Bifurcation Analysis for a Predator-Prey Model with Time Delay and Delay-Dependent Parameters

and Applied Analysis 3 2. Stability of the Equilibrium and Local Hopf Bifurcations Throughout this paper, we assume that the following condition H1 ce−dτ > d , a2 ce−dτ − d > b2d holds. The hypothesis H1 implies that system 1.1 has a unique positive equilibrium E∗ x∗, y∗ , where x∗ √ d ce−dτ − d , y ∗ a − bx∗ 1 x∗2 cx∗ . 2.1 The linearized system of 1.1 around E∗ x∗, y∗ takes the form dx t dt a1x t − b1y t , dy t dt −dy t c1x t − τ d1y t − τ , 2.2 where a1 a − 2bx∗ − 2cx∗y∗ 1 x∗2 2x∗3y∗ 1 x∗ 2 , b1 − cx ∗2 1 x∗2 , c1 e−dτ [ 2cx∗y∗ 1 x∗2 − 2x ∗3y∗ 1 x∗ 2 ] , d1 e−dτ cx∗2 1 x∗2 . 2.3 The associated characteristic equation of 2.2 is P λ, τ Q λ, τ e−λτ 0, 2.4 where P λ, τ λ2 d − a1 λ − a1d, Q λ, τ − d1λ − a1d1 − b1c1 . 2.5 When τ 0, then 2.4 becomes λ2 ( d − a1 − d0 1 ) λ b1c 1 − a1d − a1d 1 0, 2.6 where c0 1 [ 2cx∗y∗ 1 x∗2 − 2x ∗3y∗ 1 x∗ 2 ] , d0 1 cx∗ 1 x∗2 . 2.7 It is easy to obtain the following result. 4 Abstract and Applied Analysis Lemma 2.1. If the condition H2 d − a1 − d0 1 > 0, b1c 1 − a1d − a1d 1 > 0, holds, then the positive equilibrium E∗ x∗, y∗ of system 1.1 is asymptotically stable. In the following, one investigates the existence of purely imaginary roots λ iω ω > 0 of 2.4 . Equation 2.4 takes the form of a second-degree exponential polynomial in λ, which some of the coefficients of P and Q depend on τ . Beretta and Kuang 14 established a geometrical criterion which gives the existence of purely imaginary roots of a characteristic equation with delay-dependent coefficients. In order to apply the criterion due to Beretta and Kuang 14 , one needs to verify the following properties for all τ ∈ 0, τmax , where τmax is the maximum value which E∗ x∗, y∗ exists. a P 0, τ Q 0, τ / 0; b P iω, τ Q iω, τ / 0; c lim sup{|Q λ, τ /P λ, τ | : |λ| → ∞,Reλ ≥ 0} < 1; d F ω, τ |P iω, τ |2 − |Q iω, τ |2 has a finite number of zeros; e Each positive root ω τ of F ω, τ 0 is continuous and differentiable in τ whenever it exists. Here, P λ, τ and Q λ, τ are defined as in 2.5 , respectively. Let τ ∈ 0, τmax . It is easy to see that P 0, τ Q 0, τ −a1d a1d1 b1c1 / 0, 2.8 which implies that a is satisfied, and b P iω, τ Q iω, τ −ω2 iω d − a1 − a1d − iωd1 a1d1 b1c1 −ω2 − a1d a1d1 b1c1 iω d − a1 − d1 / 0. 2.9 From 2.4 , one has lim |λ|→ ∞ ∣∣∣ Q λ, τ P λ, τ ∣∣∣ lim |λ|→ ∞ ∣∣∣ − d1λ a1d1 − b1c1 λ2 d − a1 λ − a1d ∣∣∣ 0. 2.10 Therefore, c follows. Let F be defined as in d . From |P iω, τ | ( ω2 a1d )2 d − a1 ω2 ω4 ( d2 a1 ) ω2 a1d 2, |Q iω, τ | d2 1ω a1d1 b1c1 , 2.11 one obtain F ω, τ ω4 ( d2 a1 − d2 1 ) ω2 a1d − a1d1 b1c1 . 2.12 Obviously, property d is satisfied, and by implicit function theorem, e is also satisfied. Abstract and Applied Analysis 5 Now let λ iω ω > 0 be a root of 2.4 . Substituting it into 2.4 and separating the real and imaginary parts yields a1d1 b1c1 cosωτ − d1ω sinωτ ω2 a1d, d1ω cosωτ a1d1 b1c1 sinωτ d − a1 ω. 2.13and Applied Analysis 5 Now let λ iω ω > 0 be a root of 2.4 . Substituting it into 2.4 and separating the real and imaginary parts yields a1d1 b1c1 cosωτ − d1ω sinωτ ω2 a1d, d1ω cosωτ a1d1 b1c1 sinωτ d − a1 ω. 2.13 From 2.13 , it follows that sinωτ − ( ω2 a1d ) d1ω − d − a1 ω a1d1 b1c1 d2 1ω 2 a1d1 b1c1 2 , cosωτ ( ω2 a1d ) a1d1 b1c1 d − a1 ωd1ω d2 1ω 2 a1d1 b1c1 2 . 2.14 By the definitions of P and Q as in 2.5 , respectively, and applying the property a , then 2.14 can be written as sinωτ Im [ P iω, τ Q iω, τ ] , cosωτ −Re [ P iω, τ Q iω, τ ] , 2.15 which yields |P iω, τ |2 |Q iω, τ |2. Assume that I ∈ R 0 is the set where ω τ is a positive root of F ω, τ |P iω, τ | − |Q iω, τ |, 2.16 and for τ ∈ I, ω τ is not definite. Then for all τ in I, ω τ satisfied F ω, τ 0. The polynomial function F can be written as F ω, τ h ( ω2, τ ) , 2.17 where h is a second degree polynomial, defined by h z, τ z2 ( d2 a1 − d2 1 ) z a1d − a1d1 b1c1 . 