Bifurcation Analysis of a Stage-Structured Prey-Predator System with Discrete and Continuous Delays

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1 Applied Mahemaics hp://dx.doi.org/.46/am Published Online July (hp:// Bifurcaion Analysis of a Sage-Srucured Prey-Predaor Sysem wih Discree and Coninuous Delays Shunyi Li Wenwu Liu Xiangui Xue Deparmen of Mahemaics Qiannan Normal College for Naionaliies Duyun China lishunyi9845@6.com Received May 5 ; revised June 5 ; acceped June 4 Copyrigh Shunyi Li e al. This is an open access aricle disribued under he Creaive Commons Aribuion License which permis unresriced use disribuion and reproducion in any medium provided he original work is properly cied. ABSTRACT A hree-sage-srucured prey-predaor model wih discree and coninuous ime delays is sudied. The characerisic equaions and he sabiliy of he boundary and posiive equilibrium are analyzed. The condiions for he posiive equilibrium occurring Hopf bifurcaion are given by applying he heorem of Hopf bifurcaion. Finally numerical simulaion and brief conclusion are given. Keywords: Three-Sage-Srucured; Prey-Predaor Model; Time Delay; Hopf Bifurcaion. Inroducion In he naural world here are many species whose individual members have a life hisory ha akes hem hrough wo sages: immaure and maure. In 99 Aiello and Freedman [] inroduced single-species sagesrucured model wih ime delay and he sabiliy of he sysem was sudied. In 997 Wang and Chen [] inroduced single-species sage-srucured model wihou ime delay and found ha an orbially asympoically sable periodic orbi exisence. In hese papers [] he auhors assume ha he life hisory of each populaion is divided ino disincive sages: he immaure and maure members of he populaion only he maure member can reproduce hemselves. However in he naure many species go hrough hree life sages: immaure maure and old. For example many female animals lose reproducive abiliy when hey are old. A single species wih hree life hisory sage and cannibalism model have considered by S. J. Gao [4] and shown ha he sabiliy of he posiive equilibrium can change a finie number of imes a mos as ime delay is increased when he model under some parameers values. Recenly a nonauonomous hree-sage-srucured predaor-prey sysem wih ime delay have sudied by S. J. Yang and B. Shi [5] by using he coninuaion heorem of coincidence degree heory he exisence of a posiive periodic soluion is obained. And he local Hopf bifurcaion and global periodic soluions for a delayed hreesage-srucured predaor-prey considered by Li e al. [6 7].. Formulaion of he Model In his paper we consider following hree-sage-srucured prey-predaor model wih discree and coninuous ime delays e x x x x Ey x x x a x ax bx y y kex d f y d (.) s e d e ds x x x are he densiies of immaure preys maure preys and old preys populaion a ime y is he densiy of predaor populaion a ime respecively. All of he parameers are posiive is he birh rae of maure prey populaion and b are he deah rae of immaure maure and old prey populaion respecively. and a are he mauriy rae and ageing rae of he prey populaion respecively. and f are he densiy dependen coefficiens of immaure prey populaion and predaor populaion respecively. k k is he rae of conversing prey ino predaor and E is he predaion coefficien. and are he gesaion delay and densiy dependen for predaor populaion respecively. Copyrigh SciRes.

