Research Article Hopf Bifurcation of a Differential-Algebraic Bioeconomic Model with Time Delay

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1 Journal of Applied Mahemaics Volume 01, Aricle ID , 15 pages doi: /01/ Research Aricle Hopf Bifurcaion of a Differenial-Algebraic Bioeconomic Model wih Time Delay Xiaojian Zhou, 1, Xin Chen, 1 and Yongzhong Song 1 1 Jiangsu Key Laboraory for NSLSCS, Insiue of Mahemaics, School of Mahemaical Sciences, Nanjing Normal Universiy, Nanjing 1003, China School of Science, Nanong Universiy, Nanong 6008, China Correspondence should be addressed o Yongzhong Song, yzsong@njnu.edu.cn Received 7 Sepember 01; Acceped 6 Sepember 01 Academic Edior: Junjie Wei Copyrigh q 01 Xiaojian Zhou 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. We invesigae he dynamics of a differenial-algebraic bioeconomic model wih wo ime delays. Regarding ime delay as a bifurcaion parameer, we show ha a sequence of Hopf bifurcaions occur a he posiive equilibrium as he delay increases. Using he heories of normal form and cener manifold, we also give he explici algorihm for deermining he direcion of he Hopf bifurcaions and he sabiliy of he bifurcaing periodic soluions. Numerical ess are provided o verify our heoreical analysis. 1. Inroducion Predaor-prey sysems wih delay play an imporan role in populaion dynamics since Vio Volerra and James Loka proposed he seminal models of predaor-prey in he mid of 190s. In recen years, predaor-prey sysems wih delay have been sudied exensively due o heir heoreical and pracical significance. In 1973, May 1 firs proposed he delayed predaor-prey sysem ẋ x r 1 a 11 x τ a 1 y, ẏ y r a 1 x a y, 1.1 where x and y can be inerpreed as he populaion densiies of prey and predaor a ime, respecively. τ>0 is he feedback ime delay of prey species o he growh of species iself. r 1 > 0 denoes inrinsic growh rae of prey, and r > 0 denoes he deah rae of predaor. The parameers a ij i, j 1, are all posiive consans.

2 Journal of Applied Mahemaics Recenly, Song and Wei furher considered he exisence of local Hopf bifurcaions of sysem 1.1. They regarded he feedback ime delay τ as a bifurcaion parameer and invesigaed he sabiliy and he direcion of periodic soluions bifurcaing from Hopf bifurcaions, by applying he normal form heory and he cener manifold reducion developed by Hassard e al. 3. Considering he feedback ime delay of predaor species o he growh of species iself and also wih he delay τ, Yan and Li 4 sudied Hopf bifurcaion and global periodic soluions of he following modified delayed predaor-prey sysem: ẋ x r 1 a 11 x τ a 1 y, ẏ y r a 1 x a y τ. 1. They esablished he global exisence resuls of periodic soluions bifurcaing from Hopf bifurcaions using a global Hopf bifurcaion resul due o Wu 5, which was differen from ha used in Song and Wei. In 6, Yuan and Zhang also invesigaed he sabiliy of he posiive equilibrium and exisence of Hopf bifurcaion of he model 1.. Noing ha, in real siuaions, he feedback ime delay of he prey o he growh of he species iself and he feedback ime delay of he predaor o he growh of he species iself are differen. If he ime delay of he predaor o he growh of he species iself is zero, 1. becomes 1.1. Meanwhile, if he ime delay of he prey o he growh of he species iself is zero, 1. is simplified as ẋ x r 1 a 11 x a 1 y, ẏ y r a 1 x a y τ. 1.3 In 1954, Gordon 7 sudied he effec of harves effor on ecosysem from an economic perspecive and proposed he following economic heory: Ne Economic Revenue NER Toal Revenue TR Toal Cos TC. 1.4 This provides heoreical fundamen for he esablishmen of bioeconomic sysems by differenial-algebraic equaions DAEs. Le E and y represen he harves effor and he densiy of harvesed populaion, respecively. Then TC ce and TR αey, where α is uni price of harvesed populaion and c is he cos of harves effor. Considering he economic ineres m of he harves effor on he predaor, we can esablish he following algebraic equaion: E αy c m. 1.5 Recenly, a class of differenial-algebraic bioeconomic models were proposed and analyzed in For example, 8 discussed he problems of chaos and chaoic conrol for a differenial-algebraic sysem. The exisence and sabiliy of equilibrium poins, he sufficien condiions of exisence for various bifurcaions ranscriical bifurcaion, singular induced bifurcaion, and Hopf bifurcaion are invesed in However, he sabiliy and he

