STABILITY AND HOPF BIFURCATION IN A SYMMETRIC LOTKA-VOLTERRA PREDATOR-PREY SYSTEM WITH DELAYS

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1 Electronic Journal of Differential Equations Vol. 013 (013) No. 09 pp ISSN: URL: or ftp ejde.math.txstate.edu STABILITY AND HOPF BIFURCATION IN A SYMMETRIC LOTKA-VOLTERRA PREDATOR-PREY SYSTEM WITH DELAYS JING XIA ZHIXIAN YU RONG YUAN Abstract. This article concerns a symmetrical Lotka-Volterra predator-prey system with delays. By analyzing the associated characteristic equation of the original system at the positive equilibrium and choosing the delay as the bifurcation parameter the local stability and Hopf bifurcation of the system are investigated. Using the normal form theory we also establish the direction and stability of the Hopf bifurcation. Numerical simulations suggest an existence of Hopf bifurcation near a critical value of time delay. 1. Introduction Since the pioneering work of Volterra [18] the delayed predator-prey models of Lotka-Volterra type have played an important role in the development of population dynamics. For instance the predator-prey system ẋ(t) = x(t)[r 1 a 11 x(t τ) a 1 y(t)] ẏ(t) = y(t)[ r + a 1 x(t) a y(t)] (1.1) was first proposed and discussed briefly by May [11] in The dynamics of system (1.1) or systems similar to (1.1) has been widely investigated concerning boundedness of solutions stability permanence persistence periodic phenomena bifurcation and chaos (see for example [ ] and their references). In particular research on the stability of positive equilibrium and periodic solutions arising from the Hopf bifurcation is very critical which helps us to realize the law and trend for species quantity. Also extensive models of various fields (such as biology neural network engineering systems epidemic disease) have been embodied these properties (see [ ]). In mathematical biology Lotka-Volterra type predator-prey with a delay have been well studied by Song and Wei [15] Yan and Li [0] Yan and Zhang [1] and so on. In addition models involving two discrete delays are increasingly applied to the study of a variety of situations. Faria [6] considered the predator-prey system of the form ẋ(t) = x(t)[r 1 a 11 x(t) a 1 y(t τ 1 )] (1.) ẏ(t) = y(t)[ r + a 1 x(t τ ) a y(t)]. 000 Mathematics Subject Classification. 34K18 37G05 37G10 9D5. Key words and phrases. Predator-prey system; delay; stability; Hopf bifurcation; normal form. c 013 Texas State University - San Marcos. Submitted September Published January

2 JING XIA ZHIXIAN YU RONG YUAN EJDE-013/09 The author took τ as a bifurcation parameter and obtained the existence of local Hopf bifurcation. Moreover Song et. al took the sum of two delays τ 1 + τ as a parameter and showed that when τ passed through some critical values Hopf bifurcation occurs. In the present article we focus on the following symmetrical Lotka-Volterra predator-prey system: ẋ(t) = x(t)[r 1 ax(t) bx(t τ 1 ) cy(t τ )] ẏ(t) = y(t)[r ay(t) + cx(t τ 1 ) by(t τ )] (1.3) r 1 r a b c τ 1 τ are constants with a > 0 b > 0 c > 0 and τ 1 0 τ 0. The variables x(t) and y(t) denote the population densities of prey and predator at time t respectively. The initial conditions for system (1.3) are x(θ) = φ 1 (θ) τ 1 θ 0 φ 1 (0) > 0 y(θ) = φ (θ) τ θ 0 φ (0) > 0. Obviously in the case b = 0 and a b systems (1.3) is reduced to system (1.). Systems (1.3) shows that in the absence of predator species the prey species are governed by general case of Hutchinson s equation ẋ(t) = x(t)[r 1 ax(t) bx(t τ 1 )]. (1.4) (There is an extensive literature on various aspects of (1.4) see e.g. [7 16]). In the presence of predators there is a hunting term cy(t τ ) with a certain delay τ called the hunting delay. In the absence of preys the feedback cx(t τ 1 ) has a nonnegative delay which is the time of the predator maturation. For system (1.3) Saito et al studied the permanence and the global asymptotic stability of positive equilibrium including the cases b 0. Our main purpose is to study stability and Hopf bifurcation of (1.3). This article is organized as follows. In Section the local stability of the positive equilibrium of system (1.3) is addressed. In Section 3 we show that the system (1.3) with τ 1 = τ undergoes Hopf bifurcation as the delays cross some critical values. In Section 4 following the procedure of deriving normal form due to Faria and Magalhães [4 5] the direction and stability of the Hopf bifurcation are determined. Finally we give some numerical simulations to justify the theoretical analysis.. Local stability The following lemma will be needed in analyzing the characteristic equation of the linearize system. Lemma.1 ([3]). Let f(λ τ) = λ + aλ + bλe τλ + c + de τλ a b c d τ are real numbers and τ 0. Then as τ varies the sum of the multiplicities of zeros of f in the open right half-plane can change only if a zero appears on or crosses the imaginary axis. In the rest of this paper we assume that system (1.3) has a unique positive equilibrium E(x y ): and x y. x = r 1(a + b) r c (a + b) + c > 0 y = r (a + b) + r 1 c (a + b) + c > 0

