STABILITY AND HOPF BIFURCATION ANALYSIS FOR A LOTKA VOLTERRA PREDATOR PREY MODEL WITH TWO DELAYS

Transcription

1 In. J. Appl. Mah. Compu. Sci. 2 Vol. 2 No DOI: 478/v6--7- STABILITY AND HOPF BIFURCATION ANALYSIS FOR A LOTKA VOLTERRA PREDATOR PREY MODEL WITH TWO DELAYS CHANGJIN XU MAOXIN LIAO XIAOFEI HE i School of Mahemaics and Saisics Guizhou College of Finance and Economics Luchongguan Rd 269 Guiyang 554 PR China ii Faculy of Science Hunan Insiue of Engineering Fuxing Rd 88 Xiangan 44 PR China xcj43@26.com School of Mahemaics and Physics Nanhua Universiy Changsheng Rd 26 Hengyang 42 PR China maoxinliao@63.com Deparmen of Mahemaics Zhangjiajie College of Jishou Universiy Renming Rd 2 Zhangjiajie 427 PR China hexiaofei525@63.com In his paper a wo-species Loka Volerra predaor-prey model wih wo delays is considered. By analyzing he associaed characerisic ranscendenal equaion he linear sabiliy of he posiive equilibrium is invesigaed and Hopf bifurcaion is demonsraed. Some explici formulae for deermining he sabiliy and direcion of Hopf bifurcaion periodic soluions bifurcaing from Hopf bifurcaions are obained by using normal form heory and cener manifold heory. Some numerical simulaions for supporing he heoreical resuls are also included. Keywords: predaor-prey model delay sabiliy Hopf bifurcaion.. Inroducion Various mahemaical models have been esablished in he sudy of populaions since Vio Volerra and James Loka proposed seminal models of predaor-prey models in he mid 92s. Among hese models predaor-prey models play an imporan role in populaion dynamics. Many heoreicians and experimenaliss concenraed on he sabiliy of predaor-prey sysems and more specifically hey invesigaed he sabiliy of such sysems when ime delays are incorporaed ino he models. Such delayed sysems received grea aenion since ime delay may have very complicaed impac on he dynamical behavior of he sysem such as he periodic srucure bifurcaion ec. Kuang and Takeuchi 994; Xu e al. 24; Zhou e al. 28; Teramoo e al. 979; Bhaacharyya and Mukhopadhyay 26; Prajneshu Holgae 987; Gao e al. 28; Xu and Ma 28; Kar and Pahari 27; Klamka 99. May 973 firs proposed and briefly discussed he sabiliy of he following delayed predaor-prey sysem: ẋ =x[r a x τ a 2 y] ẏ =y[ r 2 + a 2 x a 22 y] x and y can be inerpreed as he populaion densiies of preys and predaors a ime respecively; τ is he feedback ime delay of he preys o he growh of he species iself; r > denoes an inrinsic growh rae of he preys and r 2 > denoes he deah rae of he predaors; he parameers a ij i j = 2 are all posiive consans. Song and Wei 25 invesigaed furher he dynamics of he sysem. Considering he delay τ as he bifurcaion parameer hey obained ha under cerain condiions he unique posiive equilibrium of he sysem is absolue sable while i is condiionally sable and here exis k swiches from sabiliy o insabiliy o sabiliy under oher condiions. Furher by using normal form heory and he cener manifold heorem hey obained he formu-