3 Journal of Applied Mahemaics 3 direcion of periodic soluions bifurcaing from Hopf bifurcaions have no been discussed here. Very recenly, Zhang e al. 15 analyzed he sabiliy and he Hopf bifurcaion of a differenial-algebraic bioeconomic sysem wih a single ime delay and Holling II ype funcional response and firs invesigaed he direcion of Hopf bifurcaion and he sabiliy of he bifurcaing periodic soluions. Based on he above economic heory and sysem 1.3, in his paper, we sudy he Hopf bifurcaion of he following differenial-algebraic bioeconomic sysem: ẋ x r 1 a 11 x a 1 y, ẏ y r a 1 x a y τ Ey, E αy c m. The res of his paper is organized as follows. In Secion, regarding ime delay τ as bifurcaion parameer, firs, we sudy he sabiliy of he equilibrium poin of sysem based on he new normal form approach proposed by Chen e al. 16. Then, following he normal form approach heory and he cener manifold heory inroduced by Hassard e al. 3, we compue he normal form for he Hopf bifurcaion of sysem 1.6 and analyze he direcion and sabiliy of he bifurcaing periodic orbis of he sysem. Numerical examples are given in Secion 3 o illusrae our heoreical resuls.. Hopf Bifurcaion In his secion, we analyze he sabiliy and he Hopf bifurcaion of he differenial-algebraic bioeconomic sysem 1.6. Furhermore, we will derive he formula for deermining he properies of he Hopf bifurcaion..1. Exisence of Hopf Bifurcaion For m>0, he sysem 1.6 has posiive equipmen poin P x,y,e, where x r 1 a 1 y /a 11, E m/ and y is he posiive roo of he following equaion: α a 11 a a 1 a 1 y ca 11 a ca 1 a 1 αr 1 a 1 αa 11 r y ma 11 cr 1 a 1 ca 11 r 0..1 In order o guaranee he exisence of he posiive equipmen poin P, some resricions mus be saisfied also: c α <y < r 1 a 1, ca 11 a ca 1 a 1 αr 1 a 1 >αa 11 r,. and ma 11 cr 1 a 1 ca 11 r < 0.

4 4 Journal of Applied Mahemaics In order o analyze he local sabiliy of he posiive equilibrium poin, we firs use he linear ransformaion x, y, E T Q u, v, E T, where Q αe 1..3 Then he sysem 1.6 yields u u r 1 a 11 u a 1 v, v v r a 1 u a v τ Ev αe v, 0 E αe v αv c m..4 According o he lieraure 16, considering he local paramerizaion of he hird equaion of he sysem.4, and we can obain he following parameric sysem of i: ẋ 1 x 1 u r 1 a 11 x 1 u a 1 x v, ẋ x v r a 1 x 1 u a x τ v E h x v αe x v,.5 where u u x 1, v v x, E E h x 1,x,.6 h : R R is a smooh mapping, and u,v, E T saisfies x,y,e T Q u,v, E T..7 Thus he linearized sysem of parameric sysem.5 a 0,0 is given as below: ẋ 1 a 11 x x 1 a 1 x x, ẋ a 1 y x 1 αe y x a y x τ..8 The characerisic equaion of he linearized sysem of parameric sysem.8 a 0,0 is given by λ pλ r sλ q e λτ 0,.9 where p a 11 x αy E /, r x y a 1 a 1 αa 11 E /, s a y,and q a 11 a x y.

