Stability and hopf bifurcation analysis on a bazykin model with delay

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Research Article

Stability and Hopf Bifurcation Analysis on

a Bazykin Model with Delay

Jianming Zhang,

1

Lijun Zhang,

1,2

and Chaudry Masood Khalique

2

1_{Department of Mathematics, School of Science, Zhejiang Sci-Tech University, Hangzhou, Zhejiang 310018, China} 2_{International Institute for Symmetry Analysis and Mathematical Modelling, Department of Mathematical Sciences,}

North-West University, Mafikeng Campus, Private Bag X2046, Mmabatho 2735, South Africa

Correspondence should be addressed to Lijun Zhang; li-jun0608@163.com Received 10 January 2014; Accepted 30 January 2014; Published 27 March 2014 Academic Editor: Hossein Jafari

Copyright © 2014 Jianming Zhang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The dynamics of a prey-predator system with a finite delay is investigated. We show that a sequence of Hopf bifurcations occurs at the positive equilibrium as the delay increases. By using the theory of normal form and center manifold, explicit expressions for determining the direction of the Hopf bifurcations and the stability of the bifurcating periodic solutions are derived.

1. Introduction

The theoretical study of predator-prey systems in mathemat-ical ecology has a long history beginning with the famous Lotka-Volterra equations because of their universal existence and importance. One of the ecological models proposed and analyzed by Bazykin [1] is ̇𝑥 1= 𝑥1(1 − 𝑥2) − 𝜀𝑥12, ̇𝑥 2= −𝛾𝑥2+_{𝑛 + 𝑥}𝑥2 2𝑥1𝑥2, (1)

where𝜀, 𝛾, and 𝑛 are positive constants and 𝑥₁ and 𝑥₂ are functions of time representing population densities of prey and predator, respectively. This system can be used to describe the dynamics of the prey-predator system when the non-linearity of predator reproduction and prey competitive are both taken into account. Bazykin [1] pointed out that for the system (1) the degenerate Bogdanov-Takens bifurcation exists when𝛾 = 4/3, 𝑛 = 1/3, and 𝜀 = 1/4 and conjectured that it is a nondegenerate codim 3 bifurcation. Kuznetsov [2] proved the conjecture is correct by using critical (generalized) eigen-vectors of the linearized matrix and its transpose. However, time delays commonly exist in biological system, information transfer system, and so on. Therefore, time delays of one type or another have been incorporated into mathematical

models of population dynamics due to maturation time, capturing time, or other reasons. In general, delay differential equations exhibit much more complicated dynamics than ordinary differential equations since a time delay may lead to changes of stability of equilibrium and the fluctuation of the populations. So far, a great deal of research has been devoted to the delayed predator-prey system. See, for example, the monographs of Cushing [3], Gopalsamy [4], and Kuang [5] for general delayed biological systems and Beretta and Kuang [6, 7], Faria [8], Gopalsamy [9, 10], May [11], Song et al. [12–14], Xiao and Ruan [15], and Liu and Yuan [16] and the references cited therein for studies on delayed prey-predator systems. In the above references, normal form and center manifold theory were one of important methods to study the stability and Hopf bifurcation of the delayed predator-prey systems. Considering the maturation time of the predator, Bazykin [1] becomes the following delayed model:

̇𝑥 1= 𝑥1(1 − 𝑥2(𝑡 − 𝜏)) − 𝜀𝑥21, ̇𝑥 2= −𝛾𝑥2+_{𝑛 + 𝑥}𝑥2 2𝑥1𝑥2. (2)

In this paper, we first discuss the effect of the time𝜏 on the stability of the positive equilibrium of the system (2). Then we investigate the existence of the Hopf bifurcation, the bifurcating direction, and the stability of the bifurcation

Volume 2014, Article ID 539684, 7 pages http://dx.doi.org/10.1155/2014/539684

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periodic solutions by the theory of normal form and center manifold. Explicit expressions for determining the direction of the Hopf bifurcations and the stability of the bifurcation periodic solutions are derived.

2. The Existence of Hopf Bifurcations

In this section, we study the existence of the Hopf bifurcations of system (2). Clearly, when−1 < 𝑛 < 0, system (2) has only one positive equilibrium, that is,