2.18 It is easy to see that h z, τ z2 ( d2 a1 − d2 1 ) z a1d − a1d1 − b1c1 2 0 2.19 has only one positive real root if the following condition H3 holds: H3 a1d < a1d1 b1c1 . 6 Abstract and Applied Analysis One denotes this positive real root by z . Hence, 2.17 has only one positive real root ω √ z . Since the critical value of τ and ω τ are impossible to solve explicitly, so one will use the procedure described in Beretta and Kuang 14 . According to this procedure, one defines θ τ ∈ 0, 2π such that sin θ τ and cos θ τ are given by the righthand sides of 2.14 , respectively, with θ τ given by 2.19 . This define θ τ in a form suitable for numerical evaluation using standard software. And the relation between the argument θ and ωτ in 2.18 for τ > 0 must be ωτ θ 2nπ , n 1, 2, . . .. Hence, one can define the maps: τn : I → R 0 given by τn τ : θ τ 2nπ ω τ , τn > 0, n 0, 1, 2, . . . , 2.20 where a positive root ω τ of F ω, τ 0 exists in I. Let us introduce the functions Sn τ : I → R, Sn τ τ − θ τ 2nπ ω τ , n 0, 1, 2, . . . , 2.21 which are continuous and differentiable in τ . Thus, one gives the following theorem which is due to Beretta and Kuang 14 . Theorem 2.2. Assume thatω τ is a positive root of 2.4 defined for τ ∈ I, I ⊆ R 0, and at some τ0 ∈ I, Sn τ0 0 for some n ∈ N0. Then, a pair of simple conjugate pure imaginary roots λ ±iω exists at τ τ0 which crosses the imaginary axis from left to right if δ τ0 > 0 and crosses the imaginary axis from right to left if δ τ0 < 0, where δ τ0 sign F ′ ω ωτ0, τ0 sign dSn τ /dτ |τ τ0 . Applying Lemma 2.1 and the Hopf bifurcation theorem for functional differential equation 5 , we can conclude the existence of a Hopf bifurcation as stated in the following theorem. Theorem 2.3. For system 1.1 , if (H1)–(H3) hold, then there exists s τ0 ∈ I such that the positive equilibrium E∗ x∗, y∗ is asymptotically stable for 0 ≤ τ < τ0 and becomes unstable for τ staying in some right neighborhood of τ0, with a Hopf bifurcation occurring when τ τ0. 3. Direction and Stability of the Hopf Bifurcation In the previous section, we obtained some conditions which guarantee that the stagestructured predator-prey model with time delay undergoes the Hopf bifurcation at some values of τ τ0. In this section, we will derive the explicit formulae determining the direction, stability, and period of these periodic solutions bifurcating from the positive equilibrium E∗ x∗, y∗ at these critical value of τ , by using techniques from normal form and center manifold theory 15 . Throughout this section, we always assume that system 1.1 undergoes Hopf bifurcation at the positive equilibrium E∗ x∗, y∗ for τ τ0, and then ±iω0 is corresponding purely imaginary roots of the characteristic equation at the equilibrium E∗ x∗, y∗ . For convenience, let τ τ0 μ, μ ∈ R. Then μ 0 is the Hopf bifurcation value of 1.1 . Thus, one will study Hopf bifurcation of small amplitude periodic solutions of 1.1 from the positive equilibrium point E∗ x∗, y∗ for μ close to 0. Abstract and Applied Analysis 7 Let u1 t x t − x∗, u2 t y t − y∗, xi t ui τt , i 1, 2 , τ τ0 μ, then system 1.1 can be transformed into an functional differential equation FDE in C C −1, 0 , R2 asand Applied Analysis 7 Let u1 t x t − x∗, u2 t y t − y∗, xi t ui τt , i 1, 2 , τ τ0 μ, then system 1.1 can be transformed into an functional differential equation FDE in C C −1, 0 , R2 as du dt Lμ ut f ( μ, ut ) , 3.1 where u t x1 t , x2 t T ∈ R2 and Lμ : C → R, f : R × C → R are given, respectively, by Lμφ ( τ0 μ ) Bφ 0 ( τ0 μ ) Gφ −1 , 3.2


