STABILITY AND HOPF BIFURCATION OF A MODIFIED DELAY PREDATOR-PREY MODEL WITH STAGE STRUCTURE

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1 Journal of Applied Analysis and Computation Volume 8, Number, April 018, Website: DOI: / STABILITY AND HOPF BIFURCATION OF A MODIFIED DELAY PREDATOR-PREY MODEL WITH STAGE STRUCTURE Jing Li 1,, Shaotao Zhu 1, Ruilan Tian,, Wei Zhang 3 and Xin Li 1 Abstract In this paper, a modified delay predator-prey model with stage structure is established, which involves the economic factor and internal competition of all the prey and predator populations. By the methods of normal form and characteristic equation, we obtain the stability of the positive equilibrium point and the sufficient condition of the existence of Hopf bifurcation. We analyze the influence of the time delay on the equation and show the occurrence of Hopf bifurcation periodic solution. The simulation gives a visual understanding for the existence and direction of Hopf bifurcation of the model. Keywords Delayed differential equation, hypernormal form, equilibrium point, stability, Hopf bifurcation. MSC(010 34C37, 35Q Introduction In the ecosystem, many systems of interest such as the predator-prey models in population dynamics involve time delay [7, 17]. The study of dynamical behavior (Hopf bifurcation, periodic solution, chaos, etc. for differential equations with delay has significant biological and mathematical meaning. Hopf bifurcation is a very important phenomenon occurs in the differential equations with delay. It seems that the existence of Hopf bifurcations for delayed differential equations can be dated back to the work of Chafee [5] in However, the first proof of the Hopf bifurcation theorem for delay differential equations under analytically computable condition was presented by Chow and Mallet-Paret in 1977 [4]. Since then there have been various contributions on the stability and existence of Hopf bifurcation for time delay equations, especially for the delayed population dynamical equations. Shao and Dai [0] established an impulsive delay predator-prey model with stage structure and Beddington-type functional response concerning ratio-dependent. They obtained the existence and global attractivity of the predator-extinction periodic solution by using the discrete dynamical system determined by the stroboscopic map. Niu and Jiang [15] researched a predator-prey the corresponding author. address:leejing@bjut.edu.cn(j. Li, tianrl@stdu.edu.cn(r. L. Tian 1 College of Applied Sciences, Beijing University of Technology, Beijing, 10014, China Department of Engineering Mechanics, Shijiazhuang Tiedao University, Shijiazhuang, , China 3 College of Mechanical Engineering and Applied Electronics Technology, Beijing University of Technology, Beijing, 10014, China

2 574 J. Li, S. T. Zhu, R. L. Tian, W. Zhang & X. Li equation with two delays and stage structure for the prey according to the theory of bifurcation analysis of neutral delay differential equations. They obtained the stability and Hopf bifurcation of the positive equilibrium, and showed how the neutral terms affect the dynamical behavior of the prey and the predator in the equation. They found that the neutral delay which makes the predator-prey equation more complicated may induces double Hopf bifurcations. Meng and Huo [13] investigated a class of Lotka-Volterra mutualistic equation with time delays of benefit and feedback. The local stability of the positive equilibrium and the existence of Hopf bifurcation were obtained by analyzing the characteristic equation. They obtained explicit formulas to determine the properties of the Hopf bifurcation by using the normal form method and center manifold theorem. Guo et al. [8] investigated a class of three dimensional delayed Gause-type predator-prey models. They presented the boundedness of solutions and the permanence of system, and gave the global asymptotically stability of the positive equilibrium under some parameter conditions by the Lyapunov function. In predator-prey models, as mentioned in [0], many species usually go through two or more life stages as they proceed from birth to death. Thus it is practical to introduce the stage structure into models. Bandyopadhyay and Banerjee [1] presented a class of predator-prey equation with time delay and stage structure as follows where x = (x J, x A, x P T, and L = ( ( d1 α r 1 0 α d 0, F (x = 0 0 d 3 ẋ = Lx + F (x, (1.1 s 1x J βx J x P 0 βx J (tx P (t s 3 x P Here x J, x A and x P represent the population density of juvenile prey, adult prey and predator respectively. They obtained the sufficient condition for the local stability of the equation (1.1 near the positive equilibrium point based on the bifurcation theory of dynamical systems. On the basis of the economic theoretical relationship of public fishery proposed in 1954 [9], Zhang et al. [] added a term of harvest E about predator x P into equation (1.1. Then it can be described by a differential-algebraic equation due to economic factor. ẋ = Lx + F (x,. m = E(px P c, (1. ( where L d1 α r 1 0 = α d 0. E represents the harvest effort about predator 0 0 d 3 E x P. p and c represent the unit price of predator and the unit cost of harvest process respectively. m is the economic profit obtained from the harvest process. They found that the increase of delay destabilized the positive equilibrium point of the system and bifurcated into small amplitude periodic solution. In [1] and [], the stage structure and economic factor were considered, but the internal competition behavior of the adult prey population was not taken into account. From this point, we add the harvest terms in all the juvenile prey, adult prey and predator, and also consider the internal competitive term in the equation.

