Stability and Hopf Bifurcation Analysis on a Bazykin Model with Delay

Abstract

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 mathematical 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 𝑥̇₁=𝑥₁1−𝑥₂−𝜀𝑥²₁,𝑥̇₂=−𝛾𝑥₂+𝑥₂𝑛+𝑥₂𝑥₁𝑥₂, (1) where 𝜀, 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 nonlinearity 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 when 𝛾=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) eigenvectors 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: 𝑥̇₁=𝑥₁1−𝑥₂(𝑡−𝜏)−𝜀𝑥²₁,𝑥̇₂=−𝛾𝑥₂+𝑥₂𝑛+𝑥₂𝑥₁𝑥₂. (2)

In this paper, we first discuss the effect of the time 𝜏 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 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, 𝐸1+𝛾𝜀−1−𝛾𝜀²−4𝜀𝛾𝑛,2𝜀1−𝛾𝜀+1−𝛾𝜀²−4𝜀𝛾𝑛2. (3) Let 𝑥₁⁽⁰⁾=1+𝛾𝜀−1−𝛾𝜀²−4𝜀𝛾𝑛;𝑥2𝜀₂₍₀₎=1−𝛾𝜀+1−𝛾𝜀²−4𝜀𝛾𝑛2, (4) then system (2) becomes 𝑥̇₁=−𝜀𝑥₁⁽⁰⁾𝑥₁−𝑥₁⁽⁰⁾−𝑥₁⁽⁰⁾𝑥₂(𝑡−𝜏)−𝑥₂⁽⁰⁾−𝑥₁−𝑥₁⁽⁰⁾𝑥₂(𝑡−𝜏)−𝑥₂⁽⁰⁾−𝜀𝑥₁−𝑥₁⁽⁰⁾²,𝑥̇₂=𝑥₂⁽⁰⁾𝑛+𝑥₂⁽⁰⁾𝑥₁−𝑥₁⁽⁰⁾+−𝛾+2𝑛𝑥₁⁽⁰⁾𝑥₂⁽⁰⁾+𝑥₁⁽⁰⁾𝑥₂⁽⁰⁾²𝑛+𝑥₂⁽⁰⁾²𝑥₂−𝑥₂⁽⁰⁾+⋯. (5) By introducing the new variables 𝑧₁(𝑡)=𝑥₁(𝑡)−𝑥₁⁽⁰⁾, 𝑧₂(𝑡)=𝑥₂(𝑡)−𝑥₂⁽⁰⁾ and denoting 𝑓(𝑥₁,𝑥₂)=−𝛾𝑥₂+(𝑥₂/(𝑛+𝑥₂))𝑥₁𝑥₂, system (5) can be rewritten in a simpler form as 𝑧̇₁(𝑡)=−𝛼₁𝑧₁(𝑡)−𝛼₂𝑧₂(𝑡−𝜏)−𝑧₁(𝑡)𝑧₂(𝑡−𝜏)−𝜀𝑧²₁(𝑡),𝑧̇₂(𝑡)=𝑟₁𝑧₁(𝑡)+𝑟₂𝑧₂(𝑡)1+𝑐𝑖!𝑗!^𝑖𝑗𝑧^𝑖₁(𝑡)𝑧^𝑗₂(𝑡), (6) where 𝛼₁=𝜀𝑥₁⁽⁰⁾,𝛼₂=𝑥₁⁽⁰⁾,𝑟₁=𝑥₂⁽⁰⁾𝑛+𝑥₂⁽⁰⁾,𝑟₂=−𝛾+2𝑛𝑥₁⁽⁰⁾𝑥₂⁽⁰⁾+𝑥₁⁽⁰⁾𝑥₂⁽⁰⁾²𝑛+𝑥₂⁽⁰⁾², (7) and 𝑐_𝑖𝑗=𝜕^𝑖+𝑗𝑓(𝑥₁⁽⁰⁾,𝑥₂⁽⁰⁾)/𝜕𝑥^𝑖₁𝜕𝑥^𝑗₂. Then the linearization of system (2) at 𝐸 is 𝑧̇₁(𝑡)=−𝛼𝑧₁(𝑡)−𝛼₂𝑧₂(),𝑧̇𝑡−𝜏₂(𝑡)=𝑟₁𝑧₁(𝑡)+𝑟₂𝑧₂(𝑡). (8) The