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 ยฉ 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 [4โ9]. 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]
๐๐
๐๐ก= ๐1๐(1 โ
๐
๐พ) โ ๐๐๐,
๐๐
๐๐ก= ๐๐โซ
๐ก
โโ
๐บ (๐ก โ ๐ ) ๐ (๐ ) ๐๐ โ ๐๐ โ ๐๐ (๐ก โ ๐)
๐ + ๐ (๐ก โ ๐)๐,
(1)
where ๐(๐ก) and ๐(๐ก) are the densities of phytoplankton andzooplankton, respectively. ๐
1, ๐พ, ๐, ๐, ๐, ๐, and ๐ are positive
constants. ๐ is toxin delay, ๐บ(๐ ) is the delay kernel and a non-negative bounded function defined on [0,โ] as follows:
โซ
โ
0
๐บ (๐ ) ๐๐ = 1, ๐บ (๐ ) = ๐๐โ๐๐
, ๐ > 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 [12โ16].
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 ๐ 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 ๐ธโ(๐โ, ๐โ), where
๐โ=
๐ + ๐ โ ๐๐ + โ(๐ + ๐ โ ๐๐)2+ 4๐๐๐
2๐,
๐โ=
๐
๐(1 โ
๐โ
๐พ) > 0,
(3)
where (๐ป1) : ๐พ > ๐
โ.Let
๐(๐ก) = โซ
๐ก
โโ
๐๐โ๐(๐กโ๐ )
๐ (๐ ) ๐๐ . (4)
By the linear chain trick technique, then system (1) can betransformed into the following system:
๐๐
๐๐ก= ๐1๐(1 โ
๐
๐พ) โ ๐๐๐,
๐๐
๐๐ก= ๐๐๐ โ ๐๐ โ ๐
๐ (๐ก โ ๐)
๐ + ๐ (๐ก โ ๐)๐,
๐๐
๐๐ก= ๐๐ (๐ก) โ ๐๐ (๐ก) .
(5)
It is easy to check that system (5) has an unique positiveequilibrium ๐ธ(๐โ, ๐โ,๐โ) with ๐โ = ๐โ provided that thecondition (๐ป
1) holds.
Let ๐ = ๐ข+๐ขโ,๐ = V+ Vโ, and๐ = ๐ค+๐โ; then system(5) can be transformed into the following system:
๏ฟฝฬ๏ฟฝ (๐ก) = โ๐
๐พ๐โ๐ข (๐ก) โ ๐๐
โV (๐ก) โ ๐๐ข (๐ก) V (๐ก) โ๐
๐พ๐ข2(๐ก) ,
Vฬ (๐ก) = โ๐๐
(๐ + ๐โ)๐โ๐ข (๐ก โ ๐) + ๐๐
โ๐ค (๐ก)
+ โ
๐+๐+๐โฅ2
๐(๐๐๐)
2๐ข๐(๐ก โ ๐) V๐๐ค๐,
๏ฟฝฬ๏ฟฝ (๐ก) = ๐๐ข (๐ก) โ ๐๐ค (๐ก) ,
(6)
where
๐(๐๐๐)
2=
1
๐!๐!๐!
๐๐+๐+๐
๐2
๐๐ข๐ (๐ก โ ๐) ๐V๐๐๐ค๐,
๐2= ๐V๐ค โ ๐V โ ๐
๐ข (๐ก โ ๐)
๐ + ๐ข (๐ก โ ๐)V.
(7)
Then linearizing system (6) at ๐ธโ(๐โ, ๐โ,๐โ) is
๏ฟฝฬ๏ฟฝ (๐ก) = โ๐
๐พ๐โ๐ข (๐ก) โ ๐๐
โV (๐ก) ,
Vฬ (๐ก) = โ๐๐
(๐ + ๐โ)๐โ๐ข (๐ก โ ๐) + ๐๐
โ๐ค (๐ก) ,
๏ฟฝฬ๏ฟฝ (๐ก) = ๐๐ข (๐ก) โ ๐๐ค (๐ก) .
(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
๐น (๐) = ๐3+ ๐2๐2+ ๐1๐ + ๐0โ [๐1๐ + ๐0] ๐โ๐๐
= 0, (9)
where
๐2= ๐ +
๐๐โ
๐พ, ๐
1=
๐๐โ๐
๐พ, ๐
0= ๐๐๐
โ๐โ๐,
๐1=
๐๐๐๐โ๐โ
(๐ + ๐โ)2, ๐
0=
๐๐๐๐โ๐โ๐
(๐ + ๐โ)2
.
(10)
Obviously, ๐2> 0, ๐
1> 0, ๐
0> 0, ๐
1> 0, and ๐
0> 0.
From (9), the following lemma is obvious.
Lemma 1. If the condition ๐ป2: ๐0= ๐0holds, then ๐ = 0 is
always a root of (9) for all ๐ โฅ 0.
