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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 Β© 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 [4β9]. 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]
ππ
ππ‘= π1π(1 β
π
πΎ) β πππ,
ππ
ππ‘= ππβ«
π‘
ββ
πΊ (π‘ β π ) π (π ) ππ β ππ β ππ (π‘ β π)
π + π (π‘ β π)π,
(1)
where π(π‘) and π(π‘) are the densities of phytoplankton andzooplankton, respectively. π
1, πΎ, π, π, π, π, and π are positive
constants. π is toxin delay, πΊ(π ) is the delay kernel and a non-negative bounded function defined on [0,β] as follows:
β«
β
0
πΊ (π ) ππ = 1, πΊ (π ) = ππβππ
, π > 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
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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 [12β16].
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 π 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 πΈβ(πβ, πβ), where
πβ=
π + π β ππ + β(π + π β ππ)2+ 4πππ
2π,
πβ=
π
π(1 β
πβ
πΎ) > 0,
(3)
where (π»1) : πΎ > π
β.Let
π(π‘) = β«
π‘
ββ
ππβπ(π‘βπ )
π (π ) ππ . (4)
By the linear chain trick technique, then system (1) can betransformed into the following system:
ππ
ππ‘= π1π(1 β
π
πΎ) β πππ,
ππ
ππ‘= πππ β ππ β π
π (π‘ β π)
π + π (π‘ β π)π,
ππ
ππ‘= ππ (π‘) β ππ (π‘) .
(5)
It is easy to check that system (5) has an unique positiveequilibrium πΈ(πβ, πβ,πβ) with πβ = πβ provided that thecondition (π»
1) holds.
Let π = π’+π’β,π = V+ Vβ, andπ = π€+πβ; then system(5) can be transformed into the following system:
οΏ½ΜοΏ½ (π‘) = βπ
πΎπβπ’ (π‘) β ππ
βV (π‘) β ππ’ (π‘) V (π‘) βπ
πΎπ’2(π‘) ,
VΜ (π‘) = βππ
(π + πβ)πβπ’ (π‘ β π) + ππ
βπ€ (π‘)
+ β
π+π+πβ₯2
π(πππ)
2π’π(π‘ β π) Vππ€π,
οΏ½ΜοΏ½ (π‘) = ππ’ (π‘) β ππ€ (π‘) ,
(6)
where
π(πππ)
2=
1
π!π!π!
ππ+π+π
π2
ππ’π (π‘ β π) πVπππ€π,
π2= πVπ€ β πV β π
π’ (π‘ β π)
π + π’ (π‘ β π)V.
(7)
Then linearizing system (6) at πΈβ(πβ, πβ,πβ) is
οΏ½ΜοΏ½ (π‘) = βπ
πΎπβπ’ (π‘) β ππ
βV (π‘) ,
VΜ (π‘) = βππ
(π + πβ)πβπ’ (π‘ β π) + ππ
βπ€ (π‘) ,
οΏ½ΜοΏ½ (π‘) = ππ’ (π‘) β ππ€ (π‘) .
(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
πΉ (π) = π3+ π2π2+ π1π + π0β [π1π + π0] πβππ
= 0, (9)
where
π2= π +
ππβ
πΎ, π
1=
ππβπ
πΎ, π
0= πππ
βπβπ,
π1=
ππππβπβ
(π + πβ)2, π
0=
ππππβπβπ
(π + πβ)2
.
(10)
Obviously, π2> 0, π
1> 0, π
0> 0, π
1> 0, and π
0> 0.
From (9), the following lemma is obvious.
Lemma 1. If the condition π»2: π0= π0holds, then π = 0 is
always a root of (9) for all π β₯ 0.
Let π0= (π/π)(π+π
β)2, π1= π1(π+πβ)2/(1βππ)πππ
βπβ,
and π2= (π+π
β)2[π2πππβπβπβ2(π+(ππ
β/πΎ))]/2ππππ
βπβ;
we have the following results.