2 98 C. Xu e al. lae for deermining he direcion of Hopf bifurcaions and he sabiliy of bifurcaing periodic soluions. Yan and Li 26 incorporaed he same delay τ ino he populaion densiy of he predaor in he second equaion of he sysem and obained he following: ẋ =x[r a x τ a 2 y] 2 ẏ =y[ r 2 + a 2 x a 22 y τ] Regarding he delay τ as he bifurcaion parameer hey invesigaed he sabiliy of he sysem 2 and sudied he properies of Hopf bifurcaion for he sysem 2 by using normal form heory and he cener manifold heorem which is differen from ha used by Song and Wei 25. Faria 2 invesigaed he sabiliy and Hopf bifurcaion of he following sysem wih wo differen delays: ẋ =x[r a x a 2 y τ 2 ] 3 ẏ =y[ r 2 + a 2 x τ a 22 y]. According o he view poin of Kuang 993 Yan and Zhang 28 considered he sabiliy and Hopf bifurcaion of he following delayed sysem: ẋ =x[r a x τ a 2 y τ] ẏ =y[ r 2 + a 2 x τ a 22 y τ]. 4 Based on 4 we consider he following sysem: ẋ =x[r a x τ a 2 y τ 2 ] ẏ =y[ r 2 + a 2 x τ 2 a 22 y τ ] 5 x and y denoe he populaion densiies of preys and predaors a ime respecively; r > denoes he inrinsic growh rae of preys and r 2 > denoes he deah rae of predaors; he parameers a ij i j = 2 are all posiive consans; τ is he gesaion periodic of preys and predaors; τ 2 in he firs equaion of he sysem 5 denoes he huning delay of predaor o prey and τ 2 in he second equaion of he sysem 5 is he delay in predaor mauraion. The biological meaning of he sysem 5 is as follows. In he absence of predaors he prey species follows he logisic equaion ẋ =x[r a x τ ]. In he presence of predaors here is a huning erm a 2 y τ 2 wih a cerain delay τ 2 called he huning delay. In he absence of prey species he predaor species follows he equaion ẏ =x[ r 2 a 22 y τ ] i.e. he number of predaors decreases. The posiive feedback a 2 x τ 2 has a posiive delay τ 2 which is he delay in predaor mauraion. We would like o poin ou ha he sysems 4 are all a special case of he sysem 5. In his paper we will sudy he sabiliy and local Hopf bifurcaion for he sysem 5. To he bes of our knowledge i is he firs ime he research of Hopf bifurcaion for he model 5 is underaken. The remainder of he paper is organized as follows. In Secion 2 we invesigae he sabiliy of he posiive equilibrium and he occurrence of local Hopf bifurcaions. In Secion 3 he direcion and sabiliy of local Hopf bifurcaion are esablished. In Secion 4 numerical simulaions are carried ou o illusrae he validiy of he main resuls. Some main conclusions are drawn in Secion Sabiliy of he posiive equilibrium and local Hopf bifurcaions In his secion we shall focus on analyzing he corresponding linearized sysem a he posiive equilibrium of he sysem 5 and invesigae he sabiliy of his equilibrium poin and he exisence of local Hopf bifurcaions occurring a he posiive equilibrium. Since any ime delay does no change he equilibrium of he sysem and according o Yan and Zhang 28 we know ha he delayed prey predaor model 5 has unique posiive equilibrium poins E x y if x = r a 22 + r 2 a 2 a a 22 + a 2 a 2 y = r