3 EJDE-013/09 STABILITY AND HOPF BIFURCATION 3 By using the transformation x(t) = x(t) x y(t) = y(t) y and dropping the bars for simplicity of notation system (1.3) can be transformed into the system ẋ(t) = ax x(t) bx x(t τ 1 ) cx y(t τ ) ax (t) bx(t)x(t τ 1 ) cx(t)y(t τ ) ẏ(t) = ay y(t) + cy x(t τ 1 ) by y(t τ ) ay (t) + cx(t τ 1 )y(t) by(t)y(t τ ). Then we obtain the linearized system ẋ(t) = ax x(t) bx x(t τ 1 ) cx y(t τ ) ẏ(t) = ay y(t) + cy x(t τ 1 ) by y(t τ ). The associated characteristic equation of system (.) is λ + λ(ax + ay + bx e λτ1 + by e λτ ) + a x y + abx y e λτ + abx y e λτ1 + (b x y + c x y )e λ(τ1+τ) = 0. Obviously λ = 0 is not a root of (.3). equation (.3) becomes (.1) (.) (.3) When τ 1 = τ = 0 the characteristic λ + λ(ax + ay + bx + by ) + a x y + abx y + b x y + c x y = 0. (.4) Letting λ 1 λ be the characteristic roots of (.4) then λ 1 + λ = (ax + ay + bx + by ) < 0 λ 1 λ = a x y + abx y + b x y + c x y > 0. Thus all roots of (.4) have negative real parts. When τ = 0 Equation (.3) becomes λ + pλ + q + se λτ1 + lλe λτ1 = 0 (.5) p = ax + ay + by q = (a + ab)x y s = (ab + b + c )x y l = bx. We first study (.5) with the delay τ 1. We consider the stable intervals of τ 1 by taking τ 1 as a parameter and applying the lemma in [3]. Secondly we investigate (.3) in the stable interval of τ 1. Choosing τ as a parameter and using the lemma in [3] once more we will find a stable interval (depending on τ 1 ) for τ. This will be the stable interval for (.3). By putting τ 1 = 0 in (.5) we obtain λ + λ(l + p) + s + q = 0. Based on the above analysis we know that two roots of the above equation have negative real parts. In addition λ = iσ is a root of (.5) if and only if σ satisfies: σ + iσp + q + se iστ1 + liσe iστ1 = 0. Separating the real and imaginary parts we have σ q = s cos στ 1 + lσ sin στ 1 pσ = s sin στ 1 lσ cos στ 1 which leads to σ 4 + σ (p q l ) + q s = 0. In this article we will use the following hypotheses: (H1) q < s (H) q > s q + l p > 0 (q + l p ) > 4(q s ).

4 4 JING XIA ZHIXIAN YU RONG YUAN EJDE-013/09 Then we easily obtain the following results. Lemma.. For (.5) we have 1. If (H1) holds then when τ 1 = τ (1) 1k Equation (.5) has a pair of purely imaginary roots ±iσ + ;. If (H) holds then when τ 1 = τ (1) 1k (τ 1 = τ () 1k ) Equation (.5) has a pair of purely imaginary roots ±iσ + (±iσ ); 3. If neither (H1) nor (H) hold then (.5) has no purely imaginary root σ± = q + l p ± (q + l p ) 4(q s ) τ (1) 1k = 1 arccos σ +s qs lpσ+ σ + s + l σ+ + kπ (.6) σ + τ () 1k = 1 arccos σ s qs lpσ σ s + l σ + kπ σ Let λ nk = µ nk (τ 1 ) + iσ nk (τ 1 )(n = 1 ) denote the roots of (.5) satisfying µ 1k (τ (1) 1k ) = 0 σ 1k(τ (1) 1k ) = σ + µ k (τ () 1k ) = 0 σ k(τ () 1k ) = σ. In what follows we can verify the following transversality conditions. Lemma.3. Assume that either (H1) or (H) holds then d Re λ 1k (τ (1) 1k ) dτ 1 > 0 d Re λ k (τ () 1k ) dτ 1 < 0. (.7) Proof. Differentiating the characteristic equation (.5) with respect to τ 1 we obtain (dλ) 1 λ + p = dτ λ(λ + pλ + q) + l λ l + λs τ 1 λ. Thus if λ = iσ + then { sign Re( dλ ) } { dτ τ1=τ = sign Re( dτ } (1) 1 1k dλ ) λ=iσ+ { σ 4 = sign + l + σ+s qs + p s l q } [(q σ+) + p σ+](σ +l + s. ) Since σ 4 +l + σ +s qs + p s l q = 1 l (q + l p ) 4(q s ) [ (q + l p ) 4(q s ) + (q + l p )] + s (q + l p ) 4(q s ) thus either (H1) or (H) implies that sign { Re( dλ dτ 1 ) } τ1=τ > 0. If λ = (1) iσ we 1k can similarly prove that { sign Re( dλ ) } dτ τ1=τ < 0. () 1 1k

