外文资料--Qualitative Analysis of a Delayed and Stage-Stuctured Predator-Prey System (1)

Qualitative Analysis of a Delayed and Stage-Stuctured Predator-Prey System Lingshu Wang1 School of Mathematics and Statistics,Hebei University of Economics&Business Shijiazhuang,P.R.China Guanghui Feng Department of Mathematics,Shijiazhuang Mechanical Engineering College,Shijiazhuang,P.R.China AbstractA delayed and stage-structured predator-prey system with Holling type-II functional response is discussed.By using the normal form theory and center manifold theorem,the linear stability of the system is investigated and Hopf bifurcations are established.Formula determining the direction of bifurcations and the stability of bifurcating periodic solutions are given.Numerical simulations are carried out to illustrate the theoretical results.Keywords-time delay;stage-structure;predator-prey system;stability;Hopf bifurcation.I.INTRODUCTION The predator-prey system is very important in population models and has been studied by many authors(see,for example,1,2,3).It is generally recognized that some kinds of time delays are inevitable in population interactions and tend to be destabilizing in the sense that longer delays may destroy the stability of positive equilibrium.Time delay due to gestation is a common example,because generally the consumption of prey by the predator throughout its past history governs the present birth rate of the predator.Recently,great attention has been received and a large body of work has been carried out on the existence of Hopf bifurcations in delayed population models(see,for example 2,5 and references cited there in).The stability of positive equilibrium and the existence and the direction of Hopf bifurcations were discussed respectively in the references mentioned above.In the natural world,there are many species whose individuals pass through two stages:immature and mature.Predator-prey systems where only immature individuals are consumed by their predators are well known.To this aim,we consider the following delay differential equations 1211121221 12222222()()()()()()()()(t)-ax(t)-1()()()()()1()x tbx tbr x ta x t y tx tb x tr xmx ta x ty ty try tmx t=+=+=+?(1)where 1()x tand 2()x trepresent the densities of the immature and the mature prey at time t,respectively;()y trepresents the 1The author was supported by the National Natural Science Foundation of China(No.10926064)and the Scientific Research Foundation of Hebei Education Department(No.2009114).density of the predator at time t;0 representing a time representing a time delay due to the gestation of the predator;the parameters 12112,a a ab b m r r rare positive constant;/(1)xmx+is the Holling type-II response function.II.STABILITY AND HOPF BIFURCATION In this section,we discuss the stability of the positive equilibrium and the existence of Hopf bifurcations for(1)with time delay as a parameter.Assume 122121111()0,()()()Hamramr bbr brar br+It is easy to check that(1)has a positive equilibrium 1*2*(,),Exxy=where 22*211*22*2*1*2*112(),bxab xr xaxrxxybramrar=+Let111*222*,xxxxxxyyy=dropping the bars,system(1)becomes 1211121121 122222*22*2*22*222*2*2222*2*2()()()()()()(t)-y(t)-ax(t)-(1)(1)()()()1()(1)()()()()(1)(1()x tbx tbr x ta rax tb x txamxmxx t y tmy xtmxmx tmxx ty tmy xty tamxmxmx t=+=+=+?()2*222*()()(1)a y x try try tmx+(2)where 21*2*2(1)raa ymx=+.The characteristic equation of system(2)at the origin is 322210210()0 (3)pppqqq e+=where 0111(),pr brbb r=+211pbrr=+,111111()()prbrbrbb=+,1*11011122*()()(1)a rybrqbb rr brmx+=+978-1-4244-4713-8/10/$25.00 2010 IEEE 1*111222*(),(1)a ryqrbrqrmx=+=+When 0,=equation(3)becomes 32221100()()0 (4)pqpqpq+=Assume 1*2111122*()()()0(1)a ryHbrbrbbmx+This assumption implies that 22000,0,pqpq+221100()()()0.pqpqpq+By Hurwitz criterion,we know that all roots of(4)have negative real part.When 0,noting that(0)i is a root of(3)if and only if 6422100,(5)hhh+=where 22220001110202,22,hpqhpqp pq q=+222221 2.hpqp=Assume 111*31122*()()22()0(1)a br yHbb rr brmx+then we can obtain 