a 2 r 2 a a a 22 + a 2 a 2 H r a 2 r 2 a >. Le x = x x ȳ = y y and sill denoe x ȳ by x y respecively. Then 5 becomes ẋ =m x+m 2 x τ +m 3 y τ 2 + m 4 xx τ +m 5 xy τ 2 6 ẏ =n y+n 2 x τ 2 +n 3 y τ + n 4 x τ 2 y+n 5 yy τ m = r a x a 2 y m 2 = a x m 3 = a 2 x m 4 = a 2 m 5 = a 2 n = r 2 + a 2 x a 22 y n 2 = a 2 y n 3 = a 22 y n 4 = a 2 n 5 = a 22. I is easy o check ha m = n =. The linearizaionofeqn.6a is ẋ =m2 x τ +m 3 y τ 2 7 ẏ =n 2 x τ 2 +n 3 y τ

3 Sabiliy and Hopf bifurcaion analysis for a Loka Volerra predaor-prey model wih wo delays 99 whose characerisic equaion is λ 2 m 2 + n 3 λe λτ + m 2 n 3 e 2λτ m 3 n 2 e 2λτ2 =. 8 In order o invesigae he disribuion of roos of he ranscendenal equaion 8 he following resul is useful. Lemma. Ruan and Wei 23 For he ranscendenal equaion P λ e λτ...e λτm = λ n + p λn + + p n λ + p n [ ] + p λn + + p n λ + p n e λτ + [ ] + p m λ n + + p m n λ + pm n e λτm = as τ τ 2 τ 3...τ m vary he sum of orders of he zeros of P λ e λτ...e λτm in he open righ half plane can change and only a zero appears on or crosses he imaginary axis. In he sequel we consider hree cases. Case A: τ = τ 2 =. Then 8 becomes λ 2 m 2 + n 3 λ + m 2 n 3 m 3 n 2 =. 9 A se of necessary and sufficien condiions ha all roos of 9 have a negaive real par is given in he following form: H2 m 2 + n 3 < m 2 n 3 m 3 n 2 >. Then he equilibrium poin E x y is locally asympoically sable when he condiion H holds. Case B: τ =τ 2 >. Then 8 becomes λ 2 + pλ + r + qe 2λτ2 = p = m 2 + n 3 r = m 2 n 3 q = m 3 n 2. For ω>iωbeing a roo of i follows ha q cos 2ωτ2 = ω 2 r q sin 2ωτ 2 = pω which leads o ω 4 +p 2 2rω 2 + r 2 q 2 =. 2 I is easy o see ha if he condiion H3 p 2 2r > r 2 q 2 > holds hen Eqn. 2 has no posiive roos. Hence all roos of have negaive real pars when τ 2 [ + under he condiions H2 and H3. If H2 and H4 p 2 2r > r 2 q 2 < hold hen 2 has a unique posiive roo ω. 2 Subsiuing ω 2 ino we obain τ 2n = arccos qω2 r } 2ω q 2 +2nπ 3 n = 2... Le λτ 2 =ατ 2 +iωτ 2 be a roo of near τ 2 = τ 2n and ατ 2n = ωτ 2n =ω. From funcional differenial equaion heory for every τ 2n n = 2... here exiss ε>such ha λτ 2 is coninuously differeniable a τ 2 for τ 2 τ 2n <ε. Subsiuing λτ 2 ino he lef-hand side of and differeniaing wih respec o τ 2 wehave dλ dτ 2 which leads o [ dreλτ dτ 2 ] Noing ha dreλ sign dτ 2 = τ 2=τ 2n } τ2=τ 2n 2λ + pe2λτ2 2qλ 2λ + pe 2λτ 2 =Re 2qλ τ 2 λ 4 } τ2=τ2n = p sin 2ω τ 2n +2ω cos 2ω τ 2n qω = p2 2r +2ω 2 2q 2 >. dλ } =sign Re = dτ 2 τ 2=τ 2n we have dreλ >. dτ 2 τ2=τ 2n According o he above analysis and Corollary 2.4 of Ruan and Wei 23 we have he following resuls. Lemma 2. For τ = assume ha H and H2 are saisfied. Then he following conclusions hold: i If H3 holds hen he posiive equilibrium E x y of he sysem 5 is asympoically sable for all τ 2. ii If H4 holds hen he posiive equilibrium E x y of he sysem 5 is asympoically sable for τ 2 <τ 2 and unsable for τ 2 >τ 2. Furhermore he sysem 5 undergoes a Hopf bifurcaion a he posiive equilibrium E x y when τ 2 = τ 2.