𝐸 (1 + 𝛾𝜀 − √(1 − 𝛾𝜀) 2_{− 4𝜀𝛾𝑛} 2𝜀 , 1 − 𝛾𝜀 + √(1 − 𝛾𝜀)2− 4𝜀𝛾𝑛 2 ) . (3) Let 𝑥(0)₁ = 1 + 𝛾𝜀 − √(1 − 𝛾𝜀) 2_{− 4𝜀𝛾𝑛} 2𝜀 ; 𝑥(0)₂ = 1 − 𝛾𝜀 + √(1 − 𝛾𝜀) 2_{− 4𝜀𝛾𝑛} 2 , (4)

then system (2) becomes

̇𝑥 1= −𝜀𝑥(0)1 (𝑥1− 𝑥(0)1 ) − 𝑥(0)1 (𝑥2(𝑡 − 𝜏) − 𝑥(0)2 ) − (𝑥1− 𝑥(0)1 ) (𝑥2(𝑡 − 𝜏) − 𝑥(0)2 ) − 𝜀(𝑥1− 𝑥(0)1 ) 2 , ̇𝑥 2= 𝑥 (0) 2 𝑛 + 𝑥(0)₂ (𝑥1− 𝑥 (0) 1 ) + (−𝛾 +2𝑛𝑥 (0) 1 𝑥(0)2 + 𝑥(0)1 (𝑥(0)2 ) 2 (𝑛 + 𝑥(0) 2 ) 2 ) (𝑥2− 𝑥(0)2 ) + ⋅ ⋅ ⋅ . (5) By introducing the new variables𝑧₁(𝑡) = 𝑥₁(𝑡) − 𝑥₁(0),𝑧₂(𝑡) = 𝑥₂(𝑡)−𝑥(0)

2 and denoting𝑓(𝑥1, 𝑥2) = −𝛾𝑥2+(𝑥2/(𝑛+𝑥2))𝑥1𝑥2,

system (5) can be rewritten in a simpler form as

̇𝑧 1(𝑡) = −𝛼1𝑧1(𝑡) − 𝛼2𝑧2(𝑡 − 𝜏) − 𝑧1(𝑡) 𝑧2(𝑡 − 𝜏) − 𝜀𝑧21(𝑡) , ̇𝑧 2(𝑡) = 𝑟1𝑧1(𝑡) + 𝑟2𝑧2(𝑡) + ∑_𝑖!𝑗!1 𝑐𝑖𝑗𝑧𝑖1(𝑡) 𝑧2𝑗(𝑡) , (6) where 𝛼1= 𝜀𝑥(0)1 , 𝛼2= 𝑥1(0), 𝑟1= 𝑥 (0) 2 𝑛 + 𝑥(0)₂ , 𝑟₂= −𝛾 + 2𝑛𝑥 (0) 1 𝑥(0)2 + 𝑥(0)1 (𝑥(0)2 ) 2 (𝑛 + 𝑥(0)₂ )2 , (7)

and𝑐_𝑖𝑗 = 𝜕𝑖+𝑗𝑓(𝑥(0)₁ , 𝑥(0)₂ )/𝜕𝑥𝑖₁𝜕𝑥𝑗₂. Then the linearization of system (2) at𝐸 is ̇𝑧 1(𝑡) = −𝛼𝑧1(𝑡) − 𝛼2𝑧2(𝑡 − 𝜏) , ̇𝑧 2(𝑡) = 𝑟1𝑧1(𝑡) + 𝑟2𝑧2(𝑡) . (8) The associated characteristic equation of (8) is given by

󵄨󵄨󵄨󵄨

󵄨󵄨󵄨󵄨𝜆 + 𝛼1 𝛼2𝑒−𝜆𝜏

−𝑟₁ 𝜆 − 𝑟₂󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨 = 0. (9) That is,

𝜆2+ (𝛼₁− 𝑟₂) 𝜆 − 𝛼₁𝑟₂+ 𝛼₂𝑟₁𝑒−𝜆𝜏= 0. (10) The equilibrium𝐸 is stable if all roots of (10) have negative real parts. Clearly, when𝜏 = 0, the characteristic equation (10) becomes

𝜆2+ (𝛼₁− 𝑟₂) 𝜆 − 𝛼₁𝑟₂+ 𝛼₂𝑟₁= 0. (11) By directly computing, we known that𝑟₂ < 0 when −1 < 𝑛 < 0. Therefore all roots of (11) have negative real parts. Obviously,𝜆 = 𝑖𝜔(𝜔> 0) is a root of (10) if and only if𝜔 satisfies

−𝜔2+ 𝑖 (𝛼₁− 𝑟₂) 𝜔 − 𝛼₁𝑟₂+ 𝛼₂𝑟₁𝑒−𝑖𝜔𝜏= 0. (12) Separating the real and imaginary parts, we have

−𝜔2− 𝛼₁𝑟₂+ 𝛼₂𝑟₁cos𝜔 𝜏 = 0, (𝛼₁− 𝑟₂) 𝜔 − 𝛼₂𝑟₁sin𝜔 𝜏 = 0,

(13) which leads to

𝜔4+ (𝛼₁2+ 𝑟₂2) 𝜔2+ 𝛼₁2𝑟2₂− 𝛼2₂𝑟2₁= 0. (14) When𝛼₁2𝑟₂2− 𝛼2₂𝑟₁2< 0, (14) has only one positive root

𝜔_∗= √− (𝛼

2

1+ 𝑟22) + √(𝛼21+ 𝑟22)2− 4 (𝛼12𝑟22− 𝛼22𝑟12)

2 .