3 Stability and Hopf bifurcation of Then the differential-algebraic equation becomes where ˆL = ( d1 α E 1 r 1 0 α d E d 3 3 E ẋ = ˆLx + F (x, m = E 3 (px P c, (1.3, F (x = ( s 1x J βx J x P s x A βx J (tx P (t s 3x P r 1 is the birth rate of juvenile prey, α is the conversion rate of juvenile prey transforms to the adult prey, β is the capture rate of juvenile prey by predator. The mortality of juvenile prey, adult prey and the predator is proportional to its own population density, the ratio coefficients are d i (i = 1,, 3. In general, the predator will not transfer the energy to the next generation immediately and we noted that the gestation time delay is τ. E i (i = 1,, 3 represent the harvest effort about juvenile prey, adult prey and predator respectively. s j (j = 1,, 3 represent the internal competitive coefficient of juvenile prey, adult prey and predator population. All of these parameters are positive. The purpose of this paper is to study the stability of the positive equilibrium point of the modified equation (1.3, and analyze the parameter conditions of the existence of Hopf bifurcation as well as the direction of bifurcation. Due to the complexity of the equation, firstly, we should simplify equation (1.3 into a simpler one by normal form (hypernormal form theory. The theories and calculation methods can refer to [11, 14, 16, 1] and references therein. And then we will focus on the stability and bifurcation analysis. According to the latest literatures, there are two common methods for studying the Hopf bifurcation of delayed differential equations. One is Lyapunov function (functional and the other is the analysis approach based on the characteristic root of the characteristic equation for the delayed differential equation [6, 1, 18]. There is no general rule for constructing Lyapunov function (functional. Here we choose the latter method to analyze the distribution of the roots of characteristic equation. We obtain the varied conditions of the real part of characteristic root and it provides theoretical basis for the property analysis of Hopf bifurcation. The rest of the present paper is organized as follows: In section, we obtain explicit expressions for the hypernormal form in terms of the original delay differential equation. This enables us to obtain not only existence but also stability and bifurcation direction. In section 3, we analyze the stability of the positive equilibrium point of equation that reduced by normal form method and obtain the existence condition of the Hopf bifurcation. In section 4, we research the properties of the Hopf bifurcation by combining the normal form theory and the center manifold method. In section 5, we present the numerical simulation of the Hopf bifurcation under given parameter conditions... Hypernormal form of equation (1.3 Based on the normal form theory, equation (1.3 is simplified in this section. Let H n be the linear space spanned by all monomials of degree n, then equation (1.3 can be expressed as the following formal series V (0 = V (0 1 + V (0,

4 576 J. Li, S. T. Zhu, R. L. Tian, W. Zhang & X. Li where V (0 m and H m (m = 1, yield V (0 1 =(r 1 x A (d 1 + α + E 1 x J xj + ( (d + E x A + αx J xa (d 3 x P + E 3 x P xp, V (0 =( s 1 x J βx J x P x J xj + ( s x A xa (βx J (t τx P (t τ s 3 x P xp. Denote the linear operator L (1, L (1 : H H, Y [Y, V (0 1 ], Y H, where operator [, ] defined by [u, v] = Du v Dv u (u, v H. Theorem.1. The first order normal form (hypernormal form of equation (1.3 with the parameter condition (d 1 + α + E 1 (d + E αr 1 = 0 is where y = (y J, y A, y P T, F (y = ẏ = ˆLy + F (y, m = E 3 (py P c, (.1 ( a1 y J +a y J y P b 1 y A c 1 y J (ty P (t s, c 1 = β αr 1 + (d 1 + α + E 1 (d + E. Proof. Let the basic vector Y in linear space H is, a 1 = s 1, a = β, b 1 = we have where and Y =(x J (x J + x A + x P + x A (x A + x P + x P xj + (x J (x J + x A + x P + x A (x A + x P + x P xa + (x J (x J + x A + x P + x A (x A + x P + x P xp, [Y, V (0 1 ] =(m + m 1 (d 3 + E 3 xj + (m αm 1 + m 1 (d + E xa + (m + m 1 (d 1 + α + E 1 r 1 m 1 xp, m 1 = x J (x J + x A + x P + x A (x A + x P + x P, m =( (x A + x J + x P (d 1 + α + E 1 + α(x A + x J + x P x J + (r 1 (x A + x J + x P (x A + x J + x P (d + E x A (x A + x J + x P (d 3 + E 3 x P. With the Maple software, under following parameter condition (d 1 + α + E 1 (d + E αr 1 = 0,

5 Stability and Hopf bifurcation of we get the complementary space C (1 to ImL (1 in H, C (1 = (x J + x J x P xj + x A xa + x J (t τx P (t τ xp. According to the proof of theorem 3.1 in paper [16], the first order normal form (.1 is the hypernormal form of equation (1.3. Theorefore, we obtain the complementary space C (1 to ImL (1 in H with the parameter condition above based on normal form theory in this section, and get the first order normal form (hypernormal form of equation (1.3. The normal form can give the critical information about not only the stability of a positive equilibrium point but also the existence and direction of bifurcation. 3. The stability of a positive equilibrium point and the existence of Hopf bifurcation In this section, we detect the positive equilibrium point of system (.1, and consider time delay τ as the bifurcation parameter. We obtain bifurcation parameter value τ 0 according to the distribution of roots of the characteristic equation of the linearized system near the positive equilibrium point, and give the sufficient condition of the existence of Hopf bifurcation near the positive equilibrium point The positive equilibrium point of equation (.1 Let the right side of equation (.1 be zero and solving these nonlinear equations. We have that system (.1 has a boundary equilibrium point P 0 (0, 0, 0. If Q 1 < 0, system (.1 has only one positive equilibrium point P + (y J, y A, y P, where y J = d 3+E 3 c 1, ya = Q b 1, yp = (d 1+α+E 1 y J a 1(y J r 1 y A a y J E 3 a 1 b 1 (d 3 + E 3 r 1 c 1Q, Q = (d + E, Q 1 = b 1 c 1 (d 1 + α + E 1 (d 3 + (d + E d 4b 3+E 3 1 c The characteristic equation of (.1 at P + and the type of the equilibrium point Let ỹ J = y J yj, ỹ A = y A ya, ỹ P = y P yp, system (.1 becomes ỹ = L 1 ỹ + L ỹ (t + F (ỹ, m = E 3 (pỹ P c, (3.1 where L 1 = ( ( ( n1 r 1 n α n 3 0, L = 0 0 0, F a1 ỹ J (ỹ = +a ỹ J ỹ P b 1ỹ A. 0 0 n 6 n 4 0 n 5 c 1 ỹ J (tỹ P (t The linearization equation of equation (3.1 at P + is ỹ = L 1 ỹ + L ỹ (t, (3. whose characteristic equation yields λ 3 q 1 λ q λ + q 3 (n 5 λ q 4 λ + q 5 e λτ = 0. (3.3

