Linearized Stability and Hopf Bifurcations for a Nonautonomous Delayed Predator-prey System

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1 SCIREA Joural of Mathematics October 0, 016 Volume 1, Issue1, October 016 Liearized Stability ad Hopf Bifurcatios for a Noautoomous Delayed Predator-prey System Li Wag, Assistat Professor School of Applied Mathematics, Xiame Uiversity of Techology, Xiame, Fujia Provice, Xiame, Chia Tel: (086) lwag@xmut.edu.c Lei Ji, PhD Faculty of Egieerig, Uiversity of Regia, Regia, SK S4S 0A, Caada Tel: (306) jile00@uregia.ca ABSTRACT I past may years, biomathematics populatio models are costructed based o plausible explicit ad implicit biological assumptios. I the case that ot eough aalysis is carried out for a well-motivated ad plausible model, the result is o or miimum isights gaied. I this study, existece of Hopf bifurcatios of a oautoomous delayed predator-prey system with stage-structure for predator is proposed. Furthermore, coditios of liearized stability ad Hopf bifurcatios for this system are established. Numerical simulatios are preseted it illustrate the feasibility of our mai result. Key words: Hopf bifurcatios; stage-structure; positive periodic solatio; liearized stability. 63

2 1. Itroductio The predator-prey system is a major mathematics cocer i biomathematics field. Over the past year, a few studies were udertake to address i predator-prey system [1-3, 6-8]. It is assumed i the model that each idividual predator admits the same ability to attack prey. However, they did ot classified idividuals of predator as belogig to either the immature or the mature ad supposed that the immature populatio does ot feed o prey. This seems reasoable for a umber of mammals, where immature predators are raised by their parets, the rate they attack at prey ad the reproductive rate ca be igored. This study is a extesio of previous efforts, emphasizig o the developmet of predator-prey system. Stage-structured models have also bee studied by several authors [4, 5, 9-11]. The model of [11] cosidered was a stage-structured model of oe species growth cosistig of immature ad mature idividuals was aalyzed. Ad the model of [11] cosidered further assumed that the time from immaturity to maturity is itself state depedet. A equilibrium aalysis ad evetual lower boud ad evetual upper boud of positive solutios for that model were give. I [9], the authors cosidered the followig predator prey system with stage-structure for predator: x( t) x( t)( r ax( t 1) by( t)), y1 ( t) kbx( t ) y( t ) ( D v1 ) y1( t), y( t) Dy1 ( t) v y( t). (1) where x(t) is the desity of prey of prey at time t; y 1 (t) is the desity of immature predator at time t; y (t) is the desity of mature predator at time t; r is the itrisic growth rate of prey, v 1 is the death rate of immature predator ad v the death rate of mature predator, costat k > 0 deotes the coefficiet i coversig prey ito ew immature predator, costat D > 0 deotes the rate of immature predator becomig mature predator. It is assumed that this rate is proportioal to the desity of immature predator. I [9], the authors studied the asymptotic behavior of system (1). Whe time delay due to gestatio of predator ad time delay from crowdig effect of prey are icorporated, they establish the coditio for the permaece of populatios ad sufficiet coditios uder which positive equilibrium of system (1) is globally stable. However, the effect of prevetio fuctios from predator itself o its growth i umber ad chagig eviromet is ot cosidered i system (1). For more complicatio coditios, such as oautoomous delayed predator-prey system with stage-structure for predator are 64