5 Journal of Applied Mahemaics 5 The second-degree ranscendenal polynomial equaion.9 has been exensively sudied and applied by many researchers, Le H1 r 1 >αy E / ; H a 1 a 1 a 11 a >αa 11 E / ; H3 eiher x y a 1 a 1 a y <a 11 x y E α/ and a 1 a 1 a 11 E α/ αy c > a 11 a or x y a 1 a 1 a y a 11 x y E α < 4x y a 1a 1 a 11E α a 11 a ;.10 H4 eiher a 1 a 1 a 11 E α/ < a 11 a or x y a 1 a 1 a y > a 11 x y E α/,and x y a 1 a 1 a y a 11 x y E α 4x y a 1a 1 a 11E α a 11 a ;.11 H5 a 1 a 1 a 11 E α/ > a 11 a,x y a 1 a 1 a y >a 11 x y E α/ αy c,and x y a 1 a 1 a y a 11 x y E α > 4x y a 1a 1 a 11E α a 11 a..1 From Lemma.1 in, we easily obain he following resuls abou he sabiliy of he posiive equilibrium and he Hopf bifurcaion of 1.6. Theorem.1. For sysem 1.6. One has he following. i If H1 H3 hold, hen he equilibrium E of he sysem 1.6 is asympoically sable for all τ 0. ii If H1, H, and H4 hold, hen he equilibrium E of he sysem 1.6 is asympoically sable when 0 τ<τ 0 and unsable when τ>τ 0. Sysem 1.6 undergoes a Hopf bifurcaion a E when τ τ j. iii If H1, H, and H5 hold, hen here is a posiive ineger k, such ha he equilibrium E swiches k imes from sabiliy o insabiliy o sabiliy; ha is, E is asympoically sable when τ [ 0,τ 0 τ 0,τ 1 τ k 1,τ k.13 and unsable when τ τ 0,τ 0 τ 1,τ 1 τ k 1,τ k 1 τ k 1,..14

6 6 Journal of Applied Mahemaics Here, ω ± x y a 1 a 1 a y a 11 x y E α τ ± j 1 ω ± arccos x y a 1 a 1 a y a y E α 11 x αy ± c 4x y a 1a 1 a 11E α a 11a a E α/ ω± a 11 a x a1 a 1 a 11 E α/ a ω ± a 11 x 1/, jπ ω ±, j 0, 1,, Denoe λ j τ α j τ iω j τ, j 0, 1,,...; hen he following ransversaliy condiions hold: d dτ Re λ j τ j > 0, d dτ Re λ j τ j < Direcion and Sabiliy of he Hopf Bifurcaion In he previous secion, we obain he condiions for Hopf bifurcaions a he posiive equilibrium P when τ τ ± n,n 0, 1,,... In his secion, we will derive he formulaes deermining he direcion, sabiliy, and period of hese periodic soluions bifurcaing from P a τ. The basic echniques used here are he normal form and he cener manifold heory developed by Hassard e al. 3. Assuming he sysem 1.6 undergoes Hopf bifurcaions a he posiive equilibrium P a τ τ n,weleiω be he corresponding purely imaginary roo of he characerisic equaion a he posiive equilibrium. In order o invesigae he direcion of Hopf bifurcaion and he sabiliy of he bifurcaing periodic soluions of sysem 1.6, we consider he parameric sysem.5 of sysem.4. Le τ, τ τ n μ; hen he parameric sysem.5 is equivalen o he following funcional differenial equaion sysem: ż 1 τ n μ a 11 x z 1 a 1 x z, ż τ n μ a 1 y z 1 αe y z a y z 1,.17 where z 1 x 1 τ,z x τ.

7 Journal of Applied Mahemaics 7 For Φ C 1, 0,R, define a family of operaors L μ Φ : τ n μ a 11 x a 1 y a 1 x αe y Φ 0 τ n μ [ ] 0 0 Φ 1, 0 a y f μ, Φ : τ n μ a 11 Φ 1 0 a 1Φ 1 0 Φ 0 a 1 Φ 1 0 Φ 0 a Φ 0 Φ 1 αe c αy c Φ 0,.18 where Φ Φ 1, Φ C. By he Riesz represenaion heorem, here exiss a funcion of bounded variaion for θ 1, 0, such ha L μ Φ 0 1 dη θ, μ Ψ θ, Ψ C..19 In fac, we can choose η θ, μ τ n μ a 11 x a 1 y a 1 x αe y δ θ τ n μ [ ] 0 0 δ θ 1. 0 a y.0 For Φ C, define dφ θ A μ, θ 1, 0, dθ Φ : 0 dη, μ Φ, θ 0, 1 R μ 0, θ 1, 0, : f, Φ, θ 0..1 Hence sysem.17 can be rewrien as Ż A μ Z R μ Z.. For Ψ C 1, 0, R, define dψ s, s 1, 0, ds A Ψ s : 0 dη T s, 0 Ψ s, s 0, 1.3