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Substituting (15) into (13), we obtain 𝜏_𝑗= 1 𝜔_∗arccos 𝜔_∗2+ 𝛼1𝑟2 𝛼₂𝑟₁ + 2𝑗𝜋 𝜔_∗ , 𝑗 = 0, 1, 2, . . . . (16) Thus, when𝜏 = 𝜏_𝑗, the characteristic equation (10) has a pair of purely imaginary roots±𝑖𝜔_∗.

Lemma 1. Let 𝜆_𝑗(𝜏) = 𝜂_𝑗(𝜏) + 𝑖𝜔_𝑗(𝜏) be the root of (10)

satisfying

𝜂𝑗(𝜏𝑗) = 0, 𝜔𝑗(𝜏𝑗) = 𝜔∗, 𝑗 = 0, 1, 2, . . . , (17)

and then

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Proof. Differentiating both sides of (10) with respect to𝜏, we obtain (2𝜆 + 𝛼₁− 𝑟₂− 𝛼₂𝑟₁𝜏𝑒−𝜆𝜏)𝑑𝜆_𝑑𝜏 = 𝛼₂𝑟₁𝜆𝑒−𝜆𝜏. (19) Therefore, sign{𝑑 (Re (𝜆)) 𝑑𝜏 }𝜏=𝜏𝑗 = sign {Re (𝑑𝜆 𝑑𝜏) −1 } 𝜏=𝜏𝑗

=sign {Re[2𝜔∗+ (𝛼1− 𝑟2)] (cos 𝜆𝜏 + 𝑖 sin 𝜆𝜏) − 𝜏_𝑖𝜔 𝑗𝛼2𝑟1

∗𝛼2𝑟1 } = sign {[[2𝜔∗+ (𝛼1− 𝑟2)] [𝛼1− 𝑟2]] 𝛼2 2𝑟12 } > 0. (20) Thus, the lemma follows.

Therefore, from Lemma1and the relations between roots of (10) and (11) [17], we have the following conclusion.

Lemma 2. When 𝜏 ∈ [0, 𝜏0), all roots of (10) have negative

real parts. When𝜏 = 𝜏₀, all roots of (10) have negative real

parts except±𝑖𝜔_∗. When𝜏 ∈ (𝜏_𝑗, 𝜏_𝑗+1], (10) has2(𝑗 + 1) roots

with positive real parts.

Furthermore, from Lemma 2, the following theorem holds.

Theorem 3. If 𝜏 ∈ [0, 𝜏₀), then the positive equilibrium 𝐸 is

asymptotically stable and unstable if𝜏 > 𝜏₀. If 𝜏 = 𝜏_𝑗, (2)

undergoes a Hopf bifurcation at𝐸.

3. Stability and Direction of

the Hopf Bifurcation

Let𝑢_𝑖 = 𝑧_𝑖(𝜏𝑡) and 𝜏 = 𝜏_𝑗 + 𝜇, where 𝜇 ∈ 𝑅. Then (2) can be written as a functional differential equation in𝐶 = 𝐶([−1, 0], 𝑅2) as

̇𝑢 (𝑡) = 𝐿𝜇(𝑢𝑡) + 𝐹 (𝜇, 𝑢𝑡) , (21)

where𝑢_𝑡(𝜃) = 𝑥(𝑡 + 𝜃) ∈ 𝐶, and 𝐿_𝜇: 𝐶 → 𝑅, 𝐹 : 𝑅 × 𝐶 → 𝑅 are given, respectively, by

𝐿_𝜇(𝜙) = (𝜏(𝑗)_{+ 𝜇) (−𝛼}1 0 𝑟1 𝑟2) ( 𝜙₁(0) 𝜙₂(0)) + (𝜏(𝑗)+ 𝜇) (0 −𝛼2 0 0 ) (𝜙𝜙1₂(−1)(−1)) , (22) 𝐹 (𝜇, 𝜙) = (𝜏(𝑗)+ 𝜇) ( −𝜙₁(0) 𝜙₂(−1) − 𝜀𝜙₁2(0) + h.o.t Σ_𝑖!𝑗!1 𝑐𝑖𝑗𝜙𝑖1(0) 𝜙2𝑗(0) + h.o.t ) , (23) where h.o.t denotes the higher order terms.

From the discussions above, we known that if 𝜇 = 0, then system (21) undergoes a Hopf bifurcation at the zero equilibrium and the associated characteristic equation of system (21) has a pair of simple imaginary roots±𝑖𝜏𝑗𝜔₀.