4 C. Xu e al. Case C: τ > τ 2 >. We consider Eqn. 8 wih τ 2 in is sable inerval. Regarding τ as a parameer wihou loss of generaliy we consider he sysem 5 under Assumpions H2 and H4. Le iωω > be a roo of 8. Then we can obain k ω 4 + k 2 ω 3 + k 3 ω 2 + k 4 = 5 k =2m 2 n 3 +cos2ωτ 2 2 +sin 2 2ωτ 2 k 2 = 2m 2 + n 3 sin2ωτ 2 [m 3 n 2 m 2 n 3 +cos2ωτ 2 + m 2 n 3 m 3 n 2 cos 2ωτ 2 ] k 3 =[m 2 + n 3 m 3 n 2 sin 2ωτ 2 ] 2 +[m 2 + n 3 m 2 n 3 m 3 n 2 cos 2ωτ 2 ] 2 k 4 = [m 2 n 3 m 3 n 2 cos 2ωτ 2 m 2 n 3 +cos2ωτ 2 + m 3 n 2 sin 2 2ωτ 2 ] 2. Wrie Hω =k ω 4 + k 2 ω 3 + k 3 ω 2 + k 4. 6 I is easy o check ha H < and lim ω + Hω = +. We can obain ha 5 has finie posiive roos ω ω 2...ω n. For every fixed ω i i = k here exiss a sequence τ j i j = } such ha 5 holds. Le =minτ j i i = 2...k; j = 2...}. 7 When τ = Eqn. 8 has a pair of purely imaginary roos ±iω for τ 2 [τ 2. In he following we assume ha [ ] dreλ H5. dτ λ=iω Thus by he general Hopf bifurcaion heorem for FDEs by Hale 977 we have he following resul on he sabiliy and Hopf bifurcaion in he sysem 5. Theorem. For he sysem 5 assume ha H H2 H4 and H5 are saisfied and τ 2 [τ 2. Then he posiive equilibrium E x y is asympoically sable when τ and he sysem 5 undergoes a Hopf bifurcaion a he posiive equilibrium E x y when τ =. 3. Direcion and sabiliy of Hopf bifurcaion In he previous secion we obained condiions for Hopf bifurcaion o occur when τ =. In his secion we shall derive explici formulae deermining he direcion sabiliy and period of hese periodic soluions bifurcaing from he posiive equilibrium E x y a hese criical values of τ by using echniques from normal form and cener manifold heory Hassard e al. 98. Throughou his secion we always assume ha he sysem 5 undergoes Hopf bifurcaion a he posiive equilibrium E x y for τ = andhen±iω denoes he corresponding purely imaginary roos of he characerisic equaion a he posiive equilibrium E x y. Wihou loss of generaliy we assume ha τ2 < τ2 τ 2. For convenience le ū i = u i τi = 2 and τ = + μ is defined by 7 and μ R; dropping he bar for he simplificaion of noaion hen he sysem 5 can be wrien as an FDE in C = C[ ] R 2 as u =L μ u +F μ u 8 u =xy T C and u θ =u + θ = x+θy+θ T CandL μ : C RF : R C R are given by and L μ φ =τ + μc +τ + μd φ τ 2 φ 2 τ 2 φ φ 2 9 F μ φ =τ + μf f 2 T 2 respecively φθ =φ θφ 2 θ T C and C = m3 n 2 m2 D = n 3 f = m 4 φ φ + m 5 φ φ 2 τ 2 f 2 = n 4 φ τ 2 φ 2 + n 5 φ 2 φ 2. From he discussion in Secion 2 we know ha if μ = hen sysem 8 undergoes Hopf bifurcaion a he posiive equilibrium E x y and he associaed characerisic equaion of he sysem 8 has a pair of simple imaginary roos ±iω. By he represenaion heorem here is a marix funcion wih bounded variaion componens ηθ μθ [ ] such ha L μ φ = dηθ μφθ for φ C. 2