6 578 J. Li, S. T. Zhu, R. L. Tian, W. Zhang & X. Li And λ 1 = γ 1, λ,3 = γ ± iγ 3, (3.4 are the roots of the equation (3.3. P n (n 1, n, n 3, n 4, n 5, n 6, P q (q 1, q, q 3, q 4, q 5 and P γ (γ 1, γ, γ 3 denote the parameter condition in equation (3.1, equation (3.3 and equation (3.4, respectively. The relations between them and the coefficients in equation (.1 are given in the appendix. According to qualitative theory of differential equation, the type of the equilibrium point P + of equation (.1 are showed in Table 1. Table 1. Characteristic equation of equation (.1 at P + and the type of P +. Characteristic equation λ 3 q 1 λ q λ + q 3 (n 5 λ q 4 λ + q 5 e λτ = 0 Parameter condition γ 1 γ < 0 γ 1 > 0, γ > 0 γ 1 < 0, γ < 0 Type of p + Saddle focus Unstable focus Stable focus Remark 3.1. Hartman Grobman theorem ensures that when the singularities of the linearized system is hyperbolic type, the trajectory of nonlinear system and corresponding linearization system keep the topology equivalence in the neighborhood of singular point. We analyze the type and the stability of singularities in nonlinear system requires the center manifold approach when the singularities are not hyperbolic type Computation of bifurcation parameter value τ 0 of equation (.1 In order to study the Hopf bifurcation of equation (.1, we denote λ = ±iω(ω > 0 as the solution of characteristic equation (3.3. Hence, iω 3 + q 1 ω iq ω + q 3 (n ω iq 4 ω + q 5 e iωτ = 0 is obtained. By separating the real part and the imaginary part leads to { ω 3 q ω + q 4 ωcos(ωτ (n ω q 5 sin(ωτ = 0, q 1 ω + q 3 + (n ω q 5 cos(ωτ + q 4 ωsin(ωτ = 0. Clearly, the solution of equations mentioned above can be written as cos(ωτ = (n q 1 q 4 ω 4 + (n q 3 q 1 q 5 q q 4 ω q 3 q 5 (n ω q 5 + ω q4, sin(ωτ = ω(n ω 4 + (n q + q 1 q 4 q 5 ω q q 5 + q 3 q 4 (n ω q 5 + ω q4, and ω is the root of the equation ω 6 + p 1 ω 4 + p ω + p 3 = 0. (3.5 We use P p (p 1, p, p 3 denote the parameter condition in equation (3.5. The relationships between P p and coefficients in equation (.1 are showed in the appendix.

7 Stability and Hopf bifurcation of Let z = ω and equation (3.5 becomes z 3 + p 1 z + p z + p 3 = 0. (3.6 Assume that equation (3.6 has three positive roots z k (k = 1,, 3. Then, equation (3.5 has positive root ω k (= z k with and where and τ (j k = 1 ω k (arccos δ 1 + jπ, (3.7 τ (j k = 1 ω k (arcsin δ + jπ, (3.8 δ 1 = (n q 1 q 4 ω 4 + (n q 3 q 1 q 5 q q 4 ω q 3 q 5 (n ω q 5 + ω q4, δ = ω(n ω 4 + (n q + q 1 q 4 q 5 ω q q 5 + q 3 q 4 (n ω q 5 + ω q4, k = 1,, 3, j = 0, 1,,. Solving equation (3.7 and (3.8, we obtain and bifurcation parameter τ 0 respectively. = (τ 0, τ 0 = min{τ 0 k } (1 k 3, 3.4. The sufficient conditions of the Hopf bifurcation of equation (.1 near P + Let z 0 = p + 3, = p 1 3p and h(z = z 3 + p 1 z + p z + p 3 in equation (3.6 and we have the following lemma. Lemma 3.1. Assuming that parameter conditions q 1 + n 5 < 0, q 3 q 5 > 0 and (q 1 + n 5 (q q 4 q 3 + q 5 > 0 hold, then (i If p 3 0 and < 0, then all the roots of the characteristic equation (3.3 have strict negative real part; (ii If p 3 < 0 or (p 3 0, 0, z 0 > 0, h(z 0 0, then all the roots of the characteristic equation (3.3 have strict negative real part with τ [0, τ 0. Proof. (i According to τ = 0, equation (3.3 can be written as λ 3 (q 1 + n 5 λ (q q 4 λ + q 3 q 5 = 0. The necessary and sufficient conditions for all the roots of the equation (3.6 have strict negative real part, according to Routh-Hurwitz criterion, is q 1 + n 5 < 0, q 3 q 5 > 0 (q 1 + n 5 (q q 4 q 3 + q 5 > 0. (3.9

8 580 J. Li, S. T. Zhu, R. L. Tian, W. Zhang & X. Li If p 3 0 and < 0, equation (3.6 has no positive real root and equation (3.3 has no pure imaginary roots. From the proof of theorem 3.1 in paper [19], we know that all roots of equation (3.3 have strict negative real parts for τ [0,. (ii If p 3 < 0(p 3 0, 0, z 1 > 0, h(z 1 0, equation (3.6 has positive real root and equation (3.3 has pure imaginary roots only when τ [0, τ 0. Let λ(τ = α(τ + iω(τ be the root of equation (3.3 with condition α(τ 0 = 0 and ω(τ 0 =. Denote that G = (ω 4 0f 1 + ω 0f + f 3 f 4, and P f1 4 (f 1, f, f 3, f 4 is the parameter condition above. The relationships between P f1 4 and coefficients in equation (.1 are showed in the appendix. Then, we have the theorem as follows. Theorem 3.1 (The sufficient condition of the existence of Hopf bifurcation. If the conditions in Lemma 3.1(ii hold and G > 0, we have (i The positive equilibrium point P + of equation (.1 is asymptotically stable when 0 τ τ 0 ; (ii The positive equilibrium point P + of equation (.1 is unstable when τ > τ 0 ; (iii Equation (3.3 has a pair of pure imaginary roots when τ = τ 0, and τ 0 is the Hopf bifurcation point of equation (.1. Proof. (i From Lemma 3.1(ii, if p 3 < 0 (p 3 0, 0, z 0 > 0, h(z 0 0, all roots of the characteristic equation (3.3 have strict negative real part with τ [0, τ 0. According to qualitative theory of differential equation, the positive equilibrium P + of equation (.1 is asymptotically stable. (ii For that λ(τ = α(τ + iω(τ is the root of equation (3.3 with condition α(τ 0 = 0 and ω(τ 0 =, we calculate dλ(τ λ=iω0,τ=τ dτ 0 = f 5 + if 6. f 7 + if 8 Denote that the parameter condition above is P f5 8 (f 5, f 6, f 7, f 8, Moreover, the relationships between P f5 8 and coefficients of equation (.1 are showed in the appendix. So, we have dreλ(τ λ=iω0,τ=τ dτ 0 = f 5f 7 + f 6 f 8 f7 + f 8 ω = 0(ω0f ω0f + f 3 (q4 + n 5 ω4 0 n 5ω0 q 5 + q5 (f 7 + f 8 = (ω0f ω0f + f 3 f 4 (f7 + f 8 G = f7 + f 8. If G > 0, we have that dreλ(τ dτ λ=iω0,τ=τ 0 > 0. Then, the Equation (3.3 has at least one characteristic root with strict positive real part when τ > τ 0, and the positive equilibrium point P + of equation (.1 is unstable according to qualitative theory of differential equation.

