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Hindawi Publishing CorporationAbstract and Applied AnalysisVolume 2013, Article ID 340174, 5 pageshttp://dx.doi.org/10.1155/2013/340174

Research ArticleHopf-Pitchfork Bifurcation in a Phytoplankton-ZooplanktonModel with Delays

Jia-Fang Zhang and Dan Zhang

School of Mathematics and Information Sciences, Henan University, Kaifeng 475001, China

Correspondence should be addressed to Jia-Fang Zhang; [email protected]

Received 21 October 2013; Accepted 12 November 2013

Academic Editor: Allan Peterson

Copyright © 2013 J.-F. Zhang and D. Zhang. This is an open access article distributed under the Creative Commons AttributionLicense, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properlycited.

The purpose of this paper is to study the dynamics of a phytoplankton-zooplankton model with toxin delay. By studying thedistribution of the eigenvalues of the associated characteristic equation, the pitchfork bifurcation curve of the system is obtained.Furthermore, on the pitchfork bifurcation curve, we find that the system can undergo aHopf bifurcation at the positive equilibrium,and we derive the critical values where Hopf-Pitchfork bifurcation occurs.

1. Introduction

The study of the dynamical interaction of zooplankton andphytoplankton is an important area of research in marineecology. Phytoplanktons are tiny floating plants that live nearthe surface of lakes and ocean. They provide food for marinelife, oxygen for human being, and also absorb half of thecarbon dioxide which may be contributing to the globalwarming [1]. Zooplanktons are microscopic animals that eatother phytoplankton. Toxins are produced by phytoplanktonto avoid predation by zooplankton. The toxin producingphytoplankton not only reduces the grazing pressure onthem but also can control the occurrence of bloom; seeChattopadhyay et al. [2] and Sarkar and Chattopadhyay [3].Phytoplankton-zooplankton models have been studied bymany authors [4–9]. In [6], models of nutrient-planktoninteraction with a toxic substance that inhibit either thegrowth rate of phytoplankton, zooplankton, or both trophiclevels are proposed and studied. In [7], authors have dealtwith a nutrient-plankton model in an aquatic environmentin the context of phytoplankton bloom. Roy [8] has con-structed a mathematical model for describing the interactionbetween a nontoxic and a toxic phytoplankton under a singlenutrient. Saha and Bandyopadhyay [9] considered a toxinproducing phytoplankton-zooplankton model in which thetoxin liberation by phytoplankton species follows a discretetime variation. Biological delay systems of one type or

another have been considered by a number of authors [10,11]. These systems governed by integrodifferential equationsexhibit much more rich dynamics than ordinary differentialsystems. For example, Das and Ray [5] investigated the effectof delay on nutrient cycling in phytoplankton-zooplanktoninteractions in the estuarine system. In this paperwe present aphytoplankton-zooplanktonmodel to investigate its dynamicbehaviors. The model we considered is based on the fol-lowing plausible toxic-phytoplankton-zooplankton systemsintroduced by Chattopadhayay et al. [2]

𝑑𝑃

𝑑𝑡= 𝑟1𝑃(1 −

𝑃

𝐾) − 𝑎𝑃𝑍,

𝑑𝑍

𝑑𝑡= 𝑏𝑍∫

𝑡

−∞

𝐺 (𝑡 − 𝑠) 𝑃 (𝑠) 𝑑𝑠 − 𝑐𝑍 − 𝑑𝑃 (𝑡 − 𝜏)

𝑒 + 𝑃 (𝑡 − 𝜏)𝑍,

(1)

where 𝑃(𝑡) and 𝑍(𝑡) are the densities of phytoplankton andzooplankton, respectively. 𝑟

1, 𝐾, 𝑎, 𝑏, 𝑐, 𝑑, and 𝑒 are positive

constants. 𝜏 is toxin delay, 𝐺(𝑠) is the delay kernel and a non-negative bounded function defined on [0,∞] as follows:

∫

∞

0

𝐺 (𝑠) 𝑑𝑠 = 1, 𝐺 (𝑠) = 𝜎𝑒−𝜎𝑠

, 𝜎 > 0. (2)

For a set of different species interacting with each otherin ecological community, perhaps the simplest and probablythe most important question from a practical point of view is

2 Abstract and Applied Analysis

whether all the species in the system survive in the long term.Therefore, the periodic phenomena of biological system areoften discussed [12–16].