By the Reiz representation theorem, there exists a func-tion𝜂(𝜃, 𝜇) of bounded variation for 𝜃 ∈ [−1, 0], such that

𝐿_𝜇𝜙 = ∫0

−1𝑑𝜂 (𝜃, 0) 𝜙 (𝜃) for 𝜙 ∈ 𝐶. (24)

In fact, we can choose

𝜂 (𝜃, 𝜇) = (𝜏(𝑗)+ 𝜇) (−𝛼1 0 𝑟1 𝑟2) 𝛿 (𝜃) − (𝜏(𝑗)+ 𝜇) (0 −𝛼2 0 0 ) 𝛿 (𝜃 + 1) , (25) where 𝛿 (𝜃) = {0, 𝜃 ̸= 0,_{1, 𝜃 = 0.} (26) For𝜙 ∈ 𝐶1([−1, 0], 𝑅2), define 𝐴 (𝜇) 𝜙 = { { { { { { { { { 𝑑𝜙 (𝜃) 𝑑𝜃 , 𝜃 ∈ [−1, 0] , ∫0 −1𝑑𝜂 (𝑠, 𝜇) 𝜙 (𝑠) , 𝜃 = 0, 𝑅 (𝜇) 𝜙 = {0, 𝜃 ∈ [−1, 0] , 𝐹 (𝜇, 𝜙) , 𝜃 = 0. (27)

Then we can rewrite (21) as

̇𝑢𝑡= 𝐴 (𝜇) 𝑢𝑡+ 𝑅 (𝜇) 𝑥𝑡, (28) where𝑢_𝑡(𝜃) = 𝑢(𝑡 + 𝜃), 𝜃 ∈ [−1, 0]. For 𝜙 ∈ 𝐶1([0, 1], 𝑅2), define 𝐴∗𝜓 (𝑠) = { { { { { { { { { −𝑑𝜓 (𝑠)_𝑑𝑠 , 𝑠 ∈ [0, 1] , ∫0 −1𝜓 (−𝑡) 𝑑𝜂 (𝑡, 0) , 𝑠 = 0 (29)

and a bilinear inner product ⟨𝜓 (𝑠) , 𝜙 (𝜃)⟩ = 𝜓 (0) 𝜙 (0) − ∫0 −1∫ 𝜃 𝜉=0𝜓 (𝜉 − 𝜃) 𝑑𝜂 (𝜃) 𝜙 (𝜉) 𝑑𝜉, (30)

where𝜂(𝜃) = 𝜂(𝜃, 0). Then 𝐴∗and𝐴(0) are adjoint operators. By the discussion of Section 2, we known that±𝑖𝜔₀𝜏₀ are eigenvalues of𝐴(0). Thus, they are also eigenvalues of 𝐴∗.

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Suppose that𝑞∗(𝑠) = 𝐷(1, 𝛼∗)𝑒𝑖𝑠𝜔0𝜏(𝑗)_{is the eigenvector of}

𝐴(0) corresponding to 𝑖𝜏(𝑗)_𝜔

0. Then,𝐴(0)𝑞(𝜃) = 𝑖𝜏(𝑗)𝜔0𝑞(𝜃).

From the definition of𝐴(0) and (25), we obtain 𝜏(𝑗)(𝑖𝜔+ 𝛼1 𝛼2𝑒−𝑖𝜔0𝜏𝑗 −𝑟₁ 𝑖𝜔₀− 𝑟₂) 𝑞 (0) = (00), (31) which yields 𝑞 (0) = (1, 𝛼)𝑇= (1, 𝑟1 𝑖𝜔₀− 𝑟₂) 𝑇 . (32)

Similarly, it can be verified that𝑞∗(𝑠) = 𝐷(1, 𝛼∗)𝑒𝑖𝑠𝜔0𝜏(𝑗)_{is the}

eigenvector of𝐴∗corresponding to−𝑖𝜔₀𝜏(𝑗), where 𝛼∗= 𝛼1− 𝑖𝜔0

𝑟₁ . (33)

Let⟨𝑞∗(𝑠), 𝑞(𝜃)⟩ = 1; that is, ⟨𝑞∗(𝑠) , 𝑞 (𝜃)⟩ = 𝐷 (1, 𝛼∗_{) (1, 𝛼)}𝑇 − ∫0 −1∫ 0 𝜉 𝐷 (1, 𝛼 ∗_{) 𝑒}−𝑖(𝜉−𝜃)𝜔0𝜏(𝑗)_{𝑑𝜂 (𝜃) (1, 𝛼)}𝑇_𝑒𝑖𝜉𝜔0𝜏(𝑗)_𝑑𝜉 = 𝐷 {1 + 𝛼𝛼∗_{− ∫}0 −1(1, 𝛼 ∗_{) 𝜃𝑒}𝑖𝜃𝜔0𝜏(𝑗)_{𝑑𝜂 (𝜃) (1, 𝛼)}𝑇_} = 𝐷 {1 + 𝛼𝛼∗_{− 𝛼} 2𝛼𝜏(𝑗)𝑒−𝑖𝜔0𝜏 (𝑗) } = 1. (34) Thus, we can choose

𝐷 = 1

1 + 𝛼𝛼∗_{− 𝛼}

2𝛼𝜏(𝑗)𝑒𝑖𝜔0𝜏(𝑗) (35)

such that⟨𝑞∗(𝑠), 𝑞(𝜃)⟩ = 1, ⟨𝑞∗(𝑠), 𝑞(𝜃)⟩ = 0.