5 Sabiliy and Hopf bifurcaion analysis for a Loka Volerra predaor-prey model wih wo delays In fac we can choose + μc + D [ θ = + μc + D θ τ 2 τ ηθ μ = + μd θ τ 2 θ =. 22 For φ C[ ] R 2 define dφθ θ< Aμφ = dθ 23 dηs μφs θ = and θ< Rφ = 24 F μ φ θ =. Then 8 is equivalen o he absrac differenial equaion u = Aμu + Rμu 25 u θ =u + θθ [ ]. For ψ C[ ] R 2 define dψs s ] A ψs = ds dηt ψ s =. For φ C[ ] R 2 and ψ C[ ] R 2 define he bilinear form ψ φ = ψφ θ ξ= ψ T ξ θdηθφξdξ ηθ = ηθ hea = A and A are adjoin operaors. From he discussions in Secion 2 we know ha ±iω are eigenvalues of A andheyare also he eigenvalues of A corresponding o iω and iω respecively. By direc compuaion we can obain qθ =α T e iω θ q s =Mα e iω s M =/K α = iω m 2 e iω m 3 e iω τ2 α = iω + m 2 e iω n 2 e iω τ2 K =+ᾱα + m 2 e iω + n2 α τ2 τ eiω 2 + m 3 ᾱτ 2 e iω τ 2 + n3 ᾱα e iω. Furhermore q sqθ =and q s qθ =. NexweusehesamenoaionasHassarde al. 98 and we firs compue he coordinaes o describe he cener manifold C a μ =.Leu be he soluion of Eqn. 8 when μ =. Define z = q u W θ =u θ 2Rezqθ}. on he cener manifold C.Wehave 26 W θ =W z zθ 27 W z zθ=w z z z 2 = W W z z + W 2 z and z and z are local coordinaes for cener manifold C in he direcions of q and q. Noing ha W is real if so is u we consider only real soluions. For soluions u C of 8 ż =iω z + q θf Wz z θ+2rezqθ} def = iω z + q F. In oher words gz z =g 2 z 2 Hence we have ż =iω z + gz z 2 + g z z + g 2 z g 2 gz z = q F z z =F u = Mτ [m 4 e iω + m5 αe iω τ 2 ] +ᾱ n 4 αe iω τ2 + n5 αe iω z 2 z 2 z M [m 4 Reαe iω } + m5 Reαe iω τ 2 } +ᾱ n 4 Reαe iω τ 2 } + n5 Re α 2 e iω } ] z z + M [m 4 e iω τ2 + m5 ᾱe iω τ2 +ᾱ n 4 ᾱe iω ] +n 5 ᾱ 2 e iω z 2 + Mτ m 4 2 W 2 ᾱeiω

6 2 C. Xu e al. + 2 W 2 τ 2 + αw 2 e iω + W + m 5 2 W 2 ᾱeiω τ 2 + W αeiω τ 2 +W 2 τ W 2 2 τ 2 [ +ᾱ n 4 2 W 2 τ 2 ᾱ + W τ τ 2 α +W e iω + W τ e iω 2 + n 5 2 W 2 2 ᾱeiω + W 2 α ] } +αw 2 eiω + 2 W 2 2 ᾱ z 2 z +. Then we obain g 2 =2 M [m 4 e iω + m5 αe iω τ 2 ] +ᾱ n 4 αe iω τ2 + n5 αe iω g =2 M [m 4 Reαe iω } + m5 Reαe iω τ 2 } +ᾱ n 4 Reαe iω τ 2 } + n5 Re α 2 e iω } ] g 2 =2 M [m 4 e iω τ2 + m5 ᾱe iω τ2 ] +ᾱ n 4 ᾱe iω + n5 ᾱ 2 e iω g 2 =2 M m 4 2 W 2 + ᾱeiω 2 W 2 2 +αw 2 + W e iω + m 5 2 W 2 ᾱeiω τ 2 + W αe iω τ 2 +W 2 τ W 2 2 τ 2 [ +ᾱ n 4 2 W 2 τ 2 ᾱ + W τ τ 2 α + 2 W 2 τ 2 2 eiω 2 + W τ e iω 2 + n 5 2 W 2 2 ᾱeiω + W 2 α +αw 2 e iω + 2 W 2 2 ᾱ ] }. For unknown W i 2 θwi θi = 2 in g 2 we sill have o compue hem. From 25 and 26 we have AW 2Re q F W qθ} θ< = AW 2Re q F qθ} + F θ= = AW + Hz z θ 29 Hz z θ =H 2 θ z2 2 + H θz z 3 + H 2 θ z Comparing he coefficiens we obain AW 2i ω W 2 = H 2 θ 3 AW θ = H θ. 32 We know ha for θ [ Hz z θ = q f qθ q f qθ = gz zqθ ḡz z qθ. 33 Comparing he coefficiens of 33 wih 3 gives H 2 θ = g 2 qθ ḡ 2 qθ 34 H θ = g qθ ḡ qθ. 35 From 3 34 and he definiion of A wege Ẇ 2 θ =2iω W 2 θ+g 2 qθ+ g 2 qθ. 36 Noing ha qθ =qe iω θ wehave W 2 θ = ig 2 ω qe iω θ + iḡ 2 3ω qe iω θ + E e 2iω θ 37 E =E E2 R2 is a consan vecor. Similarly from and he definiion of A we have Ẇ θ =g qθ+ g qθ 38 W θ = ig ω qe iω θ + iḡ ω qe iω θ + E 2 39 E 2 =E 2 E2 2 R2 is a consan vecor. In wha follows we shall seek appropriae E E 2 in respecively. From he definiion of A and i follows ha dηθw 2 θ =2iω W 2 H 2 4 and dηθw θ = H 4 ηθ =ηθ. From 3 we have H 2 = g 2 q g 2 q + 2 H H 2 T 42 H = g q g q + 2 P P 2 T 43