The primary purpose of this paper is to study the effectsof toxin delay on the dynamics of (1). That is to say, we willtake the delay 𝜏 passes through a critical value, the positiveequilibrium loses its stability and bifurcation occurs. Bystudying the distribution of the eigenvalues of the associatedcharacteristic equation, the pitchfork bifurcation curve of thesystem is obtained. Furthermore, we derive the critical valueswhere Hopf-Pitchfork bifurcation occurs.

Thepaper is structured as follows. In Section 2, we discussthe local stability of the positive solutions and the existenceof Pitchfork bifurcation. In Section 3, the conditions for theoccurrence of Hopf-Pitchfork bifurcation are determined.

2. Stability and Pitchfork Bifurcation

In this section, we focus on investigating the local stabilityand the existence of Pitchfork bifurcation of the positiveequilibrium of system (1). It is easy to see that system (1) hasa unique positive equilibrium 𝐸∗(𝑃∗, 𝑍∗), where

𝑃∗=

𝑐 + 𝑑 − 𝑏𝑒 + √(𝑐 + 𝑑 − 𝑏𝑒)2+ 4𝑏𝑐𝑒

2𝑏,

𝑍∗=

𝑟

𝑎(1 −

𝑃∗

𝐾) > 0,

(3)

where (𝐻1) : 𝐾 > 𝑃

∗.Let

𝑊(𝑡) = ∫

𝑡

−∞

𝜎𝑒−𝜎(𝑡−𝑠)

𝑃 (𝑠) 𝑑𝑠. (4)

By the linear chain trick technique, then system (1) can betransformed into the following system:

𝑑𝑃

𝑑𝑡= 𝑟1𝑃(1 −

𝑃


𝑑𝑍

𝑑𝑡= 𝑏𝑍𝑊 − 𝑐𝑍 − 𝑑

𝑃 (𝑡 − 𝜏)

𝑒 + 𝑃 (𝑡 − 𝜏)𝑍,

𝑑𝑊

𝑑𝑡= 𝜎𝑃 (𝑡) − 𝜎𝑊 (𝑡) .

(5)

It is easy to check that system (5) has an unique positiveequilibrium 𝐸(𝑃∗, 𝑍∗,𝑊∗) with 𝑃∗ = 𝑊∗ provided that thecondition (𝐻

1) holds.

Let 𝑃 = 𝑢+𝑢∗,𝑍 = V+ V∗, and𝑊 = 𝑤+𝑊∗; then system(5) can be transformed into the following system:

�̇� (𝑡) = −𝑟

𝐾𝑃∗𝑢 (𝑡) − 𝑎𝑃

∗V (𝑡) − 𝑎𝑢 (𝑡) V (𝑡) −𝑟

𝐾𝑢2(𝑡) ,

V̇ (𝑡) = −𝑑𝑒

(𝑒 + 𝑃∗)𝑍∗𝑢 (𝑡 − 𝜏) + 𝑏𝑍

∗𝑤 (𝑡)

+ ∑

𝑖+𝑗+𝑘≥2

𝑓(𝑖𝑗𝑘)

2𝑢𝑖(𝑡 − 𝜏) V𝑗𝑤𝑘,

�̇� (𝑡) = 𝜎𝑢 (𝑡) − 𝜎𝑤 (𝑡) ,

(6)

where

𝑓(𝑖𝑗𝑘)

2=

1

𝑖!𝑗!𝑘!