Let ๐0= (๐/๐)(๐+๐
โ)2, ๐1= ๐1(๐+๐โ)2/(1โ๐๐)๐๐๐
โ๐โ,
and ๐2= (๐+๐
โ)2[๐2๐๐๐โ๐โ๐โ2(๐+(๐๐
โ/๐พ))]/2๐๐๐๐
โ๐โ;
we have the following results.
Lemma 2. Suppose that the condition (๐ป2) is satisfied.
(i) If ๐ = ๐0
ฬธ= ๐1, then (9) has a single zero root.
(ii) If ๐ = ๐1
ฬธ= ๐2, then (9) has a double zero root.
Proof. Clearly, ๐ = 0 is a root to (9) if and only if ๐0= ๐0,
whichmeans๐ = ๐0. Substituting๐ = ๐
0into๐น(๐) and taking
the derivative with respect to ๐, we obtain
๐น(๐)
๐=๐0= 3๐2+ 2๐2๐ + ๐1โ [๐1(1 โ ๐๐) โ ๐๐
0] ๐โ๐๐
.
(11)
Then we can get
๐น(0)
๐=๐0= ๐1โ [๐1โ ๐๐0] . (12)
For any ๐ > 0, by solving (12), we can obtain ๐ = ๐1. If ๐ =
๐0
ฬธ= ๐1, ๐น(0) ฬธ= 0which means that ๐ = 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
๐น(๐)
๐=๐1= 6๐ + 2๐
2โ [๐1(โ2๐ + ๐
2๐) + ๐
2๐0] ๐โ๐๐
.
(13)
Then we get
๐น(0)
๐=๐1= 2๐2โ [๐2๐0โ 2๐1๐] . (14)
For any ๐ > 0, by solving (14), we can obtain ๐ = ๐2. If
๐ = ๐1
ฬธ= ๐2, ๐น(0) ฬธ= 0 which means that ๐ = 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 (๐ป2) holds if ๐ = ๐
0ฬธ= ๐1, 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 ยฑ๐๐(๐ > 0), when ๐ = ๐
0ฬธ= ๐1holds.
Substituting ๐ = ๐๐ (๐ > 0) and ๐ = ๐0into (9) and
separating the real and imaginary parts, one can get
โ๐3+ ๐1๐ โ ๐1๐ cos (๐๐) + ๐
0sin (๐๐) = 0,
โ๐2๐2+ ๐0โ ๐1๐ sin (๐๐) โ ๐
0cos (๐๐) = 0.
(15)
It is easy to see from (15) that
๐6+ ๐ท2๐4+ ๐ท1๐2+ ๐ท0= 0, (16)
where
๐ท2= ๐2
2โ 2๐1, ๐ท
1= ๐2
1โ 2๐0๐2โ ๐2
1,
๐ท0= ๐2
0โ ๐2
0.
(17)
Let ๐ง = ๐2. Then (16) can be written as
โ (๐ง) = ๐ง3+ ๐ท2๐ง2+ ๐ท1๐ง + ๐ท
0. (18)
In terms of the coefficient in โ(๐ง) define ฮ by ฮ = ๐ท22โ 3๐ท1.
It is easy to know from the characters of cubic algebraicequation that โ(๐ง) is a strictly monotonically increasingfunction ifฮ โค 0. Ifฮ > 0 and ๐งโ = (โฮโ๐ท
2)/3 < 0 orฮ > 0,
๐งโ
= (โฮ โ ๐ท2)/3 > 0 but โ(๐งโ) > 0, then โ(๐ง) has always
no positive root. Therefore, under these conditions, (9) hasno purely imaginary roots for any ๐ > 0 and this also impliesthat the positive equilibrium ๐ธ(๐โ, ๐โ,๐โ) of system (1) isabsolutely stable. Thus, we can obtain easily the followingresult on the stability of positive equilibrium ๐ธ(๐โ, ๐โ,๐โ)of system (1).
Theorem 4. Assume that (๐ป1) holds and ฮ โค 0 or ฮ > 0
and ๐งโ = (โฮ โ ๐ท2)/3 < 0 or ฮ > 0, ๐งโ > 0 and โ(๐งโ) >
0. Then the positive equilibrium ๐ธ(๐โ, ๐โ,๐โ) of system (5)is absolutely stable; namely; ๐ธ(๐โ, ๐โ,๐โ) is asymptoticallystable for any delay ๐ โฅ 0.
In what follows, we assume that the coefficients in โ(๐ง)satisfy the condition
(๐ป3) ฮ = ๐ท
2
2โ 3๐ท1> 0, ๐งโ = (โฮ โ ๐ท
2)/3 > 0, โ(๐งโ) < 0.
Then, according to Lemma 2.2 in [17], we know that (16) hasat least a positive root ๐
0; that is, the characteristic equation
(9) has a pair of purely imaginary roots ยฑ๐๐0. Eliminating
sin(๐๐) in (15), we can get that the corresponding ๐๐> 0 such
that (9) has a pair of purely imaginary roots ยฑ๐๐0, ๐๐> 0 are
given by
๐๐=
1
๐0
arccos[โ๐1๐4
0+ (๐1๐1โ ๐2๐0) ๐2
0+ ๐0๐0
๐21๐20+ ๐20
]
+2๐๐
๐0
, (๐ = 0, 1, 2, . . .) .