Lemma 2. Suppose that the condition (π»2) is satisfied.
(i) If π = π0
ΜΈ= π1, then (9) has a single zero root.
(ii) If π = π1
ΜΈ= π2, then (9) has a double zero root.
Proof. Clearly, π = 0 is a root to (9) if and only if π0= π0,
whichmeansπ = π0. Substitutingπ = π
0intoπΉ(π) and taking
the derivative with respect to π, we obtain
πΉ(π)
π=π0= 3π2+ 2π2π + π1β [π1(1 β ππ) β ππ
0] πβππ
.
(11)
Then we can get
πΉ(0)
π=π0= π1β [π1β ππ0] . (12)
For any π > 0, by solving (12), we can obtain π = π1. If π =
π0
ΜΈ= π1, πΉ(0) ΜΈ= 0which means that π = 0 is a single zero root
to (9), and hence the conclusion of (i) follows.
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Abstract and Applied Analysis 3
From (11), it follows that
πΉ(π)
π=π1= 6π + 2π
2β [π1(β2π + π
2π) + π
2π0] πβππ
.
(13)
Then we get
πΉ(0)
π=π1= 2π2β [π2π0β 2π1π] . (14)
For any π > 0, by solving (14), we can obtain π = π2. If
π = π1
ΜΈ= π2, πΉ(0) ΜΈ= 0 which means that π = 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 (π»2) holds if π = π
0ΜΈ= π1, 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 Β±ππ(π > 0), when π = π
0ΜΈ= π1holds.
Substituting π = ππ (π > 0) and π = π0into (9) and
separating the real and imaginary parts, one can get
βπ3+ π1π β π1π cos (ππ) + π
0sin (ππ) = 0,
βπ2π2+ π0β π1π sin (ππ) β π
0cos (ππ) = 0.
(15)
It is easy to see from (15) that
π6+ π·2π4+ π·1π2+ π·0= 0, (16)
where
π·2= π2
2β 2π1, π·
1= π2
1β 2π0π2β π2
1,
π·0= π2
0β π2
0.
(17)
Let π§ = π2. Then (16) can be written as
β (π§) = π§3+ π·2π§2+ π·1π§ + π·
0. (18)
In terms of the coefficient in β(π§) define Ξ by Ξ = π·22β 3π·1.
It is easy to know from the characters of cubic algebraicequation that β(π§) is a strictly monotonically increasingfunction ifΞ β€ 0. IfΞ > 0 and π§β = (βΞβπ·
2)/3 < 0 orΞ > 0,
π§β
= (βΞ β π·2)/3 > 0 but β(π§β) > 0, then β(π§) has always
no positive root. Therefore, under these conditions, (9) hasno purely imaginary roots for any π > 0 and this also impliesthat the positive equilibrium πΈ(πβ, πβ,πβ) of system (1) isabsolutely stable. Thus, we can obtain easily the followingresult on the stability of positive equilibrium πΈ(πβ, πβ,πβ)of system (1).
Theorem 4. Assume that (π»1) holds and Ξ β€ 0 or Ξ > 0
and π§β = (βΞ β π·2)/3 < 0 or Ξ > 0, π§β > 0 and β(π§β) >
0. Then the positive equilibrium πΈ(πβ, πβ,πβ) of system (5)is absolutely stable; namely; πΈ(πβ, πβ,πβ) is asymptoticallystable for any delay π β₯ 0.
In what follows, we assume that the coefficients in β(π§)satisfy the condition
(π»3) Ξ = π·
2
2β 3π·1> 0, π§β = (βΞ β π·
2)/3 > 0, β(π§β) < 0.
Then, according to Lemma 2.2 in [17], we know that (16) hasat least a positive root π
0; that is, the characteristic equation
(9) has a pair of purely imaginary roots Β±ππ0. Eliminating
sin(ππ) in (15), we can get that the corresponding ππ> 0 such
that (9) has a pair of purely imaginary roots Β±ππ0, ππ> 0 are
given by
ππ=
1
π0
arccos[βπ1π4
0+ (π1π1β π2π0) π2
0+ π0π0
π21π20+ π20
]
+2ππ
π0
, (π = 0, 1, 2, . . .) .