9 Stability and Hopf bifurcation of (iii Equation (3.3 has a pair of pure imaginary roots λ = ±i when τ = τ 0. Combine (i and (ii, we get the conclusion that τ = τ 0 is the Hopf bifurcation point of equation (.1. We get the positive equilibrium point P + of equation (.1 and discuss its type. According to the distribution of roots of the characteristic equation for the linearized system at P +, we obtain the bifurcation value τ 0 τ = t 0. Based on the differential equation qualitative theory, we get the sufficient condition of the existence of Hopf bifurcation near P The properties of the Hopf bifurcation and dreλ(τ dτ λ=iω0,τ=τ 0 across In this section, we transfer equation (.1 into abstract ordinary differential equations by Riesz representation theorem and the infinitesimal generator approach of functional differential equation. We apply normal form theory and center manifold approach [, 3, 10] to obtain the differential equations restricted to the flow of center manifold of that abstract equations. We analyze the direction, stability and periodicity of the periodic solution of Hopf bifurcation by means of the normal form obtained above Calculation of eigenvectors q(θ and q (θ Let τ = τ 0 + µ and equation (.1 exists Hopf bifurcation at µ = 0. We denote u(t = (ỹ J (t, ỹ A (t, ỹ P (t T = (u J (t, u A (t, u P (t T and u(t = u(t+θ, θ [, 0]. The equation (.1 can be rewritten as a functional differential equation with the following form where B 1 = ( n1 r 1 n α n 3 0, B = 0 0 n 6 u = L µ u + F (u, (4.1 L µ u = B 1 u(t + B u(t τ, ( , F (u = n 4 0 n 5 ( s1 u J βu J u P s u A βu J (tu P (t for any initial condition φ(θ = (φ 1 (θ, φ (θ, φ 3 (θ T C 3 with φ(0 = u(t and φ( = u(t τ, we have, L µ u = B 1 φ(0 + B φ(. (4. According to spectral decomposition theory, infinitesimal generator A(0 associated with linearized equation of equation (4. dφ(θ, θ [, 0, dθ A(0φ(θ = 0 φ(sdη(s, θ = 0, where φ C 3. And η(θ is three order matrix function with η(µ, θ = B 1 δ(θ + B δ(θ + τ, (4.3

10 58 J. Li, S. T. Zhu, R. L. Tian, W. Zhang & X. Li where δ(θ is Dirac-delta function. We also define operator R(0 as { 0, θ [, 0, R(0φ(θ = F (u, θ = 0. Then, equation (4.1 is equivalent to the abstract differential equation u = A(0u + R(0u. (4.4 Let ψ C 3, we represent the formal adjoint operator of A(0 as A and we have dψ(s, s (0, τ], A ds (0ψ(s = 0 ψ( tdη T (t, s = 0. For ψ and φ, we define the bilinear form of the inner product as follows (4.5 0 θ ψ, φ = ψ(0φ(0 ψ(ξ θφ(ξdη(θdξ. (4.6 0 According to the bilinear inner product (4.6, we have ψ, Aφ = A ψ, φ. Theorem 4.1. Let q(θ and q (θ be eigenvectors of A(0 and A (0 associated with i and i respectively, we have and where q(θ = V e iθ, q (θ = DV e iθ, q (θ, q(θ = 1, q (θ, q(θ = 0, V = ( 1, 1, ρ 1 T, ρ 1 = n 1 i + r 1 n, V = (ρ, 1, 1 T, ρ = n + r 1, n 1 + i 1 D = V T V + τe iτ V T B V. Proof. According to equation (4.3 and A(0q(θ = i q(θ, we have dq(θ = i q(θ, θ [, 0, dθ A(0q(θ = 0 q(sdη(s = i q(θ, θ = 0, and therefore q(θ = V e iω0θ, θ [, 0],

11 Stability and Hopf bifurcation of where V = (v 1, v, v 3 T C 3. From equation (4.5 and A (0q (θ = i q (θ, we have dq (s, s (0, τ], A (0q ds (s = 0 q ( sdη T (s, s = 0, and therefore q (θ = DV e iω0θ, θ [, 0], where D R. From equation (4., it follows that B 1 V + B V e iθ = i I, where I is three order unit matrix. We solve the equation above and obtain the solution as follows ( V = 1, 1, n 1 i + r 1 T. n Similarly, we have V = ( n + r 1 T., 1, 1 n 1 + i From equation (4.8, the inner product of q (θ with q(θ is 0 q (θ, q(θ = q T (0q(0 = D V T V = D V T V θ 0 0 θ 0 0 q T (ξ θq(ξdη(θdξ D V T e i(ξ θ V e iξ dη(θdξ D V T e iθ V θdη(θ = D V T V D V T (B e iτ V = D( V T V + τe iτ V T B V. Let 1 D = V T V + τe iω0τ V T B V, we have q (θ, q(θ = 1. Similarly, we have Therefore, the theorem is proved. q (θ, q(θ = 0.