(19)
Let ๐(๐) = V(๐) + ๐๐(๐) be the roots of (9) such that when๐ = ๐๐satisfying V(๐
๐) = 0 and ๐(๐
๐) = ๐0. We can claim that
sgn [๐ (Re ๐)๐๐
]
๐=๐๐
= sgn {โ (๐20)} . (20)
In fact, differentiating two sides of (9) with respect to ๐, weget
(๐๐
๐๐)
โ1
= โ(3๐2+ 2๐2๐ + ๐1) โ ๐1๐โ๐๐
+ (๐1๐ + ๐0) ๐๐โ๐๐
(๐1๐ + ๐0) ๐๐โ๐๐
= โ(3๐2+ 2๐2๐ + ๐1) ๐๐๐
(๐1๐ + ๐0) ๐
+๐1
(๐1๐ + ๐0) ๐
โ๐
๐.
(21)
Then
sgn [๐ (Re ๐)๐๐
]
๐=๐๐
= sgn[Re(๐๐๐๐
)
โ1
]
๐=๐๐0
= sgn[Re(โ(3๐2+ 2๐2๐ + ๐1) ๐๐๐
(๐1๐ + ๐0) ๐
+๐1
(๐1๐ + ๐0) ๐
โ๐
๐)]
๐=๐๐0
= sgn Re[โ(๐1โ 3๐2
0+ 2๐2๐0๐) [cos (๐
0๐๐) + ๐ sin (๐
0๐๐)]
(๐1๐0๐ + ๐0) ๐0๐
+๐1
(๐1๐0๐ + ๐0) ๐0๐]
4 Abstract and Applied Analysis
= sgn 1ฮ
{[(๐1โ 3๐2
0) cos (๐
0๐๐) โ 2๐
2๐0sin (๐
0๐๐)]
ร (๐1๐2
0)
โ [(๐1โ 3๐2
0) sin (๐
0๐๐) + 2๐
2๐0cos (๐
0๐๐)]
ร ๐0๐0โ ๐2
1๐2
0}
= sgn 1ฮ
{(3๐2
0โ ๐1) ๐0[๐1๐0cos (๐
0๐๐) โ ๐0sin (๐
0๐๐)]
โ 2๐2๐2
0[๐1๐0sin (๐
0๐๐) + ๐0cos (๐
0๐๐)] โ ๐
2
1๐2
0}
= sgn 1ฮ
[3๐6
0+ 2 (๐
2
2โ ๐1) ๐4
0+ (๐2
1โ 2๐0๐2โ ๐2
1) ๐2
0]
= sgn๐2
0
ฮ[3๐4
0+ 2๐ท2๐2
0+ ๐ท1]
= sgn๐2
0
ฮ{โ(๐2
0)} = sgn {โ (๐2
0)} ,
(22)
where ฮ = ๐21๐4
0+ ๐2
0๐2
0. It follows from the hypothesis
(๐ป3) that โ(๐2
0) ฬธ= 0 and therefore the transversality condition
holds.
Lemma5. All the roots of (9), except a zero root, have negativereal parts when ๐
1> ๐1; (i) of Lemma 2 and ๐ โ [0, ๐
0) hold.
Proof. Consider
๐3+ ๐2๐2+ (๐1โ ๐1) ๐ + ๐
0โ ๐0= ๐ (๐
2+ ๐2๐ + ๐1โ ๐1) .
(23)
It is easy to get that the roots of (23) are ๐1= 0 and ๐
2,3=
(โ๐2ยฑโ๐22โ 4(๐1โ ๐1))/2. If๐
1โ๐1> 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 (๐ป1), (๐ป2), and (๐ป
3)
are satisfied.
(i) If ๐ = ๐0
ฬธ= ๐1and ๐ โ [0, ๐
0), then the system (1)
undergoes a Pitchfork bifurcation at positive equilib-rium ๐ธโ.
(ii) If ๐ = ๐0
ฬธ= ๐1and ๐ = ๐
0, then system (1) can undergo
aHopf-Pitchfork bifurcation at the positive equilibrium๐ธโ.
4. Conclusions
In this section, we present some particular cases of system (1)as follows:
๐๐
๐๐ก= ๐1๐(1 โ
๐
๐พ) โ ๐๐๐,
๐๐
๐๐ก= ๐๐๐ โ ๐๐ โ ๐
๐ (๐ก โ ๐)
๐ + ๐ (๐ก โ ๐)๐.
(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 ๐ = 0, system (24) reduces to
๐๐
๐๐ก= ๐1๐(1 โ
๐
๐พ) โ ๐๐๐,
๐๐
๐๐ก= ๐๐๐ โ ๐๐ โ ๐
๐ (๐ก)
๐ + ๐ (๐ก)๐.
(25)
We can conclude that the positive equilibrium๐ธ(๐โ, ๐โ,๐โ)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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