2 6 S. Y. LI ET AL. Noe ha in (.) x. Tha is he asympoic behavior of x x x is linear dependen on is dependen on. Therefore we us need o sudy following subsysem x x x x Ex y x x x y y kex d fq Q y Q y Q e d a. The iniial condiions for (.) are x x y Q. 4. Local Sabiliy Analysis and Hopf Bifurcaion.. Local Sabiliy Analysis (.) Obviously sysem (.) has wo boundary equilibrium E Ex x (if condiion C : holds) and an unique posiive equilib- rium Ex x y Q (if condiion C : kex d holds) f de x x x x ke f kex d x x y Q y f. Le U x x y Q E x x y Q be any arbirary equilibrium. The linearized equaions are U AU BU (.) x ye xe A kxe d fq yf B kye and he characerisic equaion abou E is given by de ABe I. (.) T Theorem. ) E is local sable if local unsable if and E exis. ) E is local sable if kex d local unsable if kex d and E exis. Proof. ) From (.) he characerisic equaion abou is given by E d. (.) Then d 4 and are he wo oher roos of. By Rouh-Hurwiz crierion E is local sable if local unsable if and E exis. ) From (.) he characerisic equaion abou E is given by d kex x x. (.4) Then are he wo roos of x wih negaive real pars. kex d 4 by Rouh-Hurwiz crierion E is local sable if kex d local unsable if kex d and E exis... Exisence of Local Hopf Bifurcaion The characerisic equaion abou he posiive equilibrium E is given by D M N e. (.5) 4 M mmmm N n nn m x Ey x m x x Ey fy m x fy x Ey m fy x n ke x y n ke x y n ke x y When (.5) becomes o (.6) 4 m m n m n m n. and m m n m n m n. m n m mn Noe ha m n m n m m n m n m m n m n m n m m n m Copyrigh SciRes.

3 S. Y. LI ET AL. 6 if condiion C : f ke holds. By Rouh-Hurwis crierion all roos of (.6) have negaive real pars. Then he equilibrium E is local sable. Suppose i is a roo of (.5) and separaing he real and imaginary pars one can ge ha From (.7) we have namely 4 mm n n cosnsin mmn cos n n sin. (.7) n n n (.8) 4 m m mm 4 F f f f f f m m n f m m mm n nn f m m m n n n f m n. (.9) If condiion C : f ke hold hen f (.9) have a leas one posiive roo. Wihou loss of generaliy we assume (.9) have four disinc posiive roos 4 hen (.5) have four pair of imaginary roos i i. From (. 7) we have 4 m n n n mm n n n cos Thus he n n corresponding o are given by n n n 4 m n n n mm n nπ (.) And he direcion of n pass hrough he imaginary axis [8] when is given by df sign d sign 4 f f f sign. 6 4 Then sign since 4 four disinc posiive roos of (.9). Le min n 4; n. (.) According o he Hopf bifurcaion heorem for funcional differenial equaions [9] (.) can undergoes a Hopf bifurcaion a he posiive equilibrium E when. Furhermore if condiion C : 4 f f holds hen (.9) have unique posiive roo and he n n corresponding o are given by 4 mn nn mm n n n n nπ (.) and sign according o he Hopf bifurcaion heorem for funcional differenial equaions [9] (.) can undergoes a Hopf bifurcaion a he posiive equilibrium E when n n. Based on above analysis we have he following resul. Theorem. ) If condiion C : f ke holds h en here exiss a when he posiive equilibrium E of (.) is asympoically sable and unsable when is defined by (.). ) If condiion C4: f f holds hen here exiss a when he posiive equilibrium E of (.) is asympoically sable and unsable when (.) can undergoes a Hopf bifurcaion a he posiive equilibrium E when n n n is defined by (.). Remark. I mus be poined ou ha Theorem can no deermine he sabiliy and he direcion of bifurcaing periodic soluions ha is he periodic soluions may exiss eiher for or for near. To deermine he sabiliy direcion and oher properies of bifurcaing periodic soluions he normal form heory and cener manifold argumen should be considered []. 4. Numerical Simulaion We consider following sage-srucured delay sysem x.x.4x.8x.5 x y x x x y y x Q Q y Q (4.) E.5.5 a.4 k.7 d.6 f..4 X. Copyrigh SciRes.