associated characteristic equation of (8) is given by 𝜆+𝛼₁𝛼₂𝑒^−𝜆𝜏−𝑟₁𝜆−𝑟₂=0. (9) That is, 𝜆²+𝛼₁−𝑟₂𝜆−𝛼₁𝑟₂+𝛼₂𝑟₁𝑒^−𝜆𝜏=0. (10) The equilibrium 𝐸 is stable if all roots of (10) have negative real parts. Clearly, when 𝜏=0, the characteristic equation (10) becomes 𝜆²+𝛼₁−𝑟₂𝜆−𝛼₁𝑟₂+𝛼₂𝑟₁=0. (11) By directly computing, we known that 𝑟₂<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 if 𝜔 satisfies −𝜔²+𝑖𝛼₁−𝑟₂𝜔−𝛼₁𝑟₂+𝛼₂𝑟₁𝑒^{−𝑖𝜔𝜏}=0. (12) Separating the real and imaginary parts, we have −𝜔²−𝛼₁𝑟₂+𝛼₂𝑟₁𝛼cos𝜔𝜏=0,₁−𝑟₂𝜔−𝛼₂𝑟₁sin𝜔𝜏=0, (13) which leads to 𝜔⁴+𝛼²₁+𝑟²₂𝜔²+𝛼²₁𝑟²₂−𝛼²₂𝑟²₁=0. (14) When 𝛼²₁𝑟²₂−𝛼²₂𝑟²₁<0, (14) has only one positive root 𝜔_∗=−𝛼²₁+𝑟²₂+𝛼²₁+𝑟²₂²−4𝛼²₁𝑟²₂−𝛼²₂𝑟²₁2. (15) Substituting (15) into (13), we obtain 𝜏_𝑗=1𝜔_∗𝜔arccos^2∗+𝛼₁𝑟₂𝛼₂𝑟₁+2𝑗𝜋𝜔_∗,𝑗=0,1,2,…. (16) Thus, when 𝜏=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 𝜂_𝑗𝜏_𝑗=0,𝜔_𝑗𝜏_𝑗=𝜔_∗,𝑗=0,1,2,…, (17) and then 𝜂^′_𝑗𝜏_𝑗>0. (18)

Proof. Differentiating both sides of (10) with respect to 𝜏, we obtain 2𝜆+𝛼₁−𝑟₂−𝛼₂𝑟₁𝜏𝑒^−𝜆𝜏𝑑𝜆𝑑𝜏=𝛼₂𝑟₁𝜆𝑒^−𝜆𝜏. (19) Therefore, 𝑑((𝜆signRe))𝑑𝜏_{𝜏=𝜏𝑗}=signRe𝑑𝜆𝑑𝜏⁻¹_{𝜏=𝜏𝑗}=signRe2𝜔_∗+𝛼₁−𝑟₂(cos𝜆𝜏+𝑖sin𝜆𝜏)−𝜏_𝑗𝛼₂𝑟₁𝑖𝜔_∗𝛼₂𝑟₁=sign2𝜔_∗+𝛼₁−𝑟₂𝛼₁−𝑟₂𝛼²₂𝑟²₁>0. (20) Thus, the lemma follows.

Therefore, from Lemma 1 and 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. When 𝜏=𝜏₀, all roots of (10) have negative real parts except ±𝑖𝜔_∗. When 𝜏∈(𝜏_𝑗,𝜏_𝑗+1], (10) has 2(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 if 𝜏>𝜏₀. If 𝜏=𝜏_𝑗, (2) undergoes a Hopf bifurcation at 𝐸.