8 8 Journal of Applied Mahemaics and a bilinear inner produc 0 Ψ, Φ Ψ T 0 Φ 0 1 θ 0 Ψ T ξ θ dη θ Φ ξ dξ,.4 where η θ η θ, 0. Then A and A 0 are adjoin operaors. By he discussion in Secion.1, we know ha ±iω are eigenvalues of A 0. Thus hey are also eigenvalues of A. Suppose ha q θ 1,q 1 T e iωτθ and q s D q 1, 1 eiωτs are eigenvecors of A 0 and A corresponding o iωτ and iωτ, respecively. I is easy o obain q 1 iω a 11x a 1 x, D q 1 iω a y e iωτ, a 1 x q 1 q 1 τq 1a y e iωτ 1..5 Moreover, q s,q θ 1, q s, q θ 0. Now, we calculae he coordinaes o describe he cener manifold C 0 a μ 0. Using he same noaions as in 3, we define z q,z, W, θ Z θ Re { zq θ }..6 On he cener manifold C 0, we have W, θ W z, z, θ, where W z, z, θ W 0 θ z W 11 θ zz W 0 θ z,.7 and z and z are local coordinaes for C 0 in he direcion of q and q.noehaw is real if Z is real; we only consider real soluions. For Z C 0, we have ż iωτz q 0 f 0 z, z : iωτz g z, z,.8 where g z, z q 0 f 0 z, z q 0 f 0,Z g 0 θ z g 11 θ zz g 0 θ z..9

9 Journal of Applied Mahemaics 9 Moreover, we have g 0 Dτ a 1 q 1 a 11 q 1 a 1q 1 q αe c 1 αy c q 1 a q 1, e iωτθ g 11 Dτ a 1 Re q 1 a11 q 1 a 1q 1 Re q 1 a Re q 1 q 1 e iωτθ αe c αy c q 1q 1, g 0 Dτ a 1 q 1 a 11 q 1 a 1q 1 q 1 αe c αy c q 1 a q 1, eiωτθ a1q g 1 Dτ 1 a 11q 1 a 1q 1 q 1 1 W a1 q 1 a 1 q 1 q 1 a 11q 1 1 W 0 0 a 1 a q 1 e iωτθ a 1 q 1 αe c αy c q 1 W a 1 a q 1 e iωτθ a 1 q 1 αe c αy c q 1 W 0 0 a q 1 W a q 1 W Nex, we will calculae W 11 θ and W 0 θ.from. and.8, we have { { AW Re q 0 f z, z q θ }, θ 1, 0, Ẇ Ż żq żq AW Re { q 0 f z, z q θ } f, θ 0,.31 : AW H z, z, θ, where H z, z, θ H 0 θ z / H 11 θ zz H 0 θ z /. For θ 1, 0, we can ge A iωτ W 0 θ H 0 θ, AW 11 θ H 11 θ..3 On he oher hand, H z, z, θ AW Re{q 0 f z, z q θ }; hence we can obain H 0 θ g 0 q θ g 0 q θ, H 11 θ g 11 q θ g 11 q θ..33

10 10 Journal of Applied Mahemaics I follows from.3 ha Ẇ 0 θ iωw 0 θ g 0 q θ g 0 q θ, Ẇ 11 θ g 11 q θ g 11 q θ..34 Solving i, we have W 0 θ ig 0 ωτ q 0 eiωτθ ig 0 3ωτ q 0 e iωτθ Me iωτθ, W 11 θ ig 11 ωτ q 0 eiωτθ ig 11 ωτ q 0 e iωτθ N..35 Now we will seek appropriae M and N.From.9 and.31, we have H 0 θ g 0 q 0 g 0 q 0 τh 1, H 11 θ g 11 q 0 g 11 q 0 τh,.36 where a 11 a 1 q 1 H 1 αe c a 1 q 1 αy c q 1 a q 1 e iωτθ, a 11 a 1 Re q 1 H a 1 Re q 1 a Re q 1 q 1 e iωτθ αe c αy c q 1q Noing ha iωτi iωτi e iωτθ dη θ q 0 0, e iωτθ dη θ q 0 0,.38