5 EJDE-013/09 STABILITY AND HOPF BIFURCATION 5 The proof is complete. From Lemmas. and.3 we have the following result. Lemma.4. For Equation (.5) the following statements are true: 1. Assume that (H1) holds then for any τ 1 [0 τ (1) 10 ) all roots of (.5) have negative real parts and for τ 1 > τ (1) 10 Equation (.5) has at least one root with positive real parts.. Assume that (H) holds then there exists an integer k 1 > 0 such that there are k 1 switches from stability to instability to stability; that is when τ 1 (τ () 1k τ (1) 1k+1 ) k = k 1 1 all roots of (.5) have negative real parts τ () 1 1 = 0; When τ 1 (τ (1) 1k τ () 1k ) and τ 1 > τ (1) 1k 1 k = k 1 1 Equation (.5) has at least one root with positive real parts. In what follows we consider (.3) with τ 1 in its stable interval. We will take τ as a bifurcation parameter and have the following result. Lemma.5. Assume that all roots of (.5) have negative real parts then there exists τ (τ 1 ) > 0 such that when τ [0 τ (τ 1 )) all roots of (.3) have negative real parts. Proof. Since all roots of (.5) have negative real parts all roots of (.3) with τ = 0 have negative real parts. Similar to the proof in [3] let f(λ τ 1 τ ) be the left-hand side of (.3). Obviously f(λ τ 1 τ ) is analysis in λ and τ. We take τ as a bifurcation parameter when τ varies the sum of the multiplicities of zero of f(λ τ 1 τ ) in the open right half-plane can change only if a zero appears on or crosses the imaginary axis. Thus there exists τ (τ 1 ) > 0 such that when τ τ (τ 1 ) all roots of (.3) have negative real parts. The proof is complete. Following Lemma.5 we can get sufficient conditions for local stability of system (.1). Theorem.6. For system (.1) the following statements hold. 1. Suppose that (H1) holds then for any τ 1 [0 τ (1) 10 ) there is a τ (τ 1 ) > 0 such that when τ [0 τ (τ 1 )) the zero solution of system (.1) is locally asymptotically stable.. Suppose that (H) holds then for any τ 1 (τ () 1k τ (1) 1k+1 ) k = k 1 1 there is τ (τ 1 ) > 0 such that when τ [0 τ (τ 1 )) the zero solution of system (.1) is locally asymptotically stable. 3. Suppose that neither (H1) nor (H) holds then for any τ 1 0 there is a τ (τ 1 ) > 0 such that when τ [0 τ (τ 1 )) the zero solution of system (.1) is locally asymptotically stable. 3. Hopf bifurcation Let τ 1 = τ = τ. System (1.3) becomes ẋ(t) = x(t)[r 1 ax(t) bx(t τ) cy(t τ)] ẏ(t) = y(t)[r ay(t) + cx(t τ) by(t τ)]. and the characteristic equation (.3) has the form (3.1) λ + p 1 λ + q 1 + (s 1 λ + l 1 )e λτ + m 1 e λτ = 0 (3.)

6 6 JING XIA ZHIXIAN YU RONG YUAN EJDE-013/09 p 1 = ax + ay q 1 = a x y l 1 = abx y s 1 = bx + by m 1 = b x y + c x y. (3.3) Lemma 3.1. If b +c > a then there exists σ + (> 0) such that (3.) with τ = τ k has a simple pair of conjugate purely imaginary roots ±iσ + τ k = 1 arccos (l 1 s 1 p 1 )σ+ l 1 (q 1 m 1 ) σ + ( σ+ + q 1 ) m kπ k = p 1 σ + σ + Proof. Obviously λ = 0 is not a root of (3.). If λ = iσ is a root of (3.) then σ + ip 1 σ + q 1 + l 1 e iστ + iσs 1 e iστ + m 1 e iστ = 0. (3.4) Multiplying both sides of (3.4) by e iστ we have ( σ + ip 1 σ + q 1 )e iστ + (l 1 + iσs 1 ) + m 1 e iστ = 0. (3.5) Separating the real parts and imaginary parts of (3.5) leads to and thus cos στ = (l 1 s 1 p 1 )σ l 1 (q 1 m 1 ) [( σ + q 1 ) m 1 + p 1σ ] sin στ = s 1 σ 3 + [l 1 p 1 s 1 (q 1 + m 1 )]σ [( σ + q 1 ) m 1 + p 1σ ] = [(l 1 s 1 p 1 )σ l 1 (q 1 m 1 )] + [s 1 σ 3 + [l 1 p 1 s 1 (q 1 + m 1 )]σ]. Let y = σ define f(y) = [( y + q 1 ) m 1 + p 1y] [(l 1 s 1 p 1 )y l 1 (q 1 m 1 )] y[s 1 y + l 1 p 1 s 1 (q 1 + m 1 )]. (3.6) (3.7) Since f(y) is a quartic function we have f(y) + as y +. On the other hand f(0) = (q 1 m 1) l 1(q 1 m 1 ) = (q 1 m 1 ) (q 1 + m 1 + l 1 )(q 1 + m 1 l 1 ) > 0 Define F (y) = [( y + q 1 ) m 1 + p 1y] G(y) = [(l 1 s 1 p 1 )y l 1 (q 1 m 1 )] H(y) = y[s 1 y + l 1 p 1 s 1 (q 1 + m 1 )] Then f(y) = F (y) + G(y) + H(y). Assume that g(y) = ( y + q 1 ) m 1 + p 1y then g(0) = q 1 m 1 = (q 1 + m 1 )(q 1 m 1 ) = (q 1 + m 1 )x y (a b c ) In view of b + c > a we have g(0) < 0. On the other hand g(y) + as y +. Thus there is a z 0 > 0 such that g(z 0 ) = 0; i.e. F (z 0 ) = 0. It is easily to see that positive roots of F G and H are mutually different when x y. Consequently G(z 0 ) < 0 H(z 0 ) < 0. Thus f(z 0 ) < 0 which together with f(0) > 0 imply that there exists a y 0 > 0 such that f(y 0 ) = 0. Obviously there exists another positive root because f(y) + as y +. Furthermore one of the two roots is a simple root at least. Let y 0 be such a simple root and y 0 = σ. Substituting y 0 = σ into (3.6) we have τ k = 1 arccos (l 1 s 1 p 1 )σ+ l 1 (q 1 m 1 ) σ + ( σ+ + q 1 ) m kπ k = (3.8) p 1 σ σ +