3. Stability and Direction of the Hopf Bifurcation

Let 𝑢_𝑖=𝑧_𝑖(𝜏𝑡) and 𝜏=𝜏_𝑗+𝜇, where 𝜇∈𝑅. Then (2) can be written as a functional differential equation in 𝐶=𝐶([−1,0],𝑅²) as 𝑢̇(𝑡)=𝐿_𝜇𝑢_𝑡+𝐹𝜇,𝑢_𝑡, (21) where 𝑢_𝑡(𝜃)=𝑥(𝑡+𝜃)∈𝐶, and 𝐿_𝜇∶𝐶→𝑅, 𝐹∶𝑅×𝐶→𝑅 are given, respectively, by 𝐿_𝜇𝜙=𝜏^(𝑗)+𝜇−𝛼₁0𝑟₁𝑟₂𝜙₁(0)𝜙₂(0)+𝜏^(𝑗)+𝜇0−𝛼₂𝜙00¹()𝜙−1₂(),𝐹=𝜏−1𝜇,𝜙^(𝑗)+𝜇−𝜙₁(0)𝜙₂(−1)−𝜀𝜙²₁(0)Σ1+h.o.t𝑐𝑖!𝑗!^𝑖𝑗𝜙^𝑖₁(0)𝜙^𝑗₂(0)+h.o.t, (22)(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 ±𝑖𝜏^𝑗𝜔₀.

By the Reiz representation theorem, there exists a function 𝜂(𝜃,𝜇) of bounded variation for 𝜃∈[−1,0], such that 𝐿_𝜇𝜙=⁰₋₁𝑑𝜂(𝜃,0)𝜙(𝜃)for𝜙∈𝐶. (24) In fact, we can choose 𝜂𝜃,𝜇=𝜏^(𝑗)+𝜇−𝛼₁0𝑟₁𝑟₂𝛿(𝜃)−𝜏^(𝑗)+𝜇0−𝛼₂𝛿(),00𝜃+1 (25) where 𝛿(𝜃)=0,𝜃≠0,1,𝜃=0. (26) For 𝜙∈𝐶¹([−1,0],𝑅²), define 𝐴𝜇(𝜃)𝜙=𝑑𝜙[],∫𝑑𝜃,𝜃∈−1,0₀₋₁𝜙(𝑠)𝑅𝜇[],𝐹𝑑𝜂𝑠,𝜇,𝜃=0,𝜙=0,𝜃∈−1,0𝜇,𝜙,𝜃=0. (27) Then we can rewrite (21) as 𝑢̇_𝑡=𝐴𝜇𝑢_𝑡+𝑅𝜇𝑥_𝑡, (28) where 𝑢_𝑡(𝜃)=𝑢(𝑡+𝜃),𝜃∈[−1,0]. For 𝜙∈𝐶¹([0,1],𝑅²), define 𝐴^∗𝜓(𝑠)=−𝑑𝜓(𝑠)[],∫𝑑𝑠,𝑠∈0,1₀₋₁𝜓(−𝑡)𝑑𝜂(𝑡,0),𝑠=0 (29) and a bilinear inner product (𝑠)(𝜃)⟩=⟨𝜓,𝜙𝜓(0)𝜙(0)−⁰₋₁^𝜃_𝜉=0𝜓(𝜉−𝜃)𝑑𝜂(𝜃)𝜙(𝜉)𝑑𝜉, (30) where 𝜂(𝜃)=𝜂(𝜃,0). Then 𝐴^∗ and 𝐴(0) are adjoint operators. By the discussion of Section 2, we known that ±𝑖𝜔₀𝜏₀ are eigenvalues of 𝐴(0). Thus, they are also eigenvalues of 𝐴^∗.

