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