7 EJDE-013/09 STABILITY AND HOPF BIFURCATION 7 The proof is complete. Let λ(τ) = µ(τ) + iσ(τ) be the root of (3.) near τ = τ k satisfying µ(τ k ) = 0 σ(τ k ) = σ + and k = τ k is defined by (3.8). Lemma 3. ([14]). Assume that b + c > a and a > b then { sign Re( dλ } dτ ) τ=τk > 0. The proof of the above lemma is similar to that in [14 p. 549] and is omitted. By Lemma 3.1 and 3. we have the following result. Theorem 3.3. Assume that b + c > a and a > b. Then 1. When τ [0 τ 0 ) the equilibrium E = (x y ) of (3.1)is locally asymptotically stable;. When τ > τ 0 the equilibrium E = (x y ) of (3.1) is unstable; 3. System (3.1) undergoes a Hopf bifurcation at E = (x y ) and τ = τ k. 4. Direction and stability of the Hopf bifurcation In this section we shall determine the direction and stability of the bifurcating periodic solutions by employing the normal form theory duo to Faria and Magalhães [4 5]. Let z 1 (t) = x(τt) x z (t) = y(τt) y. Then system (3.1) can be rewritten as a functional differential equation in C([ 1 0] R ) ż 1 (t) = x τ[ az 1 (t) bz 1 (t 1) cz (t 1)] τz 1 (t)[az 1 (t) + bz 1 (t 1) + cz (t 1)] ż (t) = y τ[cz 1 (t 1) az (t) bz (t 1)] + τz (t)[cz 1 (t 1) and its linearized system is az 1 (t) bz (t 1)]. ż 1 (t) = τ[ ax z 1 (t) bx z 1 (t 1) cx z (t 1)] ż (t) = τ[cy z 1 (t 1) ay z (t) by z (t 1)]. The characteristic equation associated with system (4.) has the form (4.1) (4.) λ +λτ(ay +ax +bx e λ +by e λ )+τ x y (a +abe λ +b e λ +c e λ ) = 0. (4.3) By the change of variable λ = ξτ Equation (4.3) becomes ξ + p 1 ξ + q 1 + (s 1 ξ + l 1 )e ξτ + m 1 e ξτ = 0 (4.4) p 1 q 1 s 1 l 1 m 1 are defined by (3.3). Equation (4.4) is the same as (3.). Therefore from Section 3 we know that (4.4) has a simple pair of conjugate complex roots ξ(τ) = µ(τ) ± iσ(τ) which satisfy µ(τ k ) = 0 σ(τ k ) = σ + µ (τ k ) 0 τ k is given in (3.8). Thus Eq. (4.3) has two complex roots λ(τ) = τµ(τ) ± iτσ(τ) which satisfy d Re λ(τ) τ=τk = τ dτ d Re ξ(τ) τ=τk + Re ξ(τ) τ=τk = τ k µ (τ k ) 0. dτ