Suppose that 𝑞^∗(𝑠)=𝐷(1,𝛼^∗)𝑒^{𝑖𝑠𝜔0𝜏(𝑗)} is the eigenvector of 𝐴(0) corresponding to 𝑖𝜏^(𝑗)𝜔₀. Then, 𝐴(0)𝑞(𝜃)=𝑖𝜏^(𝑗)𝜔₀𝑞(𝜃). From the definition of 𝐴(0) and (25), we obtain 𝜏^(𝑗)𝑖𝜔+𝛼₁𝛼₂𝑒^{−𝑖𝜔0𝜏𝑗}−𝑟₁𝑖𝜔₀−𝑟₂𝑞(0)=00, (31) which yields 𝑞(0)=(1,𝛼)^𝑇=𝑟1,¹𝑖𝜔₀−𝑟₂^𝑇. (32) Similarly, it can be verified that 𝑞^∗(𝑠)=𝐷(1,𝛼^∗)𝑒^{𝑖𝑠𝜔0𝜏(𝑗)} is the eigenvector of 𝐴^∗ corresponding to −𝑖𝜔₀𝜏^(𝑗), where 𝛼^∗=𝛼₁−𝑖𝜔₀𝑟₁. (33) Let ⟨𝑞^∗(𝑠),𝑞(𝜃)⟩=1; that is, 𝑞^∗(𝑠)(𝜃=,𝑞)𝐷1,𝛼^∗(1,𝛼)^𝑇−⁰₋₁⁰_𝜉𝐷1,𝛼^∗𝑒^{−𝑖(𝜉−𝜃)𝜔0𝜏(𝑗)}𝑑𝜂(𝜃)(1,𝛼)^𝑇𝑒^{𝑖𝜉𝜔0𝜏(𝑗)}=𝑑𝜉𝐷1+𝛼𝛼^∗−⁰₋₁1,𝛼^∗𝜃𝑒^{𝑖𝜃𝜔0𝜏(𝑗)}𝑑𝜂(𝜃)(1,𝛼)^𝑇=𝐷1+𝛼𝛼^∗−𝛼₂𝛼𝜏^(𝑗)𝑒^{−𝑖𝜔0𝜏(𝑗)}=1. (34) Thus, we can choose 1𝐷=1+𝛼𝛼^∗−𝛼₂𝛼𝜏^(𝑗)𝑒^{𝑖𝜔0𝜏(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 manifold 𝐶₀ at 𝜇=0. Let 𝑥_𝑡 be the solution of (21) when 𝜇=0. Define𝑧(𝑡)=⟨𝑞^∗,𝑥_𝑡𝑊()⟩,𝑡,𝜃=𝑥_𝑡(𝜃)−𝑥(𝑡)𝑞(𝜃)+𝑧(𝑡)𝑞(𝜃)=𝑥_𝑡(𝜃)−2Re𝑧(𝑡)𝑞(𝜃). (36) On the center manifold 𝐶₀, we have 𝑊(𝑡,𝜃)=𝑊(𝑧(𝑡),𝑧(𝑡),𝜃), (37) where 𝑊(𝑧,𝑧,𝜃)=𝑊₂₀(𝜃)𝑧²2+𝑊₁₁(𝜃)𝑧𝑧+𝑊₀₂(𝜃)𝑧²2+𝑊₃₀(𝜃)𝑧³6+⋯, (38) where 𝑧 and 𝑧 are local coordinates for center manifold 𝐶₀ in the direction of 𝑞^∗ and 𝑞^∗. Note that 𝑊 is real if 𝑥_𝑡 is