Introduction
Over the past decade, a great many predator-prey models have been developed to describe the interaction between predator and prey.Their dynamical phenomena have been extensively studied because of the wide application in the field of biomathematics.In particular, the appearance of a cycle bifurcating from the equilibrium of an ordinary or a delayed predator-prey model with a single parameter, which is known as a Hopf bifurcation, has attracted much attention due to its theoretical and practical significance 1-5 .But most of the research literature on these models are only connected with parameters which are independent of time delay; thus, the corresponding characteristic equations are easy to deal with.While in most applications of delay differential equations in population dynamics, the need of incorporation of a time delay is often the result of existence of some stage structure 6 It is well known that time delays which occur in the interaction between predator-prey will affect the stability of a model by creating instability, oscillation, and chaos phenomena.Based on the discussion above, the main purpose of this paper is to investigate the stability and the properties of Hopf bifurcation of the model 1.1 which involves some delaydependent parameters.Recently, there are few papers on the topic that involves some delaydependent parameters, for example, Liu

1.3
It worth pointing out that Liu and Zhang 12 investigated the Hopf bifurcation of system 1.2 by choosing p not delay τ as the bifurcation parameters and Jiang and Wei 13 studied the Hopf bifurcation of system 1.3 by choosing φ not delay τ as the bifurcation parameters.In this paper, we will investigate the Hopf bifurcation by regarding the delay τ as the bifurcation parameter which is different from the papers 12, 13 .To the best of our knowledge, it is the first time to deal with the stability and Hopf bifurcation of system 1.1 .This paper is organized as follows.In Section 2, the stability of the equilibrium and the existence of Hopf bifurcation at the equilibrium are studied.In Section 3, the direction of Hopf bifurcation and the stability and periodic of bifurcating periodic solutions on the center manifold are determined.In Section 4, numerical simulations are carried out to illustrate the validity of the main results.Some main conclusions are drawn in Section 5.
Abstract and Applied Analysis 3

Stability of the Equilibrium and Local Hopf Bifurcations
Throughout this paper, we assume that the following condition The hypothesis H1 implies that system 1.1 has a unique positive equilibrium E * x * , y * , where The linearized system of 1.1 around E * x * , y * takes the form where

2.3
The associated characteristic equation of 2.2 is where When τ 0, then 2.4 becomes where It is easy to obtain the following result.
then the positive equilibrium E * x * , y * of system 1.1 is asymptotically stable.
In the following, one investigates the existence of purely imaginary roots λ iω ω > 0 of 2.4 .Equation 2.4 takes the form of a second-degree exponential polynomial in λ, which some of the coefficients of P and Q depend on τ.Beretta and Kuang 14 established a geometrical criterion which gives the existence of purely imaginary roots of a characteristic equation with delay-dependent coefficients.In order to apply the criterion due to Beretta and Kuang 14 , one needs to verify the following properties for all τ ∈ 0, τ max , where τ max is the maximum value which E * x * , y * exists.
e Each positive root ω τ of F ω, τ 0 is continuous and differentiable in τ whenever it exists.
Let τ ∈ 0, τ max .It is easy to see that which implies that a is satisfied, and b

2.9
From 2.4 , one has Therefore, c follows.Let F be defined as in d .From Obviously, property d is satisfied, and by implicit function theorem, e is also satisfied.

Abstract and Applied Analysis 5
Now let λ iω ω > 0 be a root of 2.4 .Substituting it into 2.4 and separating the real and imaginary parts yields

2.13
From 2.13 , it follows that

2.14
By the definitions of P and Q as in 2.5 , respectively, and applying the property a , then 2.14 can be written as and for τ ∈ I, ω τ is not definite.Then for all τ in I, ω τ satisfied F ω, τ 0. The polynomial function F can be written as where h is a second degree polynomial, defined by

2.18
It is easy to see that has only one positive real root if the following condition H3 holds: One denotes this positive real root by z .Hence, 2.17 has only one positive real root ω √ z .Since the critical value of τ and ω τ are impossible to solve explicitly, so one will use the procedure described in Beretta and Kuang 14 .According to this procedure, one defines θ τ ∈ 0, 2π such that sin θ τ and cos θ τ are given by the righthand sides of 2.14 , respectively, with θ τ given by 2.19 .This define θ τ in a form suitable for numerical evaluation using standard software.And the relation between the argument θ and ωτ in 2.18 for τ > 0 must be ωτ θ 2nπ, n 1, 2, . ... Hence, one can define the maps: τ n : I → R 0 given by where a positive root ω τ of F ω, τ 0 exists in I. Let us introduce the functions S n τ : which are continuous and differentiable in τ.Thus, one gives the following theorem which is due to Beretta and Kuang 14 .
Applying Lemma 2.1 and the Hopf bifurcation theorem for functional differential equation 5 , we can conclude the existence of a Hopf bifurcation as stated in the following theorem.

Direction and Stability of the Hopf Bifurcation
In the previous section, we obtained some conditions which guarantee that the stagestructured predator-prey model with time delay undergoes the Hopf bifurcation at some values of τ τ 0 .In this section, we will derive the explicit formulae determining the direction, stability, and period of these periodic solutions bifurcating from the positive equilibrium E * x * , y * at these critical value of τ, by using techniques from normal form and center manifold theory 15 .Throughout this section, we always assume that system 1.1 undergoes Hopf bifurcation at the positive equilibrium E * x * , y * for τ τ 0 , and then ±iω 0 is corresponding purely imaginary roots of the characteristic equation at the equilibrium E * x * , y * .
For convenience, let τ τ 0 μ, μ ∈ R. Then μ 0 is the Hopf bifurcation value of 1.1 .Thus, one will study Hopf bifurcation of small amplitude periodic solutions of 1.1 from the positive equilibrium point E * x * , y * for μ close to 0. where where

3.4
where Abstract and Applied Analysis where x * 2 4 .

3.6
Clearly, L μ is a linear continuous operator from C to R 2 .By the Riesz representation theorem, there exists a matrix function with bounded variation components η θ, μ , θ ∈ −1, 0 such that In fact, we can choose where δ is the Dirac delta function.
Abstract and Applied Analysis 9 3.9 Then 1.1 is equivalent to the abstract differential equation The proof is easy from 3.12 , so we omit it.By the discussions in Section 2, we know that ±iω 0 τ 0 are eigenvalues of A 0 , and they are also eigenvalues of A * corresponding to iω 0 τ 0 and −iω 0 τ 0 , respectively.We have the following result.