8 8 JING XIA ZHIXIAN YU RONG YUAN EJDE-013/09 Letting z = (z 1 z ) T then system (4.1) can further be written as ż(t) = L(τ)z t + F (z t τ) (4.5) ax L(τ)ϕ = τ ϕ 1 (0) bx ϕ 1 ( 1) cx ϕ ( 1) cy ϕ 1 ( 1) ay ϕ (0) by ϕ ( 1) aϕ F (ϕ τ) = τ 1 (0) bϕ 1 (0)ϕ 1 ( 1) cϕ 1 (0)ϕ ( 1) aϕ (0) + cϕ 1 ( 1)ϕ (0) bϕ (0)ϕ ( 1) and ϕ = (ϕ 1 ϕ ) T C([ 1 0] R ). Introducing a new parameter α = τ τ k system (4.5) is equivalent to ż(t) = L(τ k )(z t ) + F 0 (z t α) (4.6) F 0 (z t α) = L(α)z t + F (z t τ k + α). Let A 0 be the infinitesimal generator corresponding to ż(t) = L(τ k )z t. Then A 0 has a pair of purely imaginary characteristic roots ±iσ k (σ k = τ k σ + ) which are simple and no other characteristic roots with zero real part. Define Λ = { iσ k iσ k } and denote by P the invariant space of A 0 corresponding to Λ the dimension of P equals to. The phase space is C = C([ 1 0] R ) so we can decompose the space C as C = P Q by applying the formal adjoint theory for functional differential equations in [8]. Consider complex coordinates and still write as C([ 1 0] C ). Suppose that Φ 1 Φ are the eigenvectors of the operator A 0 associated with eigenvalues iσ k iσ k respectively. Thus Φ = (Φ 1 Φ ) is a basis of P Φ 1 (θ) = e iσkθ v T = e iσkθ (1 v ) T Φ (θ) = Φ 1 (θ) 1 θ 0 v = (v 1 v ) T is a vector in C and which satisfies v = iσ k + τ k ax + τ k bx e iσ k τ k cx e iσ k L(τ k )(Φ 1 ) = iσ k v T. For Φ = (Φ 1 Φ ) we note that Φ = ΦB B is a diagonal matrix of the form diag(iσ k iσ k ). Also the formal adjoint operator A 0 of A 0 has eigenvectors Ψ 1 Ψ corresponding to iσ k iσ k. Ψ(s) = col(ψ 1 (s) Ψ (s)) constructs a basis of the adjoint space P of P and Ψ 1 (s) = e iσ ks (u 1 u ) Ψ (s) = Ψ 1 (s) 0 s 1. We know Ψ 1 (0) = 0 1 Ψ 1( θ)dη(θ) therefore we can choose u = iσ k + τ k ax + τ k bx e iσ k τ k cy e iσ u 1. k In order to have (Ψ Φ) = ((Ψ j Φ i ) i j = 1 ) = I ; i.e. second order identical matrix (.. ) is a bilinear inner product form we have (ψ ϕ) = ψ(0)ϕ(0) u 1 = 0 θ 1 0 ψ(ξ θ)dη(θ)ϕ(ξ)dξ y (1 + iσ k )(y x v ) + τ kax y (1 v ).

9 EJDE-013/09 STABILITY AND HOPF BIFURCATION 9 Using that the enlarged phase space BC : {ϕ : [ 1 0] C ϕ is continuous on [ 1 0) it exists lim θ 0 ϕ(θ). The elements of BC have the form ψ = ϕ + X 0 α ϕ C α C X 0 = X 0 (θ) is given by { I θ = 0 X 0 (θ) = 0 1 θ < 0. The projection of C onto P ϕ Φ(Ψ ϕ) associated with the decomposition C = P Q is replaced by π : BC P such that π(ϕ + X 0 α) = Φ[(Ψ ϕ) + Ψ(0)α]. Thus we have the decomposition BC = P ker π. Using the decomposition z t = Φx(t) + y t x(t) C y t ker π C 1 =: Q 1 the equation (4.6) is equivalent to the system ẋ = Bx + Ψ(0)F 0 (Φx + y α) d dt y = A Q 1y + (I π)x (4.7) 0F 0 (Φx + y α). Consider the Taylor expansion as follows: Ψ(0)F 0 (Φx + y α) = 1 f 1 (x y α) + 1 3! f 1 3 (x y α) + h.o.t. (4.8) (I π)x 0 F 0 (Φx + y α) = 1 f (x y α) + 1 3! f 3 (x y α) + h.o.t. fj 1(x y α) and f j (x y α) are homogeneous polynomials in (x y α) of degree j j = 3 with coefficients in C and ker π respectively h.o.t. stands for higher order terms. Thus the normal form method gives a normal form on the center manifold of the origin for (4.7) as ẋ = Bx + 1 g1 (x 0 α) + 1 3! g1 3(x 0 α) + h.o.t (4.9) g 1 g3 1 are the second and third order terms in (x α) respectively. Always following [5] V m+p j (X) denotes the linear space of the homogeneous polynomials of degree j in m+p real variables x = (x 1... x m ) α = (α 1... α p ) with coefficients in X; that is { } V m+p j (X) = c (ql) x q α l : (q l) N m+p 0 c (ql) X. The operators M 1 j and M 1 j (ql) =j are defined by (M 1 j p)(x α) = D x p(x α)bx Bp(x α) j 3 act on Vj (C ) = Im(Mj 1) ker(m j 1 ) and ker(m 1 j ) = span{x q α l e k : (q λ) = λ k k = 1 q N 0 l N 0 (q l) = j} e 1 e is the canonical basis of C. Hence { ker(m 1 x ) = span 1 α 0 { ( ker(m3 1 x ) = span 1x x1 α 0 0 ) ( 0 x α ( 0 x 1 x ) } ) ( 0 x α ) }.