real. Here we consider only real solutions. For the solution 𝑥_𝑡∈𝐶₀ of (24), since 𝜇=0, we have 𝑧̇=𝑖𝜏^(𝑗)𝜔₀𝑧+⟨𝑞^∗(𝜃),𝐹0,𝑊(𝑧,𝑧,𝜃)+2Re𝑧𝑞(𝜃)⟩=𝑖𝜏^(𝑗)𝜔₀𝑧+𝑞^∗(0)𝐹0,𝑊(𝑧,𝑧,0)+2Re𝑧𝑞(0)=𝑖𝜏^(𝑗)𝜔₀𝑧+𝑞^∗(0)𝐹₀(𝑧,𝑧). (39) We rewrite this equation as 𝑧̇(𝑡)=𝑖𝜏^(𝑗)𝜔₀𝑧(𝑡)+𝑔(𝑧,𝑧) (40) with 𝑔(𝑧,𝑧)=𝑞^∗(0)𝐹₀(𝑧,𝑧)=𝑔₂₀𝑧²2+𝑔₁₁𝑧𝑧+𝑔₀₂𝑧²2+𝑔₂₁𝑧²𝑧2+⋯. (41) By (36), we have 𝑥_𝑡(𝜃)=(𝑥_1𝑡(𝜃),𝑥_2𝑡(𝜃))=𝑊(𝑡,𝜃)+𝑧𝑞(𝜃)+𝑧𝑞(𝜃) and 𝑞(𝜃)=(1,𝛼)^𝑇𝑒^{𝑖𝜃𝜔0𝜏(𝑗)}, and then 𝑥_1𝑡(0)=𝑧+𝑧+𝑊⁽¹⁾₂₀(0)𝑧²2+𝑊⁽¹⁾₁₁(0)𝑧𝑧+𝑊⁽¹⁾₀₂(0)𝑧²2+𝑜|(𝑧,𝑧)|³,𝑥_2𝑡(0)=𝑧𝛼𝑒^{−𝑖𝜔0𝜏(𝑗)}+𝑧𝛼𝑒^{𝑖𝜔0𝜏(𝑗)}+𝑊⁽²⁾₂₀()𝑧−1²2+𝑊⁽²⁾₁₁(−1)𝑧𝑧+𝑊⁽²⁾₀₂(−1)𝑧²2+𝑜|(𝑧,𝑧)|³. (42) It follows, together with (23), that 𝑥_2𝑡(0)=𝛼𝑧+𝛼𝑧+𝑊⁽²⁾₂₀(0)𝑧²2+𝑊⁽²⁾₀₂(0)𝑧²2𝑔(+⋯,𝑧,𝑧)=𝑞^∗(0)𝐹₀(𝑧,𝑧)=𝐷𝜏^(𝑗)1,𝛼^∗−𝑥_1𝑡(0)𝑥_2𝑡(−1)−𝜀𝑥²_1𝑡(0)Σ1+h.o.t𝑐𝑖!𝑗!^𝑖𝑗𝑥²_1𝑡(0)𝑥^𝑗_2𝑡(0)=+h.o.t𝐷𝜏^(𝑗)−𝑥_1𝑡(0)𝑥_2𝑡(−1)−𝜀𝑥²_1𝑡(0)+𝛼^∗1𝑐𝑖!𝑗!^𝑖𝑗𝑥²_1𝑡(0)𝑥^𝑗_2𝑡(0)=+h.o.t𝐷𝜏^(𝑗)−𝛼𝑒^{−𝑖𝜔0𝜏(𝑗)}−𝜀+𝛼^∗𝑐2!²⁰+𝑐₀₂𝛼²𝑧²+−𝛼𝑒^{−𝑖𝜔0𝜏(𝑗)}−𝛼𝑒^{𝑖𝜔0𝜏(𝑗)}+−2𝜀𝛼^∗𝑐₂₀+𝑐₀₂𝛼𝛼2𝑧𝑧+−𝛼𝑒^{𝑖𝜔0𝜏(𝑗)}−𝜀+𝛼^∗𝑐₂₀+2!𝛼²𝛼^∗𝑐₀₂2!