11 Journal of Applied Mahemaics 11 we have he following linear equaions: iω a 11 x a 1 y a 11 x a 1 y a 1 x iω a y αe y M H 1, a 1 x a y αe y N H..39 I is easy o ge M and N. Furhermore, we can ge he following values: C 1 0 i ωτ g 11 g 0 g0 g11 3 g 1, μ Re C 1 0 Re λ τ,.40 β Re C 1 0, T Im C 1 0 μ Im λ τ. ωτ I is well known ha, a he criical value τ, he sign of μ deermines he direcion of he Hopf bifurcaion, β deermines he sabiliy of bifurcaed periodic soluions, while T deermines he period of bifurcaed periodic soluions, respecively. In fac, we have he heorem as follows. Theorem.. i The Hopf bifurcaion is supercriical if μ > 0 and subcriical if μ < 0. ii The bifurcaed periodic soluions are unsable if β > 0 and sable if β < 0. iii The period of bifurcaed periodic soluions increases if T > 0 and decreases if T < Numerical Examples We now perform some simulaions for beer undersanding of our analyical reamen. Example 3.1. Consider he following differenial-algebraic sysem: ẋ x 6 x 1.5y, ẏ y 0.5 x y τ Ey, E y The only posiive equilibrium poin of he sysem 3.1 is P 1.5,.0, 0.5. Iiseasy o ge p.3333,q 6,r 7,s.

12 1 Journal of Applied Mahemaics x 1.5 y E y E x Figure 1: : Dynamic behavior of he differenial-algebraic sysem 3.1 wih τ 0.55 <τ 0, and he posiive equilibrium poin P is sable. When τ 0, from.9, he characerisic equaion of he linearized sysem of parameric sysem of 3.1 a P is given by λ p s λ r q By simple calculaing, p s > 0,q r 13 > 0. According o Rouh-Hurwiz sabiliy crierion, P is sable. When τ > 0, we can furher ge r q 13 > 0,s p r > 0, and s p r 4 r q > 0 which mean H1, H,and H5 hold. Furhermore, we have ω ,ω , and hence τ <τ <τ < τ 4.89 <τ <τ <τ <τ <. According o conclusion iii of Theorem.1, he equilibrium poin P is sable only for τ<τ 0, unsable for any τ>τ 0, and he Hopf bifurcaion occurs a τ τ 0. By he aid of Mahemaica, we can obain he following values according o equaliies in.40 : C i, λ τ i. 3.3 So we have μ.9586 < 0,β > 0, and T.1567 < 0. Thus from Theorem., we can conclude ha he Hopf bifurcaion of sysem 3.1 is subcriical, he bifurcaing periodic soluion exiss when τ crosses τ 0 o he lef, and he bifurcaing periodic soluion is unsable and decreases. Figure 1 shows he posiive equilibrium poin P of sysem 3.1 is locally asympoically sable when τ 0.55 <τ The periodic soluions occur from P

13 Journal of Applied Mahemaics x 1.5 y E E x y Figure : Dynamic behavior of he differenial-algebraic sysem 3.1 wih τ < τ 0, periodic soluions bifurcaing from he posiive equilibrium poin P x 1.5 y E y E x Figure 3: Dynamic behavior of he differenial-algebraic sysem 3.1 wih τ 0.57 >τ 0, and he posiive equilibrium poin P is unsable.