𝜕𝑖+𝑗+𝑘

𝑓2

𝜕𝑢𝑖 (𝑡 − 𝜏) 𝜕V𝑗𝜕𝑤𝑘,

𝑓2= 𝑏V𝑤 − 𝑐V − 𝑑

𝑢 (𝑡 − 𝜏)

𝑒 + 𝑢 (𝑡 − 𝜏)V.

(7)

Then linearizing system (6) at 𝐸∗(𝑃∗, 𝑍∗,𝑊∗) is

�̇� (𝑡) = −𝑟

𝐾𝑃∗𝑢 (𝑡) − 𝑎𝑃

∗V (𝑡) ,

V̇ (𝑡) = −𝑑𝑒

(𝑒 + 𝑃∗)𝑍∗𝑢 (𝑡 − 𝜏) + 𝑏𝑍

∗𝑤 (𝑡) ,

�̇� (𝑡) = 𝜎𝑢 (𝑡) − 𝜎𝑤 (𝑡) .

(8)

It is easy to see that the associated characteristic equation ofsystem (11) at the positive equilibrium has the following formand thus the characteristic equation of system (5) is given by

𝐹 (𝜆) = 𝜆3+ 𝑝2𝜆2+ 𝑝1𝜆 + 𝑝0− [𝑞1𝜆 + 𝑞0] 𝑒−𝜆𝜏

= 0, (9)

where

𝑝2= 𝜎 +

𝑟𝑃∗

𝐾, 𝑝

1=

𝑟𝑃∗𝜎

𝐾, 𝑝

0= 𝑎𝑏𝑃

∗𝑍∗𝜎,

𝑞1=

𝑎𝑑𝑒𝑃∗𝑍∗

(𝑒 + 𝑃∗)2, 𝑞

0=

𝑎𝑑𝑒𝑃∗𝑍∗𝜎

(𝑒 + 𝑃∗)2

.

(10)

Obviously, 𝑝2> 0, 𝑝

1> 0, 𝑝

0> 0, 𝑞

1> 0, and 𝑞

0> 0.

From (9), the following lemma is obvious.

Lemma 1. If the condition 𝐻2: 𝑝0= 𝑞0holds, then 𝜆 = 0 is

always a root of (9) for all 𝜏 ≥ 0.

Let 𝑑0= (𝑏/𝑒)(𝑒+𝑃

∗)2, 𝑑1= 𝑝1(𝑒+𝑃∗)2/(1−𝜏𝜎)𝑎𝑒𝑃

∗𝑍∗,

and 𝑑2= (𝑒+𝑃

∗)2[𝜏2𝑎𝑏𝑃∗𝑍∗𝜎−2(𝜎+(𝑟𝑃

∗/𝐾))]/2𝜏𝑎𝑒𝑃

∗𝑍∗;

we have the following results.

Lemma 2. Suppose that the condition (𝐻2) is satisfied.

(i) If 𝑑 = 𝑑0

̸= 𝑑1, then (9) has a single zero root.

(ii) If 𝑑 = 𝑑1

̸= 𝑑2, then (9) has a double zero root.

Proof. Clearly, 𝜆 = 0 is a root to (9) if and only if 𝑝0= 𝑞0,

whichmeans𝑑 = 𝑑0. Substituting𝑑 = 𝑑

0into𝐹(𝜆) and taking

the derivative with respect to 𝜆, we obtain

𝐹(𝜆)

𝑑=𝑑0= 3𝜆2+ 2𝑝2𝜆 + 𝑝1− [𝑞1(1 − 𝜏𝜆) − 𝜏𝑞

0] 𝑒−𝜆𝜏

.

(11)

Then we can get

𝐹(0)

𝑑=𝑑0= 𝑝1− [𝑞1− 𝜏𝑞0] . (12)

For any 𝜏 > 0, by solving (12), we can obtain 𝑑 = 𝑑1. If 𝑑 =

𝑑0

̸= 𝑑1, 𝐹(0) ̸= 0which means that 𝜆 = 0 is a single zero root

to (9), and hence the conclusion of (i) follows.