𝑧²+−𝛼2𝑒^{𝑖𝜔0𝜏(𝑗)}𝑊⁽¹⁾₂₀(0)−12𝑊⁽²⁾₂₀(−1)−𝜀𝑊⁽¹⁾₂₀(0)+𝛼^∗12𝑐₂₀𝑊⁽¹⁾₂₀(0)+12𝛼𝑐₀₂𝑊⁽²⁾₂₀×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^∗𝑐₂₀+𝑐₀₂𝛼²,𝑔₁₁=𝐷𝜏^(𝑗)−𝛼𝑒^{−𝑖𝜔0𝜏(𝑗)}−𝛼𝑒^{𝑖𝜔0𝜏(𝑗)}+−2𝜀𝛼^∗𝑐₂₀+𝑐₀₂𝛼𝛼2,𝑔₀₂=𝐷𝜏^(𝑗)−2𝛼𝑒^{𝑖𝜔0𝜏(𝑗)}−2𝜀+𝛼𝑐₂₀+𝛼𝑐₀₂,𝑔₂₁=−𝛼𝑒^{𝑖𝜔0𝜏(𝑗)}𝑊⁽¹⁾₂₀(0)−𝑊⁽²⁾₂₀(−1)−2𝜀𝑊⁽¹⁾₂₀(0)+𝛼^∗𝑐₂₀𝑊⁽¹⁾₂₀(0)+𝑐₀₂𝛼𝑊⁽²⁾₂₀(0)+2𝑐₂₀𝑊⁽¹⁾₁₁(0)+2𝑐₀₂𝛼𝑊⁽²⁾₁₁(0). (44) In order to determine 𝑔₂₁, we need to compute 𝑊₂₀(𝜃) and 𝑊₁₁(𝜃). From (28) and (36), we have 𝑊̇=𝑥̇_𝑡−𝑧𝑞̇−𝑧̇𝑞=𝐴𝑊−2𝑅𝑞∗(0)𝐹₀𝑞(𝜃),𝜃∈[−1,0],𝐴𝑊−2𝑅𝑞∗(0)𝐹₀𝑞(𝜃)+𝐹₀(,𝜃=0≡𝐴𝑊+𝐻𝑧,𝑧,𝜃), (45) where 𝐻(𝑧,𝑧,𝜃)=𝐻₂₀(𝜃)𝑧²2+𝐻₁₁(𝜃)𝑧𝑧+𝐻₀₂(𝜃)𝑧²2+⋯. (46) Expanding the above series and comparing the corresponding coefficients, we obtain 𝐴−2𝑖𝜏^(𝑗)𝜔₀𝑊₂₀(𝜃)=−𝐻₂₀(𝜃),𝐴𝑊₁₁(𝜃)=−𝐻₁₁(𝜃),…. (47) Following (45), we know that for 𝜃∈[−1,0], 𝐻(𝑧,𝑧,𝜃)=−𝑞^∗(0)𝐹₀𝑞(𝜃)−𝑞^∗(0)𝐹₀(𝜃)−𝑞(𝜃)=−𝑔𝑞𝑔𝑞(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 𝑊̇₂₀(𝜃)=2𝑖𝜏^(𝑗)𝜔₀𝑊₂₀(𝜃)+𝑔₂₀𝑞(𝜃)+𝑔₀₂𝑞(𝜃). (50) Solving 𝑊₂₀(𝜃), we obtain 𝑊₂₀(𝜃)=𝑖𝑔₂₀𝑞(0)𝜏^(𝑗)𝜔₀𝑒^{𝑖𝜏(𝑗)𝜔0𝜃}+𝑖𝑔₀₂𝑞(0)3𝜏^(𝑗)𝜔₀𝑒^{−𝑖𝜏(𝑗)𝜔0𝜃}+𝐸₁𝑒^{2𝑖𝜏(𝑗)𝜔0𝜃}, (51) where 𝐸₁=(𝐸₁⁽¹⁾,𝐸₁⁽²⁾)∈𝑅² is a constant vector.

Similarly, we can obtain 𝑊₁₁(𝜃)=−𝑖𝑔₁₁𝑞(0)𝜏^(𝑗)𝜔₀𝑒^{𝑖𝜏(𝑗)𝜔0𝜃}+𝑖𝑔₁₁𝑞(0)𝜏^(𝑗)𝜔₀𝑒^{−𝑖𝜏(𝑗)𝜔0𝜃}+𝐸₂, (52) where 𝐸₂=(𝐸₂⁽¹⁾,𝐸₂⁽²⁾)∈𝑅² is also a constant vector.