Using the same notations as in Hassard et al. [18] and Song et al. [19], we first compute the center manifold𝐶₀at𝜇 = 0. Let𝑥_𝑡be the solution of (21) when𝜇 = 0. Define

𝑧 (𝑡) = ⟨𝑞∗, 𝑥_𝑡⟩,

𝑊 (𝑡, 𝜃) = 𝑥𝑡(𝜃) − (𝑥 (𝑡) 𝑞 (𝜃) + 𝑧 (𝑡) 𝑞 (𝜃))

= 𝑥𝑡(𝜃) − 2 Re {𝑧 (𝑡) 𝑞 (𝜃)} .

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On the center manifold𝐶₀, we have

𝑊 (𝑡, 𝜃) = 𝑊 (𝑧 (𝑡) , 𝑧 (𝑡) , 𝜃) , (37) where 𝑊 (𝑧, 𝑧, 𝜃) = 𝑊20(𝜃)𝑧 2 2 + 𝑊11(𝜃) 𝑧𝑧 + 𝑊₀₂(𝜃)𝑧₂2 + 𝑊₃₀(𝜃)𝑧₆3 + ⋅ ⋅ ⋅ , (38)

where 𝑧 and 𝑧 are local coordinates for center manifold 𝐶₀ in the direction of𝑞∗and𝑞∗. Note that𝑊 is real if 𝑥_𝑡is real. Here we consider only real solutions. For the solution𝑥_𝑡∈ 𝐶₀ of (24), since𝜇 = 0, we have ̇𝑧 = 𝑖𝜏(𝑗)𝜔₀𝑧 + ⟨𝑞∗(𝜃) , 𝐹 (0, 𝑊 (𝑧, 𝑧, 𝜃) + 2 Re {𝑧𝑞 (𝜃)})⟩ = 𝑖𝜏(𝑗)𝜔₀𝑧 + 𝑞∗_{(0) 𝐹 (0, 𝑊 (𝑧, 𝑧, 0) + 2 Re {𝑧𝑞 (0)})} = 𝑖𝜏(𝑗)𝜔₀𝑧 + 𝑞∗_{(0) 𝐹} 0(𝑧, 𝑧) . (39) We rewrite this equation as

̇𝑧 (𝑡) = 𝑖𝜏(𝑗)_𝜔 0𝑧 (𝑡) + 𝑔 (𝑧, 𝑧) (40) with 𝑔 (𝑧, 𝑧) = 𝑞∗_{(0) 𝐹} 0(𝑧, 𝑧) = 𝑔₂₀𝑧2 2 + 𝑔11𝑧𝑧 + 𝑔02 𝑧2 2 + 𝑔21 𝑧2_𝑧 2 + ⋅ ⋅ ⋅ . (41) By (36), we have𝑥_𝑡(𝜃) = (𝑥_1𝑡(𝜃), 𝑥_2𝑡(𝜃)) = 𝑊(𝑡, 𝜃) + 𝑧𝑞(𝜃) + 𝑧 𝑞(𝜃) and 𝑞(𝜃) = (1, 𝛼)𝑇𝑒𝑖𝜃𝜔0𝜏(𝑗)_{, and then}

𝑥_1𝑡(0) = 𝑧 + 𝑧 + 𝑊₂₀(1)(0)𝑧2 2 + 𝑊11(1)(0) 𝑧𝑧 + 𝑊₀₂(1)(0)𝑧₂2 + 𝑜 (|(𝑧, 𝑧)|3) , 𝑥_2𝑡(0) = 𝑧𝛼𝑒−𝑖𝜔0𝜏(𝑗)+ 𝑧 𝛼𝑒𝑖𝜔0𝜏(𝑗)+ 𝑊(2) 20 (−1)𝑧 2 2 + 𝑊₁₁(2)(−1) 𝑧𝑧 + 𝑊₀₂(2)(−1)𝑧₂2 + 𝑜 (|(𝑧, 𝑧)|3) . (42) It follows, together with (23), that