7 Sabiliy and Hopf bifurcaion analysis for a Loka Volerra predaor-prey model wih wo delays 3 H = m 4 e iω + m5 αe iω τ2 H 2 = n 4 αe iω τ2 + n5 αe iω P = m 4 Reαe iω } + m5 Reαe iω τ 2 } P 2 = n 4 Reαe iω τ 2 } + n5 Re α 2 e iω }. Noing ha iω I iω I e iω θ dηθ q = e iω θ dηθ q = and subsiuing 37 and 42 ino 4 we have 2iω I e 2iω θ dηθ E =2 H H 2 T. Tha is 2iω m 2 e 2iω m 3 e 2iω τ 2 n 2 e 2iω τ 2 2iω n 3 e 2iω E I follows ha =2H H 2 T. E = Δ Δ E 2 = Δ 2 Δ 44 v v Δ =de 2 v 3 v 4 H v Δ =2de 2 H 2 v 4 v H Δ 2 =2de v 2 H 2 v =2iω m 2 e 2iω v2 = m 3 e 2iω τ 2 v 3 = n 2 e 2iω τ 2 v 4 =2iω n 3 e 2iω. Similarly subsiuing 38 and 43 ino 4 we have dηθ E 2 =2 P P 2 T. Tha is m2 m 3 E n 2 n 2 =2 P P 2 T. 3 I follows ha E 2 = Δ 2 Δ 2 E 2 2 = Δ 22 Δ 2 45 m2 m Δ 2 =de 3 n 2 n 3 P m Δ 2 =2de 3 P 2 n 3 m2 P Δ 22 =2de. n 2 P 2 From we can calculae g 2 and derive he following values: i c = g 2ω 2 g 2 g 2 g g μ 2 = Rec } Reλ } β 2 =2Rec T 2 = Imc } + μ 2 Imλ } ω. These formulas give a descripion of he Hopf bifurcaion periodic soluions of 8 a τ = on he cener manifold. From he discussion above we have he following resul. Theorem 2. For he sysem 5 assume ha H H2 H4 and H5 are saisfied. The periodic soluion is supercriical resp. subcriical if μ 2 > resp. μ 2 <. The bifurcaing periodic soluions are orbially asympoically sable wih an asympoical phase resp. unsable if β 2 < resp. β 2 >. The period of he bifurcaing periodic soluions increases resp. decrease if T 2 > resp. T 2 <. 4. Numerical examples In his secion we presen some numerical resuls of he sysem 5 o verify he analyical predicions obained in he previous secion. From Secion 3 we may deermine he direcion of a Hopf bifurcaion and he sabiliy of he bifurcaion periodic soluions. Le us consider he following sysem: ẋ =x[.5.5x τ y τ 2 ] 46 ẏ =y[.5+x τ 2 y τ ] which has a posiive equilibrium E x y = When τ = we can easily obain ha H2 and H4 are saisfied. For example ake n =. By some compuaion by means of Malab 7. we ge ω 59τ 2

8 4 C. Xu e al From Lemma 2 we know ha he ransversal condiion is saisfied. Thus he posiive equilibrium E = is asympoically sable for τ 2 < τ and unsable for τ 2 >τ whichisshownin Fig.. When τ 2 = τ Eqn. 46 undergoes Hopf bifurcaion a he posiive equilibrium E = i.e. a small ampliude periodic soluion occurs around E = when τ =and τ 2 is close o τ 2 =.52 which is shown in Fig. 2. Le τ 2 = and choose τ as a parameer. We have Then he posiive equilibrium is asympoically sable when τ [. The Hopf bifurcaion value of Eqn. 46 is By he algorihms derived in Secion 3 we can obain λ i c i μ x a g β T y.5. Furhermore i follows ha μ 2 > and β 2 <. Thus he posiive equilibrium E = is sable when τ < as is illusraed by compuer simulaions see Fig. 3. When τ passes hrough he criical value he posiive equilibrium E = looses is sabiliy and a Hopf bifurcaion occurs i.e. a family of periodic soluions bifurcae from he posiive equilibrium E = Since μ 2 > and β 2 < he direcion of Hopf bifurcaion is τ > and hese bifurcaing periodic soluions from E = a τ are sable which is depiced in Fig Conclusions In his paper we invesigaed he local sabiliy of he posiive equilibrium E x y and local Hopf bifurcaion of a Loka Volerra predaor-prey model wih wo delays. We showed ha if H H2 H4 and H5 are saisfied and τ 2 [τ 2 hen he posiive equilibrium E x y is asympoically sable when τ. As he delay τ increases he posiive equilibrium E x y loses is sabiliy and a sequence of Hopf bifurcaions occur a he posiive equilibrium E x y i.e. a family of periodic orbis bifurcaes from he posiive equilibrium E x y. Finally he direcion of Hopf bifurcaion and he sabiliy of he bifurcaing periodic orbis were discussed by applying normal form heory and he cener manifold heorem. A numerical example verifying our heoreical resuls was also given. y b x g c Fig.. Behavior and phase porrai of he sysem 46 wih τ = τ 2 =3.7 >τ Hopf bifurcaion occurs from he posiive equilibrium E = 2. The iniial 3 6 value is. Acknowledgmen We wish o hank he reviewers for heir valuable commens ha led o ruly significan improvemens in he manuscrip.

9 Sabiliy and Hopf bifurcaion analysis for a Loka Volerra predaor-prey model wih wo delays x.6 x a 5 5 a y y b b y y x c x c Fig. 2. Behavior and phase porrai of he sysem 46 wih τ = τ 2 =3<τ The posiive equilibrium E = 2 is asympoically sable. The iniial value 3 6 is. Fig. 3. Behavior and phase porrai of he sysem 46 wih τ 2 = 3τ =.5 <τ The posiive equilibrium E = 2 is asympoically sable. The iniial value 3 6 is. References Bhaacharyya R. and Mukhopadhyay B. 26. Spaial dynamics of nonlinear prey-predaor models wih prey migraion and predaor swiching Ecological Complexiy 32: Faria T. 2. Sabiliy and bifurcaion for a delayed predaor- prey model and he effec of diffusion Journal of Mahemaical Analysis and Applicaions 2542: Gao S.J. Chen L.S. and Teng Z.D. 28. Hopf bifurcaion and global sabiliy for a delayed predaor-prey sysem wih sage srucure for predaor Applied Mahemaics and Compuaion 222:

10 6 C. Xu e al. x y y a b x c Fig. 4. Behavior and phase porrai of he sysem 46 wih τ 2 = 3τ =.8 > Hopf bifurcaion occurs from he posiive equilibrium E = 2. The iniial 3 6 value is. Hale J Theory of Funcional Differenial Equaions Springer-Verlag Berlin. Hassard B. Kazarino D. and Wan Y. 98. Theory and Applicaions of Hopf Bifurcaion Cambridge Universiy Press Cambridge. Kar T. and Pahari U. 27. Modelling and analysis of a preypredaor sysem sage-srucure and harvesing Nonlinear Analysis: Real World Applicaions 82: Klamka J. 99. Conrollabiliy of Dynamical Sysems Kluwer Dordrech. Kuang Y Delay Differenial Equaions wih Applicaions in Populaion Dynamics Academic Press Boson MA. Kuang Y. and