, and rearranging give

x4 þ ðaðsÞ2 � bðsÞ2 � 2aðsÞx2 þ dðsÞ2 � cðsÞ2 ¼ 0: ð2:10Þ

From (2.10) we get

x2� ¼

12

b2ðsÞ � a2ðsÞ þ 2d�ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðb2ðsÞ � a2ðsÞ þ 2dðsÞÞ2 � 4ðd2ðsÞ � c2ðsÞÞ

q� �:

0 2 4 6 8 10−15

−10

−5

0

5

10

τ

S0(τ)

S1(τ)

S2(τ)

0 2 4 6 8 10−2

0

2

4

6

8

10

12

14

16

18

20

τ

τω+

θ(τ)

θ(τ)+2π

θ(τ)+4π

Fig. 1. (Left) Intersections of h(s) + 2np (n = 0,1,2) and sx+ � sx+ intersects h(s) thrice. (Right) S0(s) has three zeros.

0 50 100 150 200 250 300 350 4000

1

2

3

4

5

6

t

phyt

opla

nkto

n−Zo

opla

nkto

n po

pula

tion

τ=0.01

Z(t)

P(t)

Fig. 2. Equilibrium E+ is stable when s is small.


Define the function h(s) 2 [0,2p) such that sin h(s) and cosh(s) are given by the right-hand sides of (2.9). Then the s weseek, at which stability switches occur, are the solutions of

SnðsÞ :¼ s� hðsÞ þ 2pnxþðsÞ

; n ¼ 0;1;2; . . .

In carrying our numerical simulations, we use the package DDE23 in MATLAB. We employ the non-scaled model (1.4). sc,x+(s) and h(s) are calculated in terms of the non-scaled parameters. We fix all parameters r = 2, k = 30, b = 0.7, b1 = 0.6,d = 0.5, q = 0.2, c = 5 except for s. For these set of values we obtain sc ¼ 1

d ln kd ðb1 � q

cþkÞ

h i� 7:1480529. We see in Fig. 1, for

n = 0, h(s) intersects sx(s) exactly thrice at s � 0.08732, s � 4.418 and s � 7.1480529. For n = 1, 2 and 0 6 s < sc, h(s) + 2phas no intersection with sx+(s). For n > 2, there is no intersection and h(s) + 2np (not plotted in the figure) lies aboveh(s) + 2p.

To demonstrate the occurrence of Hopf bifurcations at intersection points of h(s) and x+(s), we choose the initial dataP(t) = 1.8 and Z(t) = 0.6 for t 2 [�s,0]. For this choice one can easily verify that the interior equilibrium point E⁄(0.5306,2.8062) is global asymptotically stable in absence of delay. For s = 0.01, the numerical simulation shows that both thephytoplankton and the zooplankton populations converge to their equilibrium values in infinite time (see Fig. 2 withs = 0.01). In order to see how the length of gestation delay s affects the dynamical behavior we increase the value of thedelay. For a small delay, the coexistence equilibrium E+ is stable. However, when s increases beyond s � 0.08732, E+ becomesunstable since a Hopf bifurcation occurs at s � 0.08732. There is a stable periodic orbit surrounding E+ (see Fig. 3 for s = 0.1).

0 500 1000 1500 20000

2

4

6

8

t

P

0 500 1000 1500 20000

2

4

6

8

10

t

Z

τ=0.1

(a)

0 1 2 3 4 5 60

1

2

3

4

5

6

7

8

P

Z

(b)Fig. 3. (a) For s = 0.1, time series of a converging solution. (b) The stable periodic solution when s = 0.1.

0 10 20 30 40 50 60 700

5

10

15

20

25

t

phyt

opla

nkto

n−Zo

opla

nkto

n po

pula

tion

τ=0.3

P(t)

Z(t)

(a)

−5 0 5 10 15 20 250

2

4

6

8

10

12

14

16

18

P

Z

τ=0.3

(b)Fig. 4. (a) For s = 0.3, time series of a converging solution. (b) A periodic-like solution when s = 0.3.

0 500 1000 1500 20000

10

20

30

t

p

0 500 1000 1500 20000

2

4

6

t

Z

(a)0 5 10 15 20 25 30

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

P

Z

(b)Fig. 5. (a) For s = 3, time series of a converging solution. (b) The periodic solution when s = 3. Equilibrium E+ regain stable when s = 4.415.

0 1000 2000 3000 4000 50000

10

20

30

t

p

0 1000 2000 3000 4000 50000

1

2

3

t

Z

τ=4.416

τ=4.416

(a)

0 20 40 60 80 1000

10

20

30

40

t

p

0 20 40 60 80 1000

0.2

0.4

0.6

0.8

t

Z

(b)

Fig. 6. (a) Periodic solution disappears and E+ regains stability For s = 4.716. (b) Equilibrium E+ disappears and E1 regains stable when s > 7.8.


For s between 0.25 and 0.30756, the periodic solution changes its shape slightly (Fig. 4). As s increases, it turns to be aperiodic solution again (see Fig. 5). Taking s � 4.16, the periodic orbit disappears and the coexistence equilibrium E+ regainsits stability and remains stable until s � 7.1480529. For s > 7.1480529, E+ disappears since the zooplankton component Z⁄(s)becomes negative (see Fig. 6).

5. Discussion

In this paper, we studied a mathematical model of toxin producing phytoplankton zooplankton system in which the graz-ing pressure of zooplankton decreases due to the release of toxic substances by toxin producing phytoplankton species. Weinduced a discrete time delay to the both of the consume response function and distribution of toxic substance term. In order


to see how the length of gestation delay and liberation delay of toxic substances by phytoplankton species affects thedynamical behavior, we have first analyzed the model without delay and obtained the parameters conditions for the exis-tence and globally asymptotically stable of the positive interior equilibrium. Next under the aids of the numerical simulationwe have investigated the delayed model system. Our numerical analysis results around the coexistent steady state revealedthat the model with gestation delay and liberation delay (liberation of toxic substances by phytoplankton species) exhibitsunstable behavior under the parameter conditions which ensure stability for the non-delayed model. Numerical results alsoshown the existence of a Hopf bifurcation around this steady state for a critical value of the delay parameter. Thus the ges-tation delay has a destabilizing effect on the system dynamics.

Our numerical example shows that the model system does not have any periodic orbit whenever 0 6 s 6 0.08732 and4.418 6 s 6 7.1480529 and all trajectories eventually approach the coexisting equilibrium point E+ starting from any pointwithin the positive quadrant of state space (Figs. 2 and 6). When s is approximately between 0.08732 and 4.418, there is adelay induced instability demonstrated in Figs. 3–5. An increase in the value of s from approximately 0.08756 to 4.418causes different system populations to oscillate periodically around E ⁄ at regular time intervals. Thus there is a range of ges-tation delay which initially imparts stability, then induces instability and ultimately leads to periodic behavior.

References

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dynamical analysis of a delay model of phytoplankton–zooplankton interaction

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