3 desired: x( t) x( t)( r ax( t 1) by( t)), y1( t) kbx( t ) y( t ) ( D v1 ) y1( t) k1y1 ( t), y( t) Dy1 ( t) v y( t) k y. () where all parameters are positio costats. I this paper, our objective is to derive the existece coditios for Hopf bifurcatio whe 1 0. Liearized stability ad Hopf bifurcatios for a three-dimesioal system are scarcely studied. Therefore, liearized stability ad Hopf bifurcatio of system () is based o evirometal factors i theory ad applicatio.. The existece of Hopf bifurcatio Let ( x, y, y ) is a positive equilibrium of system () with 1 0.The 1 the followig equatios: ( x, y, y ) satisfies 1 r ax by, kbxy ( D v1 ) y1 k1 y1, Dy1 v y k y. (3) Whe system (3) has a uique positive solutio, system () has a uique positive equilibrium. Lemma.1. Assume that r ( D v1) v (H1): (4) a kbd The system () has a uique positive equilibrium. Proof. Form (3), we have [ ad( D v ) k D kb ak v ] y ak y ak v k y ad( D v ) v rd kb Let F( y) [ ad( D v ) k D kb ak v ] y ak y ak v k y ad( D v ) v rd kb, for y 0. Sice

4 F ad D v v rd kb (0) ( 1) 0 ad 3 r b F( ) rb [ ad( D v1 ) k D kb ak1v ] ak1r ak1k r b ad( D v1 ) b v rd kb b the rb [ ad( D v ) k ak v ] ak r ak k v r b adb v ( D v ) 0, r there exists a poit (0, ) such that F( ) 0. By derivatives, it shows that F'( y) 0 for b y > 0. Hece F(y) has a uique positive zero poit, that is (3) has a uique positive solutio y. Thus from Dy1 v y ky, we obtai y 1 0. From r = ax + by, we have r by 0 x. Therefore, system () has a uique positive equilibrium. a The cosider the existece coditios of Hopf bifurcatio whe 1 0. Takig x x x, y y y, y y y ad replacig x, y1, y by x, y 1,y, respectively, ad takig =, the system () with 1 0, it becomes: x( t) ax ( t) bx y( t), y1( t) kbyx( t ) ( D v1 k1 y1 ) y1( t) kbx y( t ), y( t) Dy1 ( v k y) y. (5) Let B D v k y v k y ax, B ( D v k y )( v k y ) ax ( D v k y v k y ), B 3 kbdx, B kb Dx y akbd x 4 ( ), B ax ( D v k y )( v k y ) We assume that (H): B B, 1 (H3): B, 4 B 5 B4 > B5; 66

5 (H4): ( B B ) 3( B B B B ) The the characteristic equatio of (5) is Lettig, i ( 0),we obtai (6) 3 B1 B B3e B4e B5 0 B1 B3 si B4 cos B5 0, 3 B B3 cos B4 si 0. (7) From (7), we obtai ( B B ) ( B B B B ) B B. (8) Let z, the z ( B B) ( B B B B) z B B. (9) Set F( z) z ( B B ) z ( B B B B ) z B B, for z Sice F(0) B B 0, F( ) 0, the there exists a poit (0, ) such that F( ) Assume that ( z )( z A z A ) F( z), where A 1, A are two costats determied later. 1 The from (H3) ad (H4), we have F(z) has a uique positive zero poit z z. From (7), we have A B B 0ad 1 1 B4 B5 A 0. Therefore, that is equatio (9) has a uique positive root B1 B5 si( ) H, where B B 3 4 ta B4 arcsi H arcsi H. Thus or, 1,,. B 3 Lettig, ( ) ( ) i( ) be the roof of Eq. (6), we have the followig result. Lemma. Assume the (H), (H3) ad (H4) hold. The the followig trasversally coditio hold: dre ( ) d 0. Proof. By (6), differetiatig with respect to, derive that 67

6 d B4e B3 e d 3 B1 B B4 e B3e B3e. It is obvious that the value of d Re d at i is d Re AC1 BD1 / iw, d A B where A B 3 cos B cos B si 3 3 B 3 ( B B ) B cos, B B B si B si B cos B B si ( B ), 1 3 C B si B cos ( B ), D B cos B si ( B B ) Sice AC BD [ B 3 ( B B ) B cos ][ ( B )] [B B si ( B )][ ( B B )] ( B )( B 3 ) B3 B( )cos B B( B ) B ( B B )si [( B )( B 3 ) B B( B ) B ] [3 ( B B ) B B B B ] Therefore d Re 4 / i [3 ( 1 ) ]. B B B B B B d Sice The 4( B B ) ) 1( B B B B ]