12 584 J. Li, S. T. Zhu, R. L. Tian, W. Zhang & X. Li We define the local coordinates of the equation (4.4 in the direction of q (θ and q (θ as z(t = q, u, (4.7 where u is the solution of equation (4.4 at µ = 0. We denote the real space which differ from eigenvalue ±i of equation (4.4 is Q ±iω0. So, we obtain the following equation on the center manifold C µ=0 = W (z(t, z(t, θ of equation (4.4, W (z(t, z(t, θ = W (t, θ = u(t z(tq(θ z(t q(θ = u(t Re(z(tq(θ, (4.8 where W (t, θ Q ±iω0 and W (z(t, z(t, θ could be expressed as power series expansion about z(t and z(t, W (z(t, z(t, θ = W 0 (θ z(t + W 11 (θz(t z(t + W 0 (θ z(t +. According to equation (4.7 and (4.8, the flow of equation (14 on center manifold C µ=0 could be determined by ż(t = q, u = q, A(0u + R(0u = q, A(0u + q, R(0u = A (0q, u + q T (0F 0 = i z(t + q T (0F 0 (4.9 where F 0 = F (W (z(t, z(t, θ + z(tq(θ + z(t q(θ, 0. Hence, we need to calculate the undetermined coefficients W 0 (θ, W 11 (θ and W 0 (θ, and bring these coefficients into equation (4.9. And then, we obtain the equation restricted to the center manifold C µ=0. We denote that g(z(t, z(t = q T (0F 0 = q T (0F (W (z(t, z(t, θ + z(tq(θ + z(t q(θ, 0 =g 0 (θ z(t + g 1 z(t z(t + g 11 (θz(t z(t + g 0 (θ z(t +. (4.10 From equation (4.10, we know that, as long as the coefficients g 0, g 11, g 0 and g 1 were calculated, the equation restricted to the center manifold (4.9 will be obtained. 4.. Computation of coefficients g 0, g 11 and g 0 on center manifold C µ=0 From equation (4.8, we have Ẇ (t, θ = u(t ż(tq(θ z(t q(θ. (4.11

13 Stability and Hopf bifurcation of According to equation (4.4 and (4.9, it follows that Ẇ (t, θ = { AW (t, θ Re( q T (0F 0 q(θ, θ [, 0, AW (t, θ Re( q T (0F 0 q(θ + F 0, θ = 0. (4.1 Further more, the equation (4.1 could be expresses as where H(z(t, z(t, θ = H 0 (θ z(t Ẇ (t, θ = AW (t, θ + H(z(t, z(t, θ, + H 11 (θz(t z(t + H 0 (θ z(t On the other hand, from equation (4.13, we have Ẇ (z, z, θ =W z(t ż(t + W z(t z(t =(W 0 (θz + W 11 (θ zż + (W 0 (θz + W 11 (θ z z =(W 0 (θz + W 11 (θ z(i z + q T (0F 0 + (W 0 (θz + W 11 (θ z( i z + q T (0F 0 +. (4.13 =i W 0 (θz i W 0 (θ z +. (4.14 Substituting equation (4.14 into equation (4.1 and comparing the coefficients about the term z and z z, we have { (A iω0 IW 0 (θ = H 0 (θ, θ [, 0, (A i IW 0 (θ = g 0 q(0 + ḡ 0 q(0 F z, θ = 0, (4.15 and { AW11 (θ = H 11 (θ, θ [, 0, AW 11 (θ = g 11 q(0 + ḡ 11 q(0 F z z, θ = 0. (4.16 From (4.8, we have and u(t = u(t + θ = W (t, θ + z(tq(θ + z(t q(θ, u J (t + θ = W (1 (t, θ + z(t( 1 + z(t( 1 = W (1 0 (θz + W (1 (1 z 11 (θz z + W 0 (θ z(teiθ z(te iω0θ, u A (t + θ = W ( (t, θ + z(t( 1 + z(t( 1 = W ( 0 (θz + W ( 11 u P (t + θ = W (3 (t, θ + z(tρ 1 + z(t ρ 1 = W (3 0 (θz + W (3 11 ( z (θz z + W 0 (θ z(teiω0θ z(te iω0θ, (θz z + W (3 0 (θ z z(tρ 1e iθ z(t ρ 1 e iθ.

14 586 J. Li, S. T. Zhu, R. L. Tian, W. Zhang & X. Li For equation (4.1, we denote that s 1 u J βu z Ju P K 1 K K 3 K 4 F (φ = s u A = z z K 5 K 6 K 7 K 8 z βu J (t τu P (t τ K 9 K 10 K 11 K 1, z z where φ(j, φ(a, φ(p and K i (i = 1,,, 1 are showed in the appendix. According to equation (4.10, it follows that g(z(t, z(t = q T (0F 0 z K = D V 1 K K 3 K 4 T z z K 5 K 6 K 7 K 8 z K 9 K 10 K 11 K 1 z z = D( ρ K 1 K 5 K 9 z + D( ρ K K 6 K 10 z z + D( ρ K 3 K 7 K 11 z + D( ρ K 4 K 8 K 1 z z. Comparing the coefficients of each term with equation (4.10 yields g 0 = D( ρ K 1 K 5 K 9, g 11 = D( ρ K K 6 K 10, g 0 = D( ρ K 3 K 7 K 11. (4.17 Furthermore, coefficients g 0, g 11 and g 0 are obtained through these equations above. However, coefficient g 1 = D( ρ K 4 K 8 K 1 depends on terms W 11 (θ and W 0 (θ. We should calculate W 11 (θ and W 0 (θ in equation Computation of coefficients g 1 on center manifold C µ=0 From equation (4.1 at θ [, 0, it follows that H(z(t, z(t, θ = Re ( q T (0F 0 q(θ z = (g 0 + g z 11z z + g 0 + g z z 1 + q(θ z (ḡ 0 + ḡ z 11z z + ḡ 0 + ḡ z z 1 + q(θ. (4.18 Comparing the coefficients about the term z (4.18, we have and z z between equation (4.13 and H 0 (θ = g 0 q(θ ḡ 0 q(θ, and H 11 (θ = g 11 q(θ ḡ 11 q(θ.

15 Stability and Hopf bifurcation of From equation (4.15 at θ [, 0, it follows that Ẇ 0 (θ = i W 0 (θ H 0 (θ = i W 0 (θ + g 0 q(θ + ḡ 0 q(θ We solve equation and obtain the solution = i W 0 (θ + g 0 V e iθ + ḡ 0 V e iθ. W 0 (θ = ig 0 V e iθ + iḡ 0 3 V e i θ + M 1 e iθ. (4.19 From equation (4.16 at θ [, 0, it follows that Ẇ 11 (θ = H 11 (θ = g 11 q(θ + ḡ 11 q(θ We solve equation and obtain the solution = g 11 V e iθ + ḡ 11 V e iθ. W 11 (θ = ig 11 V e iθ + iḡ 11 V e i θ + M. (4.0 According to equation (4.16, we obtain H(z, z, 0 at θ = 0 as follows where H(z, z, 0 = Re ( q T (0F 0 q(θ + F 0 z = (g 0 + g z 11z z + g 0 + g z z 1 + q(0 z (ḡ 0 + ḡ z 11z z + ḡ 0 + ḡ z z 1 + q(0 + F 0, z = (g 0 + g z 11z z + g 0 + g z z 1 + V z (ḡ 0 + ḡ z 11z z + ḡ 0 + ḡ z z 1 + V + (U 1, U, U 3 T, (4.1 U 1 = K 1 z + K z z + K 3 z + K 4 z z, U = K 5 z + K 6 z z + K 7 z + K 8 z z, U 3 = K 9 z + K 10 z z + K 11 z + K 1 z z. Comparing the coefficients about the term z (4.13, we have and z z between equation (4.1 and H 0 (0 = g 0 q(0 ḡ 0 q(0 + (K 1, K 5, K 9 T, and H 11 (0 = g 11 q(0 ḡ 11 q(0 + (K, K 6, K 10 T.