10 10 JING XIA ZHIXIAN YU RONG YUAN EJDE-013/09 From (4.8) we have f 1 (x y α) = Ψ(0)[L(α)(Φx + y) + F (Φx + y τ k )] (4.10) aϕ F (ϕ τ k ) = τ 1 (0) bϕ 1 (0)ϕ 1 ( 1) cϕ 1 (0)ϕ ( 1) k aϕ (0) + cϕ 1 ( 1)ϕ (0) bϕ (0)ϕ ( 1) Therefore we obtain f 1 A1 x (x 0 α) = 1 α + A x α + a 0 x 1 + a 0 x Āx 1 α + Ā1x α + ā 0 x 1 + ā 0 x (4.11) here A 1 = iσ k τ k (u 1 + u v ) A = iσ k τ k (u 1 + u v ) a 0 = iσ k ( u 1 x + u v y ) a 0 = iσ k ( u 1 x + u v (4.1) y ). Since the second order terms in (α x) of the normal form on the center manifold are given by 1 g1 (x 0 α) = 1 Proj ker(m 1) f 1 (x 0 α) it follows that 1 A1 x g1 (x 0 α) = 1 α (4.13) Ā 1 x α A 1 = iσ k τ k (u 1 + u v ). We notice that { ( g3(x 1 0 α) ker(m3 1 x ) = span 1x x1 α x 1 x ) 0 } x α. However the terms O( x α ) are irrelevant to determine ( the generic ) ( Hopf ) bifurcation. Thus we only need to compute the coefficients of 1 x 0 x and 0 x 1 x. Note that 1 3! g1 3(x 0 α) = 1 3! Proj f ker(m3 1) 3 1 (x 0 α) = 1 3! Proj f s 3 1 (x 0 0) + O( x α ) { x s := span 1x 0 } 0 x 1 x and f 1 3 (x 0 0) = f 1 3 (x 0 0) + 3 [(D xf 1 )U 1 (D x U 1 )g 1 ] (x00) + 3 [(D yf 1 )h] (x00) here f 1 3 (x 0 0) is the third order terms of the equation which is obtained after computing the second terms of the normal form. Now we compute 1 3! g1 3(x 0 α) step by step. We first compute Proj s [(D x f 1 )U 1 ] (x00). Following [5] we know that U 1 (x 0) = (M 1 ) 1 P 1 If 1 (x 0 0). From (4.11) we can easily see that f 1 a0 x (x 0 0) = 1 + a 0 x ā 0 x 1 + ā 0 x.

11 EJDE-013/09 STABILITY AND HOPF BIFURCATION 11 According to the definition of M 1 the equation M 1 U 1 (x 0) = f 1 (x 0 0) can be written as the following differential equations: u 1 u 1 x 1 x u 1 = 1 (a 0 x 1 + a 0 x x 1 x iσ ) k u u x 1 x + u = 1 (ā 0 x 1 + ā 0 x x 1 x iσ ). k For the equation we can easily obtain ( 1 U 1 iσ (x 0) = k (a 0 x a 0x ) ) 1 iσ k ( 1 3ā0x 1 ā 0 x. ) Thus ( Proj s [(D x f 1 )U 1 3iσ ] (x00) = k a 0 x ) 1x 3iσ k a 0 x 1 x. From (4.13) we know g(x 1 0 0) = 0. Thus [(D x U 1 )g] 1 (x00) = 0. Next we compute Proj s [(D y f 1 )h] (x00) h = (h 1 h ) T is a second order homogeneous polynomial in (x 1 x α) with coefficients in Q 1 which has the form h = h(x 1 x α) = h 110 x 1 x + h 101 x 1 α + h 011 x α + h 00 x 1 + h 00 x + h 00 α and h = h(x 1 x α) is the unique solution in V 3 (Q 1 ) of the equation As in [5] we know that (M h)(x α) = (I π)x 0 [L(α)(Φx) + F (Φx τ k )]. (M h)(x α) = D x h(x α)bx A Q 1(h(x α)) = D x h(x α)bx ḣ(x α) X 0[L(τ k )(h(x α)) ḣ(x α)(0)] = (I π)x 0 [L(α)(Φx) + F (Φx τ k )]. thus h(x θ) is also evaluated by the system ḣ(x) D x h(x)bx = ΦΨ(0)[F (Φx τ k )] (4.14) ḣ(x)(0) L(τ k )(h(x)) = F (Φx τ k ) (4.15) ḣ denotes the derivative of h(x)(θ) relative to θ. follows that Then we have f 1 (x y 0) = Ψ(0)F (Φx + y τ k ) u1 u = ad τ 1 bd 1 d cd 1 n ū 1 ū k an 1 + cd n 1 bn 1 n d 1 = x 1 + x + y 1 (0) n 1 = v x 1 + v x + y (0) d = e iσ k x 1 + e iσ k x + y 1 ( 1) According to (4.10) it l = e iσ k v x 1 + e iσ k v x + y ( 1). [(D y f 1 )h] (x00) ad u T 1 h 1 (0) bd 1h 1 ( 1) bd h 1 (0) cd 1h ( 1) cn h 1 (0) = τ k an 1h (0) + cd h (0) + cn 1h 1 ( 1) bn 1h ( 1) bn h (0) ad ū T 1 h 1 (0) bd 1h 1 ( 1) bd h 1 (0) cd 1h ( 1) cn h 1 (0) an 1h (0) + cd h (0) + cn 1h 1 ( 1) bn 1h ( 1) bn h (0)