In what follows, we determine the constant vectors 𝐸₁ and 𝐸₂. From (47) and the definition of 𝐴, we obtain 𝑑𝜂(⁰₋₁𝜃)𝑊₂₀(𝜃)=2𝑖𝜏^(𝑗)𝜔₀𝑊₂₀(0)−𝐻₂₀(0),₀₋₁𝑑𝜂(𝜃)𝑊₁₁(𝜃)=−𝐻₁₁(𝜃), (53)(54) where 𝜂(𝜃)=𝜂(0,𝜃). From (45) and (46), we have 𝐻₂₀(0)=−𝑔₂₀𝑞(0)−𝑔₀₂𝑞(0)+2𝜏^(𝑗)−2𝛼𝑒^{−𝑖𝜔0𝜏(𝑗)}𝑐−2𝜀₂₀+𝑐₀₂𝛼²𝐻₁₁,(0)=−𝑔₁₁𝑞(0)−𝑔₁₁𝑞(0)+2𝜏^(𝑗)−2𝛼𝑒^{−𝑖𝜔0𝜏(𝑗)}−𝛼𝑒^{𝑖𝜔0𝜏(𝑗)}𝑐−2𝜀₂₀+𝑐₀₂𝛼𝛼. (55)(56) 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)𝐸₁=2𝜏^(𝑗)−2𝛼𝑒^{−𝑖𝜔0𝜏(𝑗)}𝑐−2𝜀₂₀+𝑐₀₂𝛼²; (58) that is, 2𝑖𝜔₀+𝛼₁−𝛼₂𝑒^{−2𝑖𝜔0𝜏(𝑗)}−𝑟₁2𝑖𝜔₀𝐸₁=2−2𝛼𝑒^{−𝑖𝜔0𝜏(𝑗)}𝑐−2𝜀₂₀+𝑐₀₂𝛼². (59) It follows that 𝐸₁⁽¹⁾=2𝐴−2𝛼𝑒^{−𝑖𝜔0𝜏(𝑗)}−2𝜀−𝛼₂𝑒^{−2𝑖𝜔0𝜏(𝑗)}𝑐₂₀+𝑐₀₂𝛼²2𝑖𝜔₀,𝐸₁⁽²⁾=2𝐴2𝑖𝜔₀+𝛼₁−2𝛼𝑒^{−𝑖𝜔0𝜏(𝑗)}−2𝜀−𝑟₁𝑐₂₀+𝑐₀₂𝛼², (60) where 𝐴=^{2𝑖𝜔0+𝛼1−𝛼2𝑒0𝜏−2𝑖𝜔(𝑗)}_{−𝑟12𝑖𝜔0}.

Similarly, substituting (52) and (56) into (54), we have 𝛼₁𝛼₂−𝑟₁−𝑟₂𝐸₂=2−2𝛼𝑒^{−𝑖𝜔0𝜏(𝑗)}−𝛼𝑒^{𝑖𝜔0𝜏(𝑗)}𝑐−2𝜀₂₀+𝑐₀₂𝛼𝛼. (61) Then we obtain 𝐸₂⁽¹⁾=2𝐵−2𝛼𝑒^{−𝑖𝜔0𝜏(𝑗)}−𝛼𝑒^{𝑖𝜔0𝜏(𝑗)}−2𝜀−𝛼₂𝑐₂₀+𝑐₀₂𝛼𝛼−𝑟₂,𝐸₂⁽²⁾=2𝐵𝛼₁−2𝛼𝑒^{−𝑖𝜔0𝜏(𝑗)}−𝛼𝑒^{𝑖𝜔0𝜏(𝑗)}−2𝜀−𝑟₁𝑐₂₀+𝑐₀₂𝛼𝛼, (62) where 𝐵=^𝛼1𝛼2_{−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)=𝑖2𝜔₀𝜏₀𝑔₂₀𝑔₁₁−2𝑔₁₁²−13𝑔₀₂²+𝑔₂₁2, (63) we obtain 𝐾₂𝑐=−Re₁(0)Re𝜆^′(𝜏),𝛽₂=2Re𝐶₁(0,𝑇₂)𝐶=−Im₁(0)+𝐾₂Im𝜆^′𝜏^(𝑗)𝜏^(𝑗)𝜔₀. (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 𝐾₂ and 𝑇₂ determines 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 solutions exist for 𝜏>𝜏^(𝑗)(𝜏<𝜏^(𝑗)).
(2) If 𝛽₂<0(𝛽₂>0), then the bifurcating periodic solutions are stable (unstable).
(3) If 𝑇₂>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 time 𝜏 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.