14 14 Journal of Applied Mahemaics when τ <τ as is illusraed in Figure. When τ 0.57 >τ 0 posiive equilibrium poin P becomes unsable as shown in Figure , he Acknowledgmens The auhors would like o hank he reviewers for heir valuable commens which improved he paper. This work is suppored by he Foundaion for he Auhors of he Naional Excellen Docoral Thesis Award of China 0070, he 333 Projec Foundaion of Jiangsu Province, and he Foundaion for Innovaive Program of Jiangsu Province CXZZ , CXZZ References 1 R. M. May, Time delay versus sabiliy in populaion models wih wo and hree rophic levels, Ecology, vol. 4, pp , Y. Song and J. Wei, Local Hopf bifurcaion and global periodic soluions in a delayed predaor-prey sysem, Journal of Mahemaical Analysis and Applicaions, vol. 301, no. 1, pp. 1 1, B. D. Hassard, N. D. Kazarinoff, and Y. H. Wan, Theory and Applicaions of Hopf Bifurcaion, vol. 41, Cambridge Universiy Press, Cambridge, UK, X.-P. Yan and W.-T. Li, Hopf bifurcaion and global periodic soluions in a delayed predaor-prey sysem, Applied Mahemaics and Compuaion, vol. 177, no. 1, pp , J. Wu, Symmeric funcional-differenial equaions and neural neworks wih memory, Transacions of he American Mahemaical Sociey, vol. 350, no. 1, pp , S. Yuan and F. Zhang, Sabiliy and global Hopf bifurcaion in a delayed predaor-prey sysem, Nonlinear Analysis, vol. 11, no., pp , H. S. Gordon, The economic heory of a common propery resource, he fishery, Journal of Poliical Economy, vol. 6, pp , Y. Zhang and Q. L. Zhang, Chaoic conrol based on descripor bioeconomic sysems, Conrol and Decision, vol., pp , Y. Zhang, Q. L. Zhang, and L. C. Zhao, Bifurcaions and conrol in singular biological economical model wih sage srucure, Journal of Sysems Engineering, vol., pp. 3 38, X. Zhang, Q.-L. Zhang, C. Liu, and Z.-Y. Xiang, Bifurcaions of a singular prey-predaor economic model wih ime delay and sage srucure, Chaos, Solions and Fracals, vol. 4, no. 3, pp , C. Liu, Q. Zhang, and X. Duan, Dynamical behavior in a harvesed differenial-algebraic preypredaor model wih discree ime delay and sage srucure, Journal of he Franklin Insiue. Engineering and Applied Mahemaics, vol. 346, no. 10, pp , C. Liu, Q. Zhang, X. Zhang, and X. Duan, Dynamical behavior in a sage-srucured differenialalgebraic prey-predaor model wih discree ime delay and harvesing, Journal of Compuaional and Applied Mahemaics, vol. 31, no., pp , Y. Feng, Q. Zhang, and C. Liu, Dynamical behavior in a harvesed differenial-algebraic allelopahic phyoplankon model, Inernaional Journal of Informaion & Sysems Sciences, vol. 5, no. 3-4, pp , Q. Zhang, C. Liu, and X. Zhang, Complexiy, Analysis and Conrol of Singular Bio-logical Sysems, Springer, G. Zhang, L. Zhu, and B. Chen, Hopf bifurcaion in a delayed differenial-algebraic biological economic sysem, Nonlinear Analysis, vol. 1, no. 3, pp , B. S. Chen, X. X. Liao, and Y. Q. Liu, Normal forms and bifurcaions for differenial-algebraic sysems, Aca Mahemaicae Applicaae Sinica, vol. 3, no. 3, pp , 000 Chinese. 17 K. L. Cooke and Z. Grossman, Discree delay, disribued delay and sabiliy swiches, Journal of Mahemaical Analysis and Applicaions, vol. 86, no., pp , Y. Kuang, Delay Differenial Equaions wih Applicaions in Populaion Dynamics, vol. 191 of Mahemaics in Science and Engineering, Academic Press, Boson, Mass, USA, 1993.

15 Journal of Applied Mahemaics S. Ruan, Absolue sabiliy, condiional sabiliy and bifurcaion in Kolmogorov-ype predaor-prey sysems wih discree delays, Quarerly of Applied Mahemaics, vol. 59, no. 1, pp , J. Wei and S. Ruan, Sabiliy and bifurcaion in a neural nework model wih wo delays, Physica D, vol. 130, no. 3-4, pp. 55 7, 1999.

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