4 6 S. Y. LI ET AL. Sysem (4.) has an unique posiive equilibrium poin E We solve model (4.) using funcion dde in MATLAB and compue ha f 7.64 f.5 f f According o Theorem he posiive equilibrium poin E is asympoically sable when 4.4 (see Figure ). When 4.6 he posiive equilibrium poin E is unsable and he Hopf bifurcaion occurring around he posiive equilibrium E are shown (see Figure ). The bifurcaing periodic soluion (limi cycle) of (4.) are sable when (see Figure ) he ampliudes of period oscillaory are increasing as ime delays increased. Bu oo large ime delay would make he populaion o be die ou because he populaion very close o zero as ime delay increase o some criical value. 5. Conclusion In his paper we considered a hree-sage-srucured prey-predaor sysem wih discree and coninuous ime delays and analyzed he sabiliy of he boundary and posiive equilibrium obained he condiions of he posiive equilibrium occurring Hopf bifurcaion by analyzing he characerisic equaion abou i. Numerical examples by ime-series plo have shown ha he sysem considered local asympoically sable when and sable Hopf bifurcaion periodic soluions when and near. Tha is o say ime delay can make he posiive equilibrium lose sabiliy. I is shown ha populaions can be coexisence wih periodic flucuaing under some condiions and such flucuaion is caused by he ime delay. The bifurcaing periodic soluion (limi cycle) is sable when from 5 o 4 and he ampliudes of period oscillaory are increasing as ime delays increased. Bu oo large ime delay would make he populaion o Figure. The ime-series plo show ha posiive equilibrium poin E of (4.) are asympoically sable for τ = 4.4 < τ. Figure. The ime-series plo show ha (4.) undergoes Hopf bifurcaion for τ = 4.6 > τ. Copyrigh SciRes.

5 S. Y. LI ET AL. 6 Figure. The Hopf bifurcaing periodic soluion (limi cycles) of (4.) when τ = be exinc because he populaion arbirary close o zero as ime delay increase o some criical value. These are very ineresing in mahemaics and biology. 6. Acknowledgemens This work was suppored by he Naural Science Foundaion of Guizhou Province (No. [] 6). REFERENCES [] W. Aiello and H. Freedman A Time-Delay Model of Single-Species Growh wih Sage Srucure Mahemaical Biosciences Vol. No. 99 pp doi:.6/5-5564(9)99-u [] W. Wang and L. Chen A Predaor-Prey Sysem wih Sage-Srucure for Predaor Compuers & Mahemaics wih Applicaions Vol. No pp doi:.6/s898-(97)56-4 [] S. Liu L. Chen and R. Agarwal Recen Progress on Sage Srucured Populaion Dynamics Mahemaical and Compuer Modelling Vol. 6 No. - pp doi:.6/s ()79- [4] S. Gao Global Sabiliy of Three-Sage-Srucured Single-Species Growh Model Journal Xiniang Universiy Vol. 8 No. pp (in Chinese). [5] S. Yang and B. Shi. Periodic Soluion for a Three-Sage- Srucured Predaor-Prey Sysem wih Time Delay Journal of Mahemaical Analysis and Applicaions Vol. 4 No. 8 pp doi:.6/.maa.7..5 [6] S. Li and X. Xue Hopf Bifurcaion in a Three-Sage- Srucured Prey-Predaor Sysem wih Predaor Densiy Dependen Communicaions in Compuer and Informaion Science Vol. 88 pp doi:.7/ _86 [7] S. Li Y. Xue and W. Liu Hopf Bifurcaion and Global Periodic Soluions for a Three-Sage-Srucured Prey- Journal of Predaor Sysem wih Delays Inernaional Informaiona and Sysems Sciences Vol. 8 No. pp [8] Z. Wang A Very Simple Crierion for Characerizing he Crossing Direcion of Time-delayed Sysems wih Delay-Dependen Parameers Inernaional Journal of Copyrigh SciRes.

6 64 S. Y. LI ET AL. Bifurcaion and Chaos Vol. No. Aricle ID: 548. doi:.7/ [9] J. Hale Theory of Funcional Differenial Equaions Springer New York 977. [] B. Hassard N. Kazarinoff and Y. Wan Theory and Applicaions of Hopf Bifurcaion Cambridge Universiy Press Cambridge 98. Copyrigh SciRes.