16 588 J. Li, S. T. Zhu, R. L. Tian, W. Zhang & X. Li and From equation (4.4 and (4.15, we calculate the integrals as following at θ = 0, 0 W 0 (θdη(θ =i W 0 (0 H 0 (0 0 =i W 0 (0 + g 0 q(0 + ḡ 0 q(0 (K 1, K 5, K 9 T =i ( ig 0 V + iḡ 0 3 V + M1 + g 0 V + ḡ 0 V (K 1, K 5, K 9 T, (4. W 11 (θdη(θ = H 11 (0 = g 11 q(0 + ḡ 11 q(0 (K, K 6, K 10 T = g 11 V + ḡ 11 V (K, K 6, K 10 T. (4.3 Because that q(θ is characteristic vector of A(0 associated with i, when θ = 0, from equation (4.3 we have A(0q(0 = 0 From equation (4.4, it follows that ( 0 i I So, for η = η, we have q(sdη(s = i q(0. (4.4 e iω0s dη(s q(0 = 0. ( 0 i I e iω0s dη(s q(0 = 0. Hence, from equations (4.19 and (4., we obtain ( 0 1(K1 M 1 = i I e dη(s iω0s, K 5, K 9 T. (4.5 Similarly, from equations (4.0 and (4.3, we also obtain ( 0 1(K M = dη(s, K 6, K 10 T. (4.6 Moreover, M 1 = ( M (1 1, M ( 1, M (3 T 1 and M = ( M (1, M (, M (3 T, (j M i (i = 1,, j = 1,, 3 are showed in the appendix. Because of equations (4.19 and (4.0, we calculate W (i (i 0 (θ and W 11 (θ(i = 1,, 3 and obtain the coefficient g 1 = D( ρ K 4 K 8 K 1, (4.7 where W (i (i 0 (θ, W 11 (θ(i = 1,, 3 are showed in the appendix. We calculate all these coefficients and bring g 0, g 11, g 0 and g 1 into equation (4.9. Therefore, the equation can be written as ż(t = i z + g(z, z = i z + g 0 z + g 11z z + g 0 z + g 1 z z +.

17 Stability and Hopf bifurcation of The properties of Hopf bifurcation of equation (.1 and corresponding algorithm Based on the Hopf bifurcation theory of dynamical systems, equation (4.9 can be transformed into ẇ = i w + c 1 (0w w + O( w 4, (4.8 ( where c 1 (0 = i ω g11 0 g 0 g 11 g g 1. We denote the first Lyapunov coefficient of equation (4.8 as and l 1 (0 = Rec 1(0 = 1 ω0 Re(ig 0 g 11 + g 1, µ = l 1 (0 Reλ (0, where µ and l 1 (0 determines the direction and stability of Hopf bifurcation respectively, we have the following theorem. Theorem 4. (The properties of the Hopf bifurcation. If µ > 0 (µ < 0, the Hopf bifurcation of equation (.1 is supercritical (subcritical, and there exits bifurcation periodic solution when τ > τ 0 (τ < τ 0 ; If l 1 (0 < 0 (l 1 (0 > 0, the bifurcation periodic solution of equation (.1 is stable (unstable. Combined with analysis the properties of Hopf bifurcation periodic solution of equation (.1, we give the algorithm process as follows: Rewrite the equation (3.1 in abstract form u = A(0u+R(0u and obtain the local coordinate on the direction of q (θ and q (θ of equation (4.4 according to the theory of spectral decomposition of equation; Obtain the equation (4.9 restricted on the center manifold C µ=0 and calculate the coefficients; Calculate µ and l 1 (0, research the bifurcation direction and stability of Hopf bifurcation periodic solution of equation (.1. The corresponding algorithm process is shown in Figure 1. Figure 1. Algorithm process. The spectral decomposition theory of equation is basic approach for studying bifurcation problems by applying center manifold method. We give the equation

18 590 J. Li, S. T. Zhu, R. L. Tian, W. Zhang & X. Li which restricted on the flow of center manifold of equation (.1 by using of spectral decomposition theory and center manifold method, and obtain the algorithm shows the properties such as stability and direction of Hopf bifurcation periodic solution of equation ( Numerical simulations In this section, we carry out the numerical simulation of the Hopf bifurcation periodic solution of the equation (.1. We obtain τ 0 and dreλ(τ dτ λ=iω0,τ=τ 0 according to lemma 3.1 and theorem 3.1, and calculate µ and l 1 (0 to determines the direction and stability of Hopf bifurcation respectively. We choose the parameters in equation (1.3 as P = (r 1, α, β, d, s, E, where r 1 = 80, α = 0.3, β = 1., d = (d 1, d, d 3 = (1.5, 1, 0.6, s = (s 1, s, s 3 = (1., 0.3, 0.8, E = (E 1, E, E 3 = (1.5, 1.8,. Under this parameter condition, equation (.1 has a unique positive equilibrium point P + = (.167, 0.7,.057. For the ecological (biological equation, the positive equilibrium point means that all the speices in the equation are exist. We also get that = 0.361, τ 0 = 7.650, p 3 = < 0, G = > 0, dreλ(τ dτ λ=iω0,τ=τ 0 > 0. All the characteristic roots of the characteristic equation (3.3 have strictly negative real part with τ [0, τ 0. Hence, the positive equilibrium point P + of equation (.1 is asymptotically stable as showed in Figure. Figure shows that the juvenile prey, adult prey and predator can reach a stable state after a period of time in the harvest condition. The characteristic equation (3.3 has, at least, one characteristic root with strictly positive real part at τ > τ 0. Hence, the positive equilibrium point P + of equation (.1 is unstable. The coefficients of the center manifold C µ=0 are obtained as follows g 0 = i, g 11 = i, g 0 = i, g 1 = i. From equation (4.8, we obtained c 1 (0 = i, µ < 0 and l 1 (0 > 0. According to theorem 4., we know that this Hopf bifurcation is subcritical and there exists the bifurcation periodic solution at τ < τ 0 in equation (.1 as showed in Figure 3. Based on the algorithm in section 4, we obtain that these coefficients of equation which restricted on the flow of center manifold C µ=0, and give µ and l 1 (0 to determines the direction and stability of Hopf bifurcation respectively in this section. Numerical simulation shows that there exists subcritical Hopf bifurcation near P + in equation (.1. Hopf bifurcation periodic solution exists when τ < τ 0 and disappears when τ τ Conclusion The response of a biological and financial equations to a particular input is often not immediate but is delayed. Even though a small time delay may leads to complex dynamic behavior.