Abstract and Applied Analysis 3

From (11), it follows that

𝐹(𝜆)

𝑑=𝑑1= 6𝜆 + 2𝑝

2− [𝑞1(−2𝜏 + 𝜏

2𝜆) + 𝜏

2𝑞0] 𝑒−𝜆𝜏

.

(13)

Then we get

𝐹(0)

𝑑=𝑑1= 2𝑝2− [𝜏2𝑞0− 2𝑞1𝜏] . (14)

For any 𝜏 > 0, by solving (14), we can obtain 𝑑 = 𝑑2. If

𝑑 = 𝑑1

̸= 𝑑2, 𝐹(0) ̸= 0 which means that 𝜆 = 0 is a double

zero root to (9), and hence the conclusion of (ii) follows.Thiscompletes the proof.

From Lemma 2, we have the following result.

Theorem 3. Suppose that (𝐻2) holds if 𝑑 = 𝑑

0̸= 𝑑1, then,

the system (5) undergoes a Pitchfork bifurcation at the positiveequilibrium.

3. Hopf-Pitchfork Bifurcation

In the following, we consider the case that (9) not only has azero root, but also has a pair of purely imaginary roots ±𝑖𝜔(𝜔 > 0), when 𝑑 = 𝑑

0̸= 𝑑1holds.

Substituting 𝜆 = 𝑖𝜔 (𝜔 > 0) and 𝑑 = 𝑑0into (9) and

separating the real and imaginary parts, one can get

−𝜔3+ 𝑝1𝜔 − 𝑞1𝜔 cos (𝜔𝜏) + 𝑞

0sin (𝜔𝜏) = 0,

−𝑝2𝜔2+ 𝑝0− 𝑞1𝜔 sin (𝜔𝜏) − 𝑞

0cos (𝜔𝜏) = 0.

(15)

It is easy to see from (15) that

𝜔6+ 𝐷2𝜔4+ 𝐷1𝜔2+ 𝐷0= 0, (16)

where

𝐷2= 𝑝2

2− 2𝑝1, 𝐷

1= 𝑝2

1− 2𝑝0𝑝2− 𝑞2

1,

𝐷0= 𝑝2

0− 𝑞2

0.

(17)

Let 𝑧 = 𝜔2. Then (16) can be written as

ℎ (𝑧) = 𝑧3+ 𝐷2𝑧2+ 𝐷1𝑧 + 𝐷

0. (18)

In terms of the coefficient in ℎ(𝑧) define Δ by Δ = 𝐷22− 3𝐷1.

It is easy to know from the characters of cubic algebraicequation that ℎ(𝑧) is a strictly monotonically increasingfunction ifΔ ≤ 0. IfΔ > 0 and 𝑧∗ = (√Δ−𝐷

2)/3 < 0 orΔ > 0,

𝑧∗

= (√Δ − 𝐷2)/3 > 0 but ℎ(𝑧∗) > 0, then ℎ(𝑧) has always

no positive root. Therefore, under these conditions, (9) hasno purely imaginary roots for any 𝜏 > 0 and this also impliesthat the positive equilibrium 𝐸(𝑃∗, 𝑍∗,𝑊∗) of system (1) isabsolutely stable. Thus, we can obtain easily the followingresult on the stability of positive equilibrium 𝐸(𝑃∗, 𝑍∗,𝑊∗)of system (1).

Theorem 4. Assume that (𝐻1) holds and Δ ≤ 0 or Δ > 0

and 𝑧∗ = (√Δ − 𝐷2)/3 < 0 or Δ > 0, 𝑧∗ > 0 and ℎ(𝑧∗) >

0. Then the positive equilibrium 𝐸(𝑃∗, 𝑍∗,𝑊∗) of system (5)is absolutely stable; namely; 𝐸(𝑃∗, 𝑍∗,𝑊∗) is asymptoticallystable for any delay 𝜏 ≥ 0.