12 1 JING XIA ZHIXIAN YU RONG YUAN EJDE-013/09 Thus d 1 = x 1 + x d = e iσ k x 1 + e iσ k x n 1 = v x 1 + v x n = e iσ k v x 1 + e iσ k v x. Proj s [(D y f 1 c3 x )h)] (x00) = 1x c 3 x 1 x (4.16) c 3 = u T τ k Ah (0) + Ah 1 00(0) + Bh 1 110( 1) + Bh 1 00( 1) + Ch 110( 1) + Ch 00( 1) Dh 1 110( 1) + Dh 1 00( 1) + Eh 110(0) + Eh 00(0) + F h 110( 1) + F h 00( 1) and A = a be iσ k ce iσ k v B = b C = c D = cv E = av + ce iσ k be iσ k v F = bv. Thus we only need to compute h 110 (θ) and h 00 (θ). From (4.14) and (4.15) we know that h 110 = (h h 110) T and h 00 = (h 1 00 h 00) T are respectively the solution of the following two systems ( 0 ḣ 110 = 0) (4.17) a1 ḣ 110 (0) L(τ k )(h 110 ) = τ k b 1 and ḣ 00 iσ k h 00 = a Φ 1 Φ 0 ā 0 a ḣ 00 (0) L(τ k )(h 00 ) = τ k b a 1 = a be iσ k be iσ k ce iσk v ce iσ k v = 0 b 1 = av v + ce iσk v + ce iσ k v be iσ k v v be iσ k v v = 0 Solving (4.17) and (4.18) we obtain and a = a be iσ k cv e iσ k b = av + ce iσ k v bv e iσ k. (4.18) h 110 (θ) = c 1 (4.19) h 00 (θ) = 1 iσ k (a 0 e iσ kθ v + 1 3ā0e iσ kθ v) + e iσ kθ c (4.0) c 1 = (c (1) 1 c() 1 )T c = (c (1) c() )T and c (1) 1 = a 1(a + b)y b 1 cx x y [(a + b) + c c () 1 = a 1cy + b 1 (a + b)x ] x y [(a + b) + c ]

13 EJDE-013/09 STABILITY AND HOPF BIFURCATION 13 ( ) c (1) = τ k a (iσ k + τ k ay + τ k by e iσ k ) τk b cx e iσ k ( τk c x y e 4iσ k + (iσ k + τ k ax ) + τ k bx e iσ k )(iσ k + τ k ay + τ k by e iσ k ) c () = ( ) τ k b (iσ k + τ k ax + τ k bx e iσ k ) + τk a cy e iσ k ( τk c x y e 4iσ k + (iσ k + τ k ax ) + τ k bx e iσ k )(iσ k + τ k ay + τ k by e iσ k ). Finally we compute Proj s f3 1 (x 0 0). From (4.8) we can see that f3 1 (x 0 0) = 0. Summarizing the above steps we obtain 1 A3 x 3! g1 3(x 0 0) = 1x Ā 3 x 1 x A 3 = 1 6iσ k a c 3. (4.1) Accordingly the normal form of (4.9) has the form ẋ = Bx + 1 g1 (x 0 α) + 1 3! g1 3(x 0 α) + h.o.t. A1 x = Bx + 1 α A3 x + 1x Ā 1 x α Ā 3 x 1 x + O( x α + x 4 ). The normal form (4.9) relative to P can be written in real coordinates (w 1 w ) through the change of variables x 1 = w 1 iw x = w 1 + iw. Followed by the use of polar coordinates (ρ ξ) w 1 = ρ cos ξ w = ρ sin ξ this formal form becomes ρ = k 1 αρ + k ρ 3 + O(α ρ + (ρ α) 4 ) ξ = σ k + O( (ρ α ) (4.) k 1 = Re A 1 k = Re A 3. Following [] we know that the sign of k 1 k determines the direction of the bifurcation and the sign of k determines the stability of the nontrivial periodic solution bifurcating from Hopf bifurcation. Therefore we have the following theorem Theorem 4.1. The flow of (4.6) on the center manifold of the origin at τ = τ k is given by (4.). Hopf bifurcation is supercritical if k 1 k < 0 and subcritical if k 1 k > 0. Moreover the nontrivial periodic solution is stable if k < 0 and unstable if k > 0. We remark that by the same method and calculation for the nonsymmetric system we can also obtain a similar conclusion. Thus the symmetry does not affect the emergence of periodic solutions. 5. Numerical simulations In the section we will give numerical simulations to illustrate the conclusions obtained. Consider system (3.1) with r 1 = 3/ r = 1 a = 1/ b = 1/3 c = 1;