19 Stability and Hopf bifurcation of initial value =.00 initial value =.10 initial value = YJ t ( initial value = 0.00 initial value = 0.0 initial value = YA t ( initial value =.00 initial value =.10 initial value = YP t (3 Figure. The positive equilibrium point P + of the equation (.1 is asymptotically stable. (1,(,(3 shows the population of juvenile prey, adult prey and predator variety process about time t respectively. In this paper, we have detected the harvest term about juvenile prey, adult prey and predator as well as the internal competition, based on the result obtained in papers [1, ]. We analyze the stability of the positive equilibrium point of equa-

20 59 J. Li, S. T. Zhu, R. L. Tian, W. Zhang & X. Li (1 ( Figure 3. Subcritical Hopf bifurcation of equation (.1 near P +. (1 τ < τ 0. ( τ τ 0. tion (.1, and obtain the existence condition of Hopf bifurcation. When τ passes through the critical value τ 0, the equation loses its stability and generates Hopf bifurcation. We calculate the bifurcation value τ 0 and dreλ(τ dτ λ=iω0,τ=τ 0 across τ = τ 0. Based on differential equation qualitative theory, we present the sufficient condition of the existence of Hopf bifurcation near P +. By using of spectral decomposition theory and center manifold method, we obtain the equation which restricted on the flow of center manifold of equation (.1, and also present the algorithm which shows the properties such as stability and direction of Hopf bifurcation periodic solution of equation (.1. Numerical simulation indicates that equation (.1 exists subcritical Hopf bifurcation near P + under certain parameter conditions. Hopf bifurcation periodic solution exists when τ < τ 0 and disappears when τ τ 0. We have obtained the critical value of time delay which could affect the stable coexistence of both prey and predator species at the equilibrium point. However, the conditions to preserve stability or generate Hopf bifurcation are dependent upon the system parameters (e.g. harvest efforts, internal competition. It would be more interesting to give a thorough discussion about these parameters and their implications on the global behavior of the solutions. Furthermore, it is necessary to impose some control to prevent the possible abnormal oscillation in population density which can be carried out in a separate paper. Acknowledgements The research project is supported by National Natural Science Foundation of China ( , , , , and and also supported by Beijing Natural Science Foundation (11700, 11001, the International Science and Technology Cooperation Program of China (014DFR61080, Natural Science Foundation for Outstanding Young Researcher in Hebei Province of China(A , the Funding Project for Academic Human Resources Development in Institutions of Higher Learning under the Jurisdiction of Beijing Municipality(PHRIHLB, Beijing Key Laboratory on Nonlinear Vibrations and Strength of Mechanical Structures, College of Mechanical Engineering, Beijing University of Technology, Beijing 10014, P.R. China.

21 Stability and Hopf bifurcation of References [1] M. Bandyopadhyay and S. Banerjee, A stage-structured prey-predator model with discrete time delay, Appl. Math. Comput., 006, 18(, [] P. L. Buono and J. Belair, Restrictions and unfolding of double Hopf bifurcation in functional differential equations, J. Differ. Equations, 003, 189(l, [3] S. N. Chow and J. K. Hale, Methods of bifurcation theory, New York: Springer- Verlag, 198. [4] S. N. Chow and J. Mallet-Paret, Integral averaging and bifurcation, J. Differ. Equations, 1977, 6, [5] N. Chafee, A bifurcation problem for a functional differential equation of finitely retarded type, J. Math. Anal. Appl., 1971, 35, [6] D. J. Fan, L. Hong and J. J. Wei, Hopf bifurcation analysis in synaptically coupled HR neurons with two time delays, Nonlinear Dynam., 010, 6(1, [7] S. A. Gourley and R. S. Liu, Delay equation models for populations that experience competition at immature life stages, J. Differ. Equations, 015, 59(5, [8] S. Guo, W. H. Jiang and H. B. Wang, Global analysis in delayed ratio-dependent Gause-type predator-prey models, J. Appl. Anal. Comput., 017, 7(3, [9] H. S. Gordon, The economic theory of a common-property resource: the fishery, J. Polit. Econ., 1954, 6(, [10] B. D. Hassard, N. D. Kazarinoff and Y. H. Wan, Theory and application of Hopf bifurcation, Cambridge University Press, Cambridge, [11] J. Li, L. N. Zhang and D. Wang, Unique normal form of a class of 3 dimensional vector fields with symmetries, J. Differ. Equations, 014, 57, [1] X. L. Li and J. J. Wei, On the zeros of a fourth degree exponential polynomial with application to a neural network model with delays, Chaos Soliton Fract., 005, 6(, [13] X. Y. Meng and H. F. Huo, Bifurcation analysis of a Lotka-Volterra mutualistic system with multiple delays, Abstr. Appl. Anal., 014. DOI: /014/ [14] J. Murdock, Hypernormal form theory: foundations and algorithms, J. Differ. Equations, 004, 05, [15] B. Niu and W. H. Jiang, Hopf bifurcation induced by neutral delay in a predatorprey system, Int. J. Bifurcat. Chaos, 013, 3(11. DOI: /S [16] J. P. Peng and D. Wang, A sufficient condition for the uniqueness of normal forms and unique normal forms of generalized Hopf singularities, Int. J. Bifurcat. Chaos, 004, 14, [17] D. Riad, K. Hattaf and N. Yousfia, Dynamics of a delayed business cycle model with general investment function, Chaos Soliton Fract., 016, 85,