In what follows, we assume that the coefficients in ℎ(𝑧)satisfy the condition

(𝐻3) Δ = 𝐷

2

2− 3𝐷1> 0, 𝑧∗ = (√Δ − 𝐷

2)/3 > 0, ℎ(𝑧∗) < 0.

Then, according to Lemma 2.2 in [17], we know that (16) hasat least a positive root 𝜔

0; that is, the characteristic equation

(9) has a pair of purely imaginary roots ±𝑖𝜔0. Eliminating

sin(𝜔𝜏) in (15), we can get that the corresponding 𝜏𝑘> 0 such

that (9) has a pair of purely imaginary roots ±𝑖𝜔0, 𝜏𝑘> 0 are

given by

𝜏𝑘=

1

𝜔0

arccos[−𝑞1𝜔4

0+ (𝑝1𝑞1− 𝑝2𝑞0) 𝜔2

0+ 𝑝0𝑞0

𝑞21𝜔20+ 𝑞20

]

+2𝑘𝜋

𝜔0

, (𝑘 = 0, 1, 2, . . .) .

(19)

Let 𝜆(𝜏) = V(𝜏) + 𝑖𝜔(𝜏) be the roots of (9) such that when𝜏 = 𝜏𝑘satisfying V(𝜏

𝑘) = 0 and 𝜔(𝜏

𝑘) = 𝜔0. We can claim that

sgn [𝑑 (Re 𝜆)𝑑𝜏

]

𝜏=𝜏𝑘

= sgn {ℎ (𝜔20)} . (20)

In fact, differentiating two sides of (9) with respect to 𝜏, weget

(𝑑𝜆

𝑑𝜏)

−1

= −(3𝜆2+ 2𝑝2𝜆 + 𝑝1) − 𝑞1𝑒−𝜆𝜏

+ (𝑞1𝜆 + 𝑞0) 𝜏𝑒−𝜆𝜏

(𝑞1𝜆 + 𝑞0) 𝜆𝑒−𝜆𝜏

= −(3𝜆2+ 2𝑝2𝜆 + 𝑝1) 𝑒𝜆𝜏

(𝑞1𝜆 + 𝑞0) 𝜆

+𝑞1

(𝑞1𝜆 + 𝑞0) 𝜆

−𝜏

𝜆.

(21)

Then

sgn [𝑑 (Re 𝜆)𝑑𝜏

]

𝜏=𝜏𝑘

= sgn[Re(𝑑𝜆𝑑𝜏

)

−1

]

𝜆=𝑖𝜔0

= sgn[Re(−(3𝜆2+ 2𝑝2𝜆 + 𝑝1) 𝑒𝜆𝜏

(𝑞1𝜆 + 𝑞0) 𝜆

+𝑞1

(𝑞1𝜆 + 𝑞0) 𝜆

−𝜏

𝜆)]

𝜆=𝑖𝜔0

= sgn Re[−(𝑝1− 3𝜔2

0+ 2𝑝2𝜔0𝑖) [cos (𝜔

0𝜏𝑘) + 𝑖 sin (𝜔

0𝜏𝑘)]

(𝑞1𝜔0𝑖 + 𝑞0) 𝜔0𝑖

+𝑞1

(𝑞1𝜔0𝑖 + 𝑞0) 𝜔0𝑖]

4 Abstract and Applied Analysis

= sgn 1Λ

{[(𝑝1− 3𝜔2

0) cos (𝜔

0𝜏𝑘) − 2𝑝

2𝜔0sin (𝜔

0𝜏𝑘)]

× (𝑞1𝜔2

0)

− [(𝑝1− 3𝜔2

0) sin (𝜔

0𝜏𝑘) + 2𝑝

2𝜔0cos (𝜔

0𝜏𝑘)]

× 𝑞0𝜔0− 𝑞2

1𝜔2

0}

= sgn 1Λ

{(3𝜔2

0− 𝑝1) 𝜔0[𝑞1𝜔0cos (𝜔

0𝜏𝑘) − 𝑞0sin (𝜔

0𝜏𝑘)]