14 14 JING XIA ZHIXIAN YU RONG YUAN EJDE-013/09 that is ẋ(t) = x(t)[ 3 1 x(t) 1 x(t τ) y(t τ)] 3 ẏ(t) = y(t)[1 1 y(t) + x(t τ) 1 (5.1) 3 y(t τ)]. In this case system (5.1) has unique positive equilibrium (x y ) = ( ). By means of the software Maple we know that y 0 = is a simple root of (3.7). By computation we obtain σ + = τ 0 = σ 0 = τ 0 σ + = Applying Theorems 3.3 and 4.1 we have the following theorem. Theorem 5.1. The characteristic equation of (5.1) at (x y ) = ( ) has a simple pair of purely imaginary roots ±iσ 0 for τ = τ 0 and the other roots has nonzero real parts. System (5.1) with τ 0 = has a supercritical Hopf bifurcation and the nontrivial periodic solution bifurcating from Hopf bifurcation of (5.1) is stable on the center manifold. Proof. From Theorem 3.3 we know that the equilibrium (x y ) is locally asymptotically stable when τ [ ) and is unstable when τ > In the following we will consider the direction and stability of Hopf bifurcation at τ = τ 0 = By means of software Maple we obtain the following values: ( ) v1 1 u i v = = u = = i i and v a 0 = i u c 3 = i. According to (4.1) and (4.1) we obtain A 1 = i a 0 = i A 3 = i. Thus k 1 = k = Applying Theorem 4.1 we obtain that system (5.1) has a supercritical Hopf bifurcation at τ = τ 0 and the nontrivial periodic solution associated with Hopf bifurcation at τ = τ 0 is stable on the center manifold. The proof is complete. The numerical simulations for τ = are shown in Figure 1. Acknowledgements. The authors would like to thank the anonymous referees for their carefully reading of the original manuscript and for their valuable comments and suggestions for improving the results as well as the exposition of this article. Z. Yu was supported by grant from the National Natural Science Foundation of China. R. Yuan was supported by National Natural Science Foundation of China and RFDP. References [1] E. Beretta Y. Kuang; Convergence results in a well-known delayed predator-prey systems J. Math. Anal. Appl. 04 (1996) [] S. N. Chow J. K. Hale; Methods of Bifurcation Theory Springer-Verlag New York 198. [3] K. L. Cooke Z. Grossman; Discrete delay distributed delay and stability switches J. Math. Anal. Appl. 86 (198) [4] T. Faria L. T. Magalhães; Normal forms for retarded functional differential equations and applications to Bogdanov-Takens singularity J. Differential Equations 1 (1995) 01-4.

15 EJDE-013/09 STABILITY AND HOPF BIFURCATION 15 x 0. a1 x b t 1.6 a t 1.6 b y 1.4 y 1.4 y t x a3 y t x b3 Figure 1. Trajectories for system (5.1) with τ = 4.41 (left column) and τ = 4.49 (right column) on the t-x plane (top row) t-y plane (middle row) and x-y plane (bottom row) [5] T. Faria L. T. Magalhães; Normal forms for retarded functional differential equations with parameters and applications to Hopf bifurcation J. Differential Equations 1 (1995) [6] T. Faria; Stability and bifurcation for a delayed predator-prey model and the effect of diffusion J. Math. Anal. Appl. 54 (001) [7] K. Gopalsamy; Stability and oscillation in delayed differential equations of population dynamics Dordrecht:Kluwer Academic Press 199. [8] J. K. Hale; Theory of Functional Differential Equations Springer-Verlag New York [9] Y. Kuang; Delay Differential Equations with Application in Population Dynamics Academic Press New York [10] Z. H. Liu R. Yuan; Stability and bifurcation in a harmonic oscillator with delays Chaos Solitons and Fractals 3 (005)

16 16 JING XIA ZHIXIAN YU RONG YUAN EJDE-013/09 [11] R. M. May; Time delay versus stability in population models with two and three trophic levels Ecology 4 (1973) [1] S. Nakaoka Y. Saito Y. Takeuchi; Stability delay and chaotic behavior in a Lotka-Volterra predator-prey system Math. Biosci. Engineer. 3 (006) [13] S. G. Ruan; Absolute stability conditional stability and bifurcation in Kolmogorov-type predator-prey systems with discrete delays Quarterly of Applied Mathematics 59 (001) [14] Y. Saito; Permanence and global stability for general Lotka-Volterra predator-prey systems with distributed delays Nonlinear Anal. 47 (001) [15] Y. L. Song J. J. Wei; Local Hopf bifurcation and global periodic solutions in a delayed predarot-prey systems J. Math. Anal. Appl. 301 (005) 1-1. [16] C. J. Sun M. A. Han Y. P. Yang; Analysis of stability and Hopf bifurcation for a delayed logistic equation Chaos Solitons and Fractals 31 (007) [17] C. Sun Y. Lin M. Han ; Stability and Hopf bifurcation for an epidemic model with delay Chaos Solitons and Fractals 30 (006) [18] V. Volterra; Lecons sur la théorie mathematique de la lutte pour la vie Gauthier-Villars Paris [19] J. J. Wei S. G. Ruan; Stability and bifurcation in a neural network model with two delays Physica D 130 (1990) [0] X. P. Yan W.T. Li; Hopf bifurcation and global periodic solutions in a delayed predator-prey system Applied Mathematics and Computation 177 (006) [1] X. P. Yan C. H. Zhang; Hopf bifurcation in a delayed Lotka-Volterra predator-prey system Nonlinear Anal.: Real World Appl. 9 (008) Jing Xia School of Mathematical Sciences Peking University Beijing China address: xiajing005@mail.bnu.edu.cn Zhixian Yu College of Science University of Shanghai for Science and Technology Shanghai China address: yuzx090@yahoo.com.cn Rong Yuan School of Mathematical Sciences Beijing Normal University Beijing China address: ryuan@bnu.edu.cn