𝑥_2𝑡(0) = 𝛼𝑧 + 𝛼 𝑧 + 𝑊₂₀(2)(0)𝑧₂2 + 𝑊₀₂(2)(0)𝑧₂2 + ⋅ ⋅ ⋅ , 𝑔 (𝑧, 𝑧) = 𝑞∗_{(0) 𝐹} 0(𝑧, 𝑧) = 𝐷𝜏(𝑗)(1, 𝛼∗_{) (}−𝑥1𝑡(0) 𝑥2𝑡(−1) − 𝜀𝑥 2 1𝑡(0) + h.o.t Σ_𝑖!𝑗!1 𝑐_𝑖𝑗𝑥2 1𝑡(0) 𝑥𝑗2𝑡(0) + h.o.t ) = 𝐷𝜏(𝑗)_(−𝑥 1𝑡(0) 𝑥2𝑡(−1) − 𝜀𝑥21𝑡(0) +𝛼∗_∑ 1 𝑖!𝑗!𝑐𝑖𝑗𝑥21𝑡(0) 𝑥2𝑡𝑗 (0) + h.o.t)

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= 𝐷𝜏(𝑗){(−𝛼𝑒−𝑖𝜔0𝜏(𝑗)_{− 𝜀 +}𝛼∗ 2! (𝑐20+ 𝑐02𝛼2)) 𝑧2 + ( − 𝛼𝑒−𝑖𝜔0𝜏(𝑗)_{− 𝛼𝑒}𝑖𝜔0𝜏(𝑗)_{− 2𝜀} +𝛼∗_(𝑐 20+𝑐02₂𝛼𝛼)) 𝑧𝑧 + (−𝛼𝑒𝑖𝜔0𝜏(𝑗)− 𝜀 +𝛼∗𝑐20 2! + 𝛼2_𝛼∗_𝑐 02 2! ) 𝑧2 + (−𝛼 2𝑒𝑖𝜔0𝜏 (𝑗) 𝑊₂₀(1)(0) −1 2𝑊20(2)(−1) − 𝜀𝑊₂₀(1)(0) + 𝛼∗₍1 2𝑐20𝑊20(1)(0) +1 2𝛼𝑐02𝑊20(2)) × 𝑐₂₀𝑊(1) 11 (0) +𝛼𝑐02𝑊(2)11(0) ) 𝑧2𝑧 + ⋅ ⋅ ⋅ } . (43) Comparing the coefficients with (41), we have

𝑔₂₀= 𝐷𝜏(𝑗)(−2𝛼𝑒−𝑖𝜔0𝜏(𝑗)_{− 2𝜀 + 𝛼}∗_(𝑐 20+ 𝑐02𝛼2)) , 𝑔11= 𝐷𝜏(𝑗)(−𝛼𝑒−𝑖𝜔0𝜏 (𝑗) − 𝛼𝑒𝑖𝜔0𝜏(𝑗)_{− 2𝜀} +𝛼∗_(𝑐 20+𝑐02₂𝛼𝛼)) , 𝑔₀₂= 𝐷𝜏(𝑗)(−2𝛼𝑒𝑖𝜔0𝜏(𝑗)_{− 2𝜀 + 𝛼 (𝑐} 20+ 𝛼𝑐02)) , 𝑔₂₁ = −𝛼𝑒𝑖𝜔0𝜏(𝑗)_𝑊(1) 20 (0) − 𝑊20(2)(−1) − 2𝜀𝑊20(1)(0) + 𝛼∗_(𝑐 20𝑊20(1)(0) + 𝑐02𝛼𝑊20(2)(0)) + 2𝑐20𝑊11(1)(0) + 2𝑐₀₂𝛼𝑊₁₁(2)(0) . (44) In order to determine𝑔₂₁, we need to compute𝑊₂₀(𝜃) and 𝑊11(𝜃). From (28) and (36), we have

̇ 𝑊 = ̇𝑥_𝑡− ̇𝑧𝑞 − ̇𝑧 𝑞 ={_{ { 𝐴𝑊 − 2𝑅 {𝑞 ∗ (0)𝐹₀𝑞 (𝜃)} , 𝜃 ∈ [−1, 0] , 𝐴𝑊 − 2𝑅 {𝑞 ∗ (0)𝐹₀𝑞 (𝜃)} + 𝐹0, 𝜃 = 0 ≡ 𝐴𝑊 + 𝐻 (𝑧, 𝑧, 𝜃) , (45) where 𝐻 (𝑧, 𝑧, 𝜃) = 𝐻20(𝜃)𝑧 2 2 + 𝐻11(𝜃) 𝑧𝑧 + 𝐻02(𝜃) 𝑧2 2 + ⋅ ⋅ ⋅ . (46)

Expanding the above series and comparing the correspond-ing coefficients, we obtain

(𝐴 − 2𝑖𝜏(𝑗)𝜔₀) 𝑊₂₀(𝜃) = −𝐻₂₀(𝜃) ,

𝐴𝑊₁₁(𝜃) = −𝐻₁₁(𝜃) , . . . . (47) Following (45), we know that for𝜃 ∈ [−1, 0],

𝐻 (𝑧, 𝑧, 𝜃) = −𝑞∗_(0)𝐹

0𝑞 (𝜃) − 𝑞∗(0) 𝐹0𝑞(𝜃)

= −𝑔𝑞 (𝜃) − 𝑔𝑞(𝜃).