7 This completes the proof of Lemma.. d Re / 0. By above discussio,the Lemma below is derived. Lemma.3 If (H1)-(H4) hold, the there exists real sequece{ }, = 0, 1,,, satisfyig, 1,ad arcsi such that H, 0,1,,..., (i) all root of Eq (6) have strictly egative real parts for (0, 0) ; (ii) whe 0, Eq (6) has a pair of pure imagiary root i0 which are simply, ad all other roots have egative parts; (iii) whe 0, Eq (6) has at least oe root with strictly positive positio real part. Applyig Lemma.3 above ad Theorem 1.1 i [5], we have Theorem.1 If (H1)-(H4) hold, the system () whe 1 0 has a Hopf bifucatio at ; = 0,1,,. Theorem. I system () with 1 0, let (H1)-(H4) hold. (1) If 0 0, the ( x, y, y ) is asymptotically stable. 1 () If 0, the ( x, y, y ) is ustable. 1 Moreover, the characteristic equatio (6) always has a root i such Coclusios I this study, predator-prey system with stage-structure for predator has bee developed. It has bee combied with the effect of prevetio fuctios from predator itself o its growth i umber ad chagig eviromet o the oautoomous delayed predator-prey system. The existece of Hopf bifurcatios of the system () is proposed. Furthermore, coditios of liearized stability ad Hopf bifurcatios for this system are established. We would like to metio here that a iterestig but maybe challegig problem associated 69

8 with the study of system () should be the uiqueess ad global stability of positive solutio. We leave this to future work. Ackowledgemets This research was supported by Natioal Natural Sciece Foudatio of Chia-NSAF (Nos ), Xiame Uiversity of Techology Foreig Sciece ad Techology Cooperatio ad Commuicatio Foudatio (E ), ), Xiame Uiversity of Techology High-level persoel Foudatio (YKJ14038), Fujia Class A Foudatio (JA144). The writers are very grateful to the editor ad the aoymous reviewers for their isightful commets ad suggestios. Refereces [1] B.S.Goh, Global stability i two species iteractios, J.Math.Biol., 3(1976), [] A.Hasbigs, Global stability of two species system, J.Math.Biol., 5(1978), [3] X.Z.He, Stability delay i a predator system, J.Math.Aal.Appl., 198(1996), [4] Zhegqiu Zhag ad Xiawu Zeg, O a periodic stage-structure model, Applied Mathematics Letter, 003, 16(7), [5] Zhegqiu Zhag, Ju Wu ad Zhicheg Wag, Periodic solutio of oautoomous satge-structured cooperative system, Computers Math.Appl., 47(004), [6] Yogku Li,Periodic solutio for delay Lotha-Volterra competitio system, J.Math.Aal.Appl., 46(000), [7] K.Gopalsamy, M.R.Kuleovic ad G.Ladas, Evirometal periodicity ad time delays i a food-limited populatio model, J.Math.Aal.Appl., 147(1990). [8] B.G.Zhag Ad K.Gopalsamy, Global attractively ad oscillatios i a periodic delay logistic equatio, J.Math.Aal.Appl., 150(1990), [9] Wedi Wag, A periodic-prey system with stable-structure for predator, Computers Math.Applic., 33(8),1997, [10] W.G.Aiello, H.I.Fredma, A time delay model of sigle-species growth stage-structure, Math.Biosic. 101(1990), [11] W.G.Aiello, H.I.Fredma ad J.Wu, Aalysis of a model represetig stage-structure populatio growig with state depedet time delay, J.Appl.Math., 5(199),