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23 Stability and Hopf bifurcation of Appendix Section 3. (1 Relationships between parameter condition P n (n 1, n, n 3, n 4, n 5, n 6 and coefficients in equation (.1 are shown as follows. n 1 = (d 1 + α + E 1 + a 1 y J + a y P, n = a y J, n 3 = (d + E s y A, n 4 = c 1 y P, n 5 = c 1 y J, n 6 = (d 3 + E 3. ( Relationships between parameter condition P q (q 1, q, q 3, q 4, q 5 and coefficients in equation (.1 are shown as follows. q 1 = n 1 + n 3 + n 6, q = αr 1 n 1 n 3 n 1 n 6 n 3 n 6, q 3 = αn 6 r 1 n 1 n 3 n 6, q 4 = n 1 n 5 + n 3 n 5 n n 4, q 5 = n 1 n 3 n 5 n n 3 n 4 n 5 αr 1. (3 Relationships between parameter condition P γ (γ 1, γ, γ 3 and coefficients in equation (.1 are shown as follows. γ 1 = Q q 1, γ = 1 Q q 1, γ 3 = 1 3 (Q 5 + 6Q 6, Q 5 where Q 3 = 1 6 Q 5 6Q 6 Q 5, Q 4 = 1q 3 1q 3 3q 11 54q 1 q q 3 1q q 3 0, Q 5 = 3 36q 1 q 108q 3 + 8q Q 4, Q 6 = 1 3 q 1 9 q 1, with ( n 5 Q q ( 1 q4 Q q 1 + q5 = 0, ( ( 1 n 5 Q q ( ( 1 3 Q5 + 6Q 5 ( 1 q 4 Q 4 Q q 1 + q5 = 0, ( 1 n 5 Q q ( 1 ( 1 3 Q5 + 6Q 6 (1 ( q4 3 Q5 + 6Q 6 = 0. Q 5 Q 5

24 596 J. Li, S. T. Zhu, R. L. Tian, W. Zhang & X. Li Section 3.3 Relationships between parameter condition P p (p 1, p, p 3 and coefficients in equation (.1 are shown as follows. Section 3.4 p 1 = n 5 q q 1, p = q 4 n 5 q 5 q q 1 q 3, p 3 = q 5 q 3. Relationships between parameter condition P f1 8 (f 1, f, f 3, f 4, f 5, f 6, f 7, f 8 and coefficients in equation (.1 are shown as follows. f 1 =n 5(q + q 1 q 3 + n 5 (q 5 q 1 + 4q 5 q q 1q 4 3q 5 q 4q, f =n 5q 3 4q 5q q 1q 5, f 3 = n 5 q 5 q 3 q 1 q 3 q 5 + q 3q 4 q 5q, f 4 =q 4ω 0 + n 5ω 4 0 n 5 ω 0q 5 + q 5, f 5 = q 4 ω 0 cos( τ 0 + (n 5 ω 3 0 q 5 sin( τ 0, f 6 =(n 5 ω 3 0 q 5 cos( τ 0 + q 4 ω 0 sin( τ 0, f 7 = 3ω 0 q + ( 0 n 5 ω 0 + τ 0 q 5 cos( τ 0 (n 5 + τ 0 q 4 sin( τ 0, f 8 = q 1 (n 5 + τ 0 q 4 cos( τ 0 ( 0 n 5 ω 0 + τ 0 q 5 sin( τ 0. Section 4. φ J (0 = z z + W (1 0 (0z + W (1 11 φ P (0 = zρ 1 z ρ 1 + W (3 0 (0z φ A (0 = z z + W ( 0 (0z + W ( W ( 11 φ J ( = ze iω0τ ze iω0τ + W (1 0 (z (1 z (0z z + W 0 (0 +, (3 z (0z z + W 0 (0 +, ( z (0z z + W 0 (0 +, + W (1 11 φ P ( = zρ 1 e iτ z ρ 1 e iτ + W (3 0 (z φ A ( = ze iτ ze iτ + W ( 0 (z and K 1 = s 1 βρ 1, K = s 1 β(ρ 1 + rho 1, K 3 = s 1 β ρ 1, K 4 = s 1 ( W (1 0 ( β W (3 11 (0 W (1 11 (0 + W ( W ( 11 (3 W 0 (0 (0 ρ 1 W (1 11 (0 ρ 1 (1 z (z z + W 0 ( +, (3 z (z z + W 0 ( +, ( z (z z + W 0 ( +, 0 (0, W (1

25 Stability and Hopf bifurcation of Section 4.3 K 5 = s, K 6 = s, K 7 = s, ( ( ( K 8 = s W 0 (0 W 11 (0, K 9 =βρ 1 e iω0τ, K 10 =β(ρ 1 + ρ 1, K 11 =β ρ 1 e iω0τ, ( ( (3 (1 K 1 =β W 11 ( W 11 (ρ 1 e i τ ( + W (3 0 ( W (1 0 ( ρ 1 e iτ. and M (1 1 = K 1m 4 m 5 r 1 K 5 m 5 + n K 9 m 4 m 3 m 4 m 5 + r 1 αm 5 n n 4 m 4 e iτ, 1 = K 5 + αm (1 1, M (3 1 = n 4e iω0τ M (1 1 + K 9, m 4 m 5 M ( = K n 3 (n n 6 K 6 r 1 (n n 6 + K 10 n n 3, αr 1 (n n 6 n n 3 n 4 n n 3 (n n 6 M (1 = K 6 + αm (1, M (3 = n 4M (1 + K 10, n 3 (n n 6 M ( where m 3 = i n 1, m 4 = i n 3, m 5 = i n 6 +n e iω0τ. From equation (4.1, we obtain W (i (i 0 (θ and W 11 (θ(i = 1,, 3, W (1 0 (θ = ig 0 e iθ iḡ 0 3 e iθ + M (1 1 eiθ, W ( 0 (θ = ig 0 e iθ iḡ 0 3 e iθ + M ( 1 eiθ, W (3 0 (θ = ig 0 e iω0θ iḡ 0 3 e iω0θ + M (3 1 eiω0θ. From equation(4., we have W (1 11 (θ = ig 11 e iθ iḡ 11 e iθ + M (1, W ( 11 (θ = ig 11 e iθ iḡ 11 e iθ + M (, W (3 11 (θ = ig 11 e iω0θ iḡ 11 e iω0θ + M (3.