(48)

Comparing the coefficients with (46), we get 𝐻₂₀(𝜃) = −𝑔₂₀𝑞 (𝜃) − 𝑔02𝑞 (𝜃),

𝐻₁₁(𝜃) = −𝑔₁₁𝑞 (𝜃) − 𝑔11𝑞(𝜃).

(49) Substituting these relations into (47), we obtain

̇ 𝑊₂₀(𝜃) = 2𝑖𝜏(𝑗)𝜔₀𝑊₂₀(𝜃) + 𝑔20𝑞 (𝜃) + 𝑔02𝑞(𝜃). (50) Solving𝑊₂₀(𝜃), we obtain 𝑊₂₀(𝜃) =𝑖𝑔20𝑞 (0) 𝜏(𝑗)_𝜔 0 𝑒 𝑖𝜏(𝑗)_𝜔 0𝜃 +𝑖𝑔02𝑞(0) 3𝜏(𝑗)_𝜔 0𝑒 −𝑖𝜏(𝑗)_𝜔 0𝜃_{+ 𝐸} 1𝑒2𝑖𝜏 (𝑗)_𝜔 0𝜃_, (51)

where𝐸₁= (𝐸(1)₁ , 𝐸₁(2)) ∈ 𝑅2is a constant vector. Similarly, we can obtain

𝑊11(𝜃) =−𝑖𝑔_𝜏_(𝑗)11𝑞 (0)_𝜔 0 𝑒 𝑖𝜏(𝑗)_𝜔 0𝜃 +𝑖𝑔11𝑞(0) 𝜏(𝑗)_𝜔 0 𝑒 −𝑖𝜏(𝑗)_𝜔 0𝜃+ 𝐸 2, (52)

where𝐸₂= (𝐸(1)₂ , 𝐸₂(2)) ∈ 𝑅2is also a constant vector. In what follows, we determine the constant vectors𝐸₁and 𝐸₂. From (47) and the definition of𝐴, we obtain

∫0 −1𝑑𝜂 (𝜃) 𝑊20(𝜃) = 2𝑖𝜏 (𝑗)_𝜔 0𝑊20(0) − 𝐻20(0) , (53) ∫0 −1𝑑𝜂 (𝜃) 𝑊11(𝜃) = −𝐻11(𝜃) , (54)

where𝜂(𝜃) = 𝜂(0, 𝜃). From (45) and (46), we have 𝐻₂₀(0) = −𝑔₂₀𝑞 (0) − 𝑔02𝑞(0) + 2𝜏(𝑗)(−2𝛼𝑒 −𝑖𝜔0𝜏(𝑗)_{− 2𝜀} 𝑐₂₀+ 𝑐₀₂𝛼2 ) , (55) 𝐻₁₁(0) = −𝑔₁₁𝑞 (0) − 𝑔11𝑞(0) + 2𝜏(𝑗)(−2𝛼𝑒−𝑖𝜔0𝜏 (𝑗) − 𝛼𝑒𝑖𝜔0𝜏(𝑗)− 2𝜀 𝑐₂₀+ 𝑐₀₂𝛼𝛼 ) . (56)

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Substituting (51) and (55) into (53) and noticing that (𝑖𝜏(𝑗)𝜔₀𝐼 − ∫0 −1𝑒 𝑖𝜃𝜔0𝜏(𝑗)_{𝑑𝜂 (𝜃)) 𝑞 (0) = 0,} (−𝑖𝜏(𝑗)𝜔₀𝐼 − ∫0 −1𝑒 −𝑖𝜃𝜔0𝜏(𝑗)_{𝑑𝜂 (𝜃)) 𝑞 (0) = 0,} (57) we get (2𝑖𝜏(𝑗)𝜔₀𝐼 − ∫0 −1𝑒 2𝑖𝜃𝜔0𝜏(𝑗)_{𝑑𝜂 (𝜃)) 𝐸} 1 = 2𝜏(𝑗)(−2𝛼𝑒−𝑖𝜔0𝜏 (𝑗) − 2𝜀 𝑐₂₀+ 𝑐₀₂𝛼2 ) ; (58) that is, (2𝑖𝜔0+ 𝛼1 −𝛼2𝑒−2𝑖𝜔0𝜏 (𝑗) −𝑟₁ 2𝑖𝜔₀ ) 𝐸1= 2 (−2𝛼𝑒 −𝑖𝜔0𝜏(𝑗)− 2𝜀 𝑐₂₀+ 𝑐₀₂𝛼2 ) . (59) It follows that 𝐸(1)₁ = _𝐴2 󵄨󵄨󵄨󵄨_{󵄨󵄨󵄨󵄨} 󵄨 −2𝛼𝑒−𝑖𝜔0𝜏(𝑗)− 2𝜀 −𝛼 2𝑒−2𝑖𝜔0𝜏 (𝑗) 𝑐₂₀+ 𝑐₀₂𝛼2 2𝑖𝜔₀ 󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨_󵄨, 𝐸(2)₁ = 2 𝐴󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨_󵄨2𝑖𝜔0+ 𝛼1 −2𝛼𝑒 −𝑖𝜔0𝜏(𝑗)− 2𝜀 −𝑟₁ 𝑐₂₀+ 𝑐₀₂𝛼2 󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨 󵄨, (60) where𝐴 =󵄨󵄨󵄨󵄨_󵄨󵄨2𝑖𝜔0+𝛼1−𝛼2𝑒−2𝑖𝜔0𝜏(𝑗) −𝑟1 2𝑖𝜔0 󵄨󵄨󵄨󵄨󵄨󵄨.