3.18
That is, Therefore, we can easily obtain

3.20
And so

3.27
In the sequel, one will verify that q * s , q θ 1.In fact, from 3.12 , we have

3.28
Next, we use the same notations as those in Hassard et al. 15 , and we first compute the coordinates to describe the center manifold C 0 at μ 0. Let u t be the solution of 1.1 when μ 0. Define z t q * , u t , W t, θ u t θ − 2 Re z t q θ , 3.29 on the center manifold C 0 , and we have where and z and z are local coordinates for center manifold C 0 in the direction of q * and q * .Noting that W is also real if u t is real, we consider only real solutions.For solutions u t ∈ C 0 of 1.1 ,

3.32
That is, ż t iω 0 z g z, z , 3.33 where Hence, we have where

3.37
From 3.34 and 3.35 , we have

3.38
and we obtain Abstract and Applied Analysis

3.39
For unknown in g 21 , we still need to compute them.From 3.10 and 3.29 , we have def AW H z, z, θ ,
Similarly, from 3.44 and 3.47 and the definition of A, we have Ẇ11 θ g 11 q θ g 11 q θ , 3.50 T is a constant vector.
In what follows, one will seek appropriate E 1 , E 2 in 3.49 and 3.51 , respectively.It follows from the definition of A and 3.46 and 3.47 that where η θ η 0, θ .From 3.43 , we have where

3.55
From 3.44 , we have where

3.59
That is, Hence, where

3.68
From 3.49 and 3.51 , we can calculate g 21 and derive the following values:

3.69
These formulae give a description of the Hopf bifurcation periodic solutions of 1.1 at τ τ 0 on the center manifold.From the discussion above, we have the following result.

Numerical Examples
In this section, we present some numerical results to verify the analytical predictions obtained in the previous section.As an example, we consider the following special case of system 1.1 with the parameters a 0.5, b 0.5, c 2, 2, and d 0.2.Then system 1.1 becomes ẋ which has a positive equilibrium E * x * , y * 1.4928, 1.3217 .By some complicated computation by means of Matlab 7.0, we get only one critical values of the delay τ 0 ≈ 1.7355, λ τ 0 ≈ 0.2035 − 0.5423i.Thus, we derive c 1 0 ≈ −1.3122 − 5.0131i, μ 2 ≈ 0.6177, β 2 ≈ −3.3326, T 2 ≈ 9.3042.We obtain that the conditions indicated in Theorem 2.3 are satisfied.Furthermore, it follows that μ 2 > 0 and β 2 < 0. Thus, the positive equilibrium E * x * , y * is stable when τ < τ 0 which is illustrated by the computer simulations see Figures 1 a -1 d .When τ passes through the critical value τ 0 , the positive equilibrium E * x * , y * loses its stability and a Hopf bifurcation occurs, that is, a family of periodic solutions bifurcations   from the positive equilibrium E * x * , y * .Since μ 2 > 0 and β 2 < 0, the direction of the Hopf bifurcation is τ > τ 0 , and these bifurcating periodic solutions from E * x * , y * at τ 0 are stable, which are depicted in Figures 2 a -2 d .

Conclusions
In this paper, the main object is to investigate the local stability and Hopf bifurcation and also to study the stability of bifurcating periodic solutions and some formulae for determining the direction of Hopf bifurcation for a stage-structured predator-prey model with time delay and delay.By choosing the delay as a bifurcation parameter, It is shown that under certain condition, the positive equilibrium E * x * , y * of system 1.1 is asymptotically stable for all τ ∈ 0, τ 0 and unstable for τ > τ 0 and under another condition; when the delay τ increases, the equilibrium loses its stability and a sequence of Hopf bifurcations occur at the positive equilibrium E * x * , y * , that is, a family of periodic orbits bifurcate from the positive equilibrium E * x * , y * .At the same time, using the normal form theory and the center manifold theorem, the direction of Hopf bifurcation and the stability of the bifurcating periodic orbits are discussed.Finally, numerical simulations are carried out to validate the theorems obtained.

3 . 11 For
φ ∈ C −1, 0 , R 2 and ψ ∈ C 0, 1 , R 2 * , define the bilinear form 0 .We have the following result on the relation between the operators A A 0 and A * .Lemma 3.1.A A 0 and A * are adjoint operators.
and y t stand for prey and predator density at time t, respectively.a, b, c, are real positive parameters and the time delay τ is a positive constant.Wang et al. 11 obtained the conditions that guarantee the system asymptotically stable and permanent.For more knowledge about the model, one can see 11 .
and Zhang 12 investigated the stability and Hopf bifurcation of the following SIS model with nonlinear birth rate: Let u 1 t x t − x * , u 2 t y t − y * , x i t u i τt , i 1, 2 , τ τ 0 μ, then system 1.1 can be transformed into an functional differential equation FDE in C C −1, 0 , R 2 as T ∈ R 2 and L μ : C → R, f : R × C → R are given, respectively, by