Takeuchi Y Predaor-prey dynamics in models of prey dispersal in wo-pach environmens Mahemaical Biosciences 2: Li K. and Wei J. 29. Sabiliy and Hopf bifurcaion analysis of a prey-predaor sysem wih wo delays Chaos Solions & Fracals 425: May R.M Time delay versus sabiliy in populaion models wih wo and hree rophic levels Ecology 42: Prajneshu Holgae P A prey-predaor model wih swiching effec Journal of Theoreical Biology 25: Ruan S. and Wei J. 23. On he zero of some ranscendenial funcions wih applicaions o sabiliy of delay differenial equaions wih wo delays Dynamics of Coninuous Discree and Impulsive Sysems Series A : Song Y.L. and Wei J. 25. Local Hopf bifurcaion and global periodic soluions in a delayed predaor-prey sysem Journal of Mahemaical Analysis and Applicaions 3: 2. Teramoo E.I. Kawasaki K. and Shigesada N Swiching effecs of predapion on compeiive prey species Journal of Theoreical Biology 793: Xu R. Chaplain M.A.J. and Davidson F.A. 24. Periodic soluions for a delayed predaor-prey model of prey dispersal in wo-pach environmens Nonlinear Analysis: Real World Applicaions 5: Xu R. and Ma Z.E. 28. Sabiliy and Hopf bifurcaion in a raio-dependen predaor-prey sysem wih sage srucure Chaos Solions & Fracals 383: Yan X.P. and Li W.T. 26. Hopf bifurcaion and global periodic soluions in a delayed predaor-prey sysem Applied Mahemaics and Compuaion 77: Yan X.P. and Zhang C.H. 28. Hopf bifurcaion in a delayed Loka Volerra predaor-prey sysem Nonlinear Analysis: Real World Applicaions 9: Zhou X.Y. Shi X.Y. and Song X.Y. 28. Analysis of nonauonomous predaor-prey model wih nonlinear diffusion and ime delay Applied Mahemaics and Compuaion 96:

11 Sabiliy and Hopf bifurcaion analysis for a Loka Volerra predaor-prey model wih wo delays 7 Changjin Xu received he M.Sc. degree from he Kunming Universiy of Science and Technology Kunming China in 24 and he Ph.D. degree from Cenral Souh Universiy Changsha China in 2. He is also a lecurer a he Faculy of Science Hunan Insiue of Engineering Xiangan China. His research ineress focus on he sabiliy and bifurcaion heory of differenial equaions wih delays. Maoxin Liao received he M.Sc. degree from he Changsha Universiy of Science and Technology Changsha in 25 and he Ph.D. degree from Cenral Souh Universiy Changsha China in 2. He is also an associae professor a he School of Mahemaics and Physics Universiy of Souh China Hengyang China. His research ineress focus on he sabiliy and bifurcaion heory of differenial equaions wih delays. Xiaofei He is a Ph.D. suden a he School of Mahemaical Sciences and Compuing Technology Cenral Souh Universiy Changsha China and an associae professor a he College of Mahemaical and Compuer Science Jishou Universiy Hunan Jishou China. He received his M.S. from Hunan Normal Universiy Changsha in 28. His research ineress focus on he sabiliy and bifurcaion heory of differenial equaions wih delays. Received: 6 July 2 Revised: 8 Ocober 2