Similarly, substituting (52) and (56) into (54), we have ( 𝛼1 𝛼2 −𝑟₁ −𝑟₂) 𝐸2= 2 (−2𝛼𝑒 −𝑖𝜔0𝜏(𝑗)− 𝛼𝑒𝑖𝜔0𝜏(𝑗)− 2𝜀 𝑐₂₀+ 𝑐₀₂𝛼𝛼 ) . (61) Then we obtain 𝐸(1)₂ = 2 𝐵󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨_󵄨−2𝛼𝑒 −𝑖𝜔0𝜏(𝑗)_{− 𝛼𝑒}𝑖𝜔0𝜏(𝑗)_{− 2𝜀} _−𝛼 2 𝑐20+ 𝑐02𝛼𝛼 −𝑟2 󵄨󵄨󵄨󵄨 󵄨󵄨󵄨󵄨 󵄨, 𝐸(2)₂ = 2 𝐵󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨_󵄨𝛼1 −2𝛼𝑒 −𝑖𝜔0𝜏(𝑗)− 𝛼𝑒𝑖𝜔0𝜏(𝑗)− 2𝜀 −𝑟₁ 𝑐₂₀+ 𝑐₀₂𝛼𝛼 󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨_󵄨, (62) where𝐵 = 󵄨󵄨󵄨󵄨𝛼1 𝛼2 −𝑟1−𝑟2󵄨󵄨󵄨󵄨.

Therefore, all 𝑔_𝑖𝑗 in (41) have been expressed in terms of the parameters and the delay given in (2). Substituting expressions of 𝑔₀₂, 𝑔₁₁, 𝑔₂₀, and 𝑔₂₁ into the following relations, 𝐶₁(0) = _2𝜔𝑖 0𝜏0(𝑔20𝑔11− 2󵄨󵄨󵄨󵄨𝑔11󵄨󵄨󵄨󵄨 2₋1 3󵄨󵄨󵄨󵄨𝑔02󵄨󵄨󵄨󵄨 2₊𝑔21 2 ) , (63) we obtain 𝐾₂= −Re{𝑐1(0)} Re{𝜆󸀠(𝜏)}, 𝛽2= 2 Re {𝐶1(0)} , 𝑇₂= −Im{𝐶1(0)} + 𝐾2Im{𝜆 󸀠_(𝜏(𝑗)_)} 𝜏(𝑗)_𝜔 0 . (64)

We follow the idea in Hassard et al. [18] and Song et al. [19], which implies that the direction of the Hopf bifurcation is determined by the sign of 𝛽₂, and the stability of the bifurcating periodic solutions is determined by the sign of 𝐾₂and𝑇₂determines the period of the bifurcating periodic solution. Thus we have the following.

Theorem 4. (1) If 𝐾2> 0 (𝐾2< 0), then the Hopf bifurcation

is supercritical (subcritical) and the bifurcating periodic solu-tions exist for𝜏 > 𝜏(𝑗) (𝜏 < 𝜏(𝑗)).

(2) If𝛽₂ < 0 (𝛽₂ > 0), then the bifurcating periodic solu-tions are stable (unstable).

(3) If𝑇₂ > 0 (𝑇₂ < 0), then the periodic of the bifurcating periodic solutions increase (decrease).

4. Conclusions

In the paper, we focused on the effect of the maturation time of the predator in Bazykin [1]. We first discussed the effect of the time𝜏 on the stability of the positive equilibrium of the system (2), and then we investigated the existence of the Hopf bifurcation, the bifurcating direction, and the stability of the bifurcating periodic solutions by the normal form and center manifold. In fact, we can also incorporate other time delays such as capturing time into the mathematical model and look at their dynamics by other methods. In this regard, we can obtain other complicated and interesting results.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This work is supported by the Natural Science Foundation of China (no. 11101371) and the Scientific Research Foundation of Zhejiang Sci-Tech University (13062176-Y). The authors would like to thank the anonymous reviewers for their suggestions and comments.

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Stability and hopf bifurcation analysis on a bazykin model with delay

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