− 2𝑝2𝜔2

0[𝑞1𝜔0sin (𝜔

0𝜏𝑘) + 𝑞0cos (𝜔

0𝜏𝑘)] − 𝑞

2

1𝜔2

0}

= sgn 1Λ

[3𝜔6

0+ 2 (𝑝

2

2− 𝑝1) 𝜔4

0+ (𝑝2

1− 2𝑝0𝑝2− 𝑞2

1) 𝜔2

0]

= sgn𝜔2

0

Λ[3𝜔4

0+ 2𝐷2𝜔2

0+ 𝐷1]

= sgn𝜔2

0

Λ{ℎ(𝜔2

0)} = sgn {ℎ (𝜔2

0)} ,

(22)

where Λ = 𝑞21𝜔4

0+ 𝑞2

0𝜔2

0. It follows from the hypothesis

(𝐻3) that ℎ(𝜔2

0) ̸= 0 and therefore the transversality condition

holds.

Lemma5. All the roots of (9), except a zero root, have negativereal parts when 𝑝

1> 𝑞1; (i) of Lemma 2 and 𝜏 ∈ [0, 𝜏

0) hold.

Proof. Consider

𝜆3+ 𝑝2𝜆2+ (𝑝1− 𝑞1) 𝜆 + 𝑝

0− 𝑞0= 𝜆 (𝜆

2+ 𝑝2𝜆 + 𝑝1− 𝑞1) .

(23)

It is easy to get that the roots of (23) are 𝜆1= 0 and 𝜆

2,3=

(−𝑝2±√𝑝22− 4(𝑝1− 𝑞1))/2. If𝑝

1−𝑞1> 0, all the roots of (23),

except a zero root, have negative real parts. We complete theproof.

Summarizing the previous discussions, we have the fol-lowing result.

Theorem 6. Suppose that the conditions (𝐻1), (𝐻2), and (𝐻

3)

are satisfied.

(i) If 𝑑 = 𝑑0

̸= 𝑑1and 𝜏 ∈ [0, 𝜏

0), then the system (1)

undergoes a Pitchfork bifurcation at positive equilib-rium 𝐸∗.

(ii) If 𝑑 = 𝑑0

̸= 𝑑1and 𝜏 = 𝜏

0, then system (1) can undergo

aHopf-Pitchfork bifurcation at the positive equilibrium𝐸∗.

4. Conclusions

In this section, we present some particular cases of system (1)as follows:

𝑑𝑃

𝑑𝑡= 𝑟1𝑃(1 −

𝑃


𝑑𝑍

𝑑𝑡= 𝑏𝑍𝑃 − 𝑐𝑍 − 𝑑

𝑃 (𝑡 − 𝜏)

𝑒 + 𝑃 (𝑡 − 𝜏)𝑍.

(24)

From [2], we know that the system (24) undergoes a Hopfbifurcation at the positive equilibrium. In this paper, weget the condition that (9) has a zero root and also get theconditions that (9) has double zero roots. Furthermore, weobtain the conditions that (9) has a single zero root and apair of purely imaginary roots. Under this condition, system(1) undergoes a Hopf-Pitchfork bifurcation at the positiveequilibrium. Especially, when 𝜏 = 0, system (24) reduces to

𝑑𝑃

𝑑𝑡= 𝑟1𝑃(1 −

𝑃


𝑑𝑍

𝑑𝑡= 𝑏𝑍𝑃 − 𝑐𝑍 − 𝑑

𝑃 (𝑡)

𝑒 + 𝑃 (𝑡)𝑍.

(25)

We can conclude that the positive equilibrium𝐸(𝑃∗, 𝑍∗,𝑊∗)is locally asymptotically stable in the absence of toxin delay.

Acknowledgments

The authors are grateful to the referees for their valuablecomments and suggestions on the paper. The research of theauthors was supported by the Fundamental Research Fund ofHenan University (2012YBZR032).

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