(19)
Let π(π) = V(π) + ππ(π) be the roots of (9) such that whenπ = ππsatisfying V(π
π) = 0 and π(π
π) = π0. We can claim that
sgn [π (Re π)ππ
]
π=ππ
= sgn {β (π20)} . (20)
In fact, differentiating two sides of (9) with respect to π, weget
(ππ
ππ)
β1
= β(3π2+ 2π2π + π1) β π1πβππ
+ (π1π + π0) ππβππ
(π1π + π0) ππβππ
= β(3π2+ 2π2π + π1) πππ
(π1π + π0) π
+π1
(π1π + π0) π
βπ
π.
(21)
Then
sgn [π (Re π)ππ
]
π=ππ
= sgn[Re(ππππ
)
β1
]
π=ππ0
= sgn[Re(β(3π2+ 2π2π + π1) πππ
(π1π + π0) π
+π1
(π1π + π0) π
βπ
π)]
π=ππ0
= sgn Re[β(π1β 3π2
0+ 2π2π0π) [cos (π
0ππ) + π sin (π
0ππ)]
(π1π0π + π0) π0π
+π1
(π1π0π + π0) π0π]
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4 Abstract and Applied Analysis
= sgn 1Ξ
{[(π1β 3π2
0) cos (π
0ππ) β 2π
2π0sin (π
0ππ)]
Γ (π1π2
0)
β [(π1β 3π2
0) sin (π
0ππ) + 2π
2π0cos (π
0ππ)]
Γ π0π0β π2
1π2
0}
= sgn 1Ξ
{(3π2
0β π1) π0[π1π0cos (π
0ππ) β π0sin (π
0ππ)]
β 2π2π2
0[π1π0sin (π
0ππ) + π0cos (π
0ππ)] β π
2
1π2
0}
= sgn 1Ξ
[3π6
0+ 2 (π
2
2β π1) π4
0+ (π2
1β 2π0π2β π2
1) π2
0]
= sgnπ2
0
Ξ[3π4
0+ 2π·2π2
0+ π·1]
= sgnπ2
0
Ξ{β(π2
0)} = sgn {β (π2
0)} ,
(22)
where Ξ = π21π4
0+ π2
0π2
0. It follows from the hypothesis
(π»3) that β(π2
0) ΜΈ= 0 and therefore the transversality condition
holds.
Lemma5. All the roots of (9), except a zero root, have negativereal parts when π
1> π1; (i) of Lemma 2 and π β [0, π
0) hold.
Proof. Consider
π3+ π2π2+ (π1β π1) π + π
0β π0= π (π
2+ π2π + π1β π1) .
(23)
It is easy to get that the roots of (23) are π1= 0 and π
2,3=
(βπ2Β±βπ22β 4(π1β π1))/2. Ifπ
1βπ1> 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 (π»1), (π»2), and (π»
3)
are satisfied.
(i) If π = π0
ΜΈ= π1and π β [0, π
0), then the system (1)
undergoes a Pitchfork bifurcation at positive equilib-rium πΈβ.
(ii) If π = π0
ΜΈ= π1and π = π
0, then system (1) can undergo
aHopf-Pitchfork bifurcation at the positive equilibriumπΈβ.
4. Conclusions
In this section, we present some particular cases of system (1)as follows:
ππ
ππ‘= π1π(1 β
π
πΎ) β πππ,
ππ
ππ‘= πππ β ππ β π
π (π‘ β π)
π + π (π‘ β π)π.
(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 π = 0, system (24) reduces to
ππ
ππ‘= π1π(1 β
π
πΎ) β πππ,
ππ
ππ‘= πππ β ππ β π
π (π‘)
π + π (π‘)π.
(25)
We can conclude that the positive equilibriumπΈ(πβ, πβ,πβ)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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