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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
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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 [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
𝐾) − 𝑎𝑃𝑍,
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.
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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]
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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
𝐾) − 𝑎𝑃𝑍,
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
𝐾) − 𝑎𝑃𝑍,
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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