111*111*0111222*2*()()22()0(1)(1)a br rya br ryhbbrr brmxmx+=+222111()20hbrbb=+Hence,(5)has only one positive real root 0.Let 53202 112001 00 102422202010200()()1arcsin(2)2),0,1,2,(6)jqp qp qqpqp qqqq qqjj+=+=?then(3)has a pair of purely imaginary roots 0.i Lemma 1 For equation(3),if 1()H,2()Hand 3()H holds,then we have the following transversal condition 0Re0.idd=Proof Differentiating both sides of(3)aboutyields 111210202()()pqddppqq+=+012201022102Re0ihddqq=+=+Therefore,001ReRe0iiddsignsigndd=.From Lemma1 and the results in 4,we have Theorem 1 For system(12),If 1()H,2()Hand 3()Hare satisfied,then the following results hold (i)when 00,),the zero solution is asymptotically stable;(ii)when 0,the zero solution is unstable;(iii)(0,1,)jj=?are the values of Hopf bifurcations.III.DIRECTION AND STABILITY OF HOPF BIFURCATION In this section,we study the direction of bifurcations and the stability of bifurcating periodic solutions.The method we used here is based on the normal form theory and center manifold theory introduced by Hassard et al.in 5.Now,we re-scale the time by 0,=ts =+,then system(1)can be rewritten as ()10211121201122222*2*2122*2*22*2022*()()()()()()()()(t)-y(t)-ax(t)(1)()()()-(1)(1()(1)()()()1(1)x tbxtbr x ta rxtb x txamxxt y tmy xtamxmxmxta y xty try try tmx=+=+=+?22*2*2222*2*2(1)(1)(1)(1)(1)(1(1)mxxty tmy xtamxmxmxt+(7)For 3012(,)(1,0,),TCR =define operator 012()(0)(1),LBB=+(8)111112()000brbarBbar+=22*22*0000000(1)Ba yrmx=+222*23*202122*2*222*23*2222*2*20(1)(0)(0)(0)(,)()(0)(1)(1(0)(1)(1)(1)(1)(1)(1(1)mxmyfaamxmxmmxmyamxmxm+=+By the Riesz representation theorem,there exists a matrix whose components are bounded variation functions 3(,):1,0,R such that 01(,)()Ld =.Define 01(),-1,0)()(,)(),=0Pdss =?,0,-1,0)()(,),0Rf =Hence,system(7)can be rewritten as =p()U+R()U (9)tttU?where 12U=(x,x,y).T For 10,1,C define*01(),-1,0)()(,0)(),=0TssPsdtts=?For 3*(0,1,),CC3*(0,1,(),CC define 010,(0)(0)()()()TTdd =where()(,0).=Then,(0)PP=and *P are adjoint operators.By discussion in Section II and transformation,ts=we know that 00i are eigenvalues of P.Thus,they are also eigenvalues of*.P Direct computation yields the following result.Lemma 2 0023()(1,)iTqq qe =and 00*23()(1,)isTq sDq qe=are eigenvectors of P and*P corresponding to 00i and 00i,respectively,and*(),()1,(),()0,qqqq=where 1102briqb+=，212011031()()a bbaibriqa rb+=*1102briqb+=，002*2*1110032 1*(1)()()imxbbbriiqea b y+=1*2*2233032322*1()(1)a yDq qq qq qrqmx=+Now we compute the coordinates to describe the center manifold 0Cat 0.=Let tUbe the solution of(9)when 0,=and define*(),W(t,)=()2Re()().ttz tq UUz t q=(10)On 0C,we have W(t,)=(),(),)W z t z t,where 22201102(,)()()()22zzW z zWWzzW=+?(11)zand zare local coordinates for 0Cin the direction of *q and*q.For the solution 0tUC,since 0=,then *000()()(0)(,).z tiz tqf z z=+?(12)We rewrite(12)as 00()()(,)z tiz tg z z=+?with 22220110221(,)222zzz zg z zgg zzgg=+?(13)Hence*0(,)(0)(,)(0)(0,)tg z zqfz zqfU=.Substitute ()tU into above and comparing with(13),we get 002*22*3*2231220032*222(1)()2(1)iqmxqmy qa q ea qgDmxaq q+=+*23122*23*23221132*2202()(1)()2)(1)2a qa qmxq qq qmy q qgmxaq q qD+=+002*2*231223202032*2*22(1)()2(1)iqmxqmy qa q ea qgDmxaq q+=+0000*(2)(2)210202112232*(2)(2)1*202112*222*(3)(3)(2)(2)20211311202*33(2)(2)20211212(0)2(0)(1)1(0)2(0)(1)(21(0)(0)(0)(0)(2(1)2(1)iigDaqq Wq Wmxa qmy q Wq WmxqWq Wq Wq Wa qmyq Weq We =+000000002*2(3)(3)(2)202113113(2)201)(1)(21(1)(1)(1)2(1)(14)iiiimxqWeq Weq WeqWe +Now,we compute 20()Wand 11()W.By(9),we get*0*002Re(0)(),-1,0)=2Re(0)(),0 tWUzqzqAWqF qAWqF qF=+=?2201102 =AW+H(z,z,)=AW+h()()()22zzhzzh+?(15)For 1,0),we can get 0020201111(A-2i)W()(),AW()().(16)hh=By(15),we can get *0(,)2Re(0)()(,)()(,)()H z zqF qg z z qg z z q=Comparing the coefficients with(13),we can obtain 202002111111()()(),h()()().hg qg qg qg q=On the other hand,by(16),we get 20002020()2()()WiWh=?.Solving it,we have 00000022002200000()(0)(0)3iiiigigWqeqeEe =+Similarly,we can get 00001111110000()(0)(0)iiiggWqeqeFi =+In what follows,we seek appropriate Eand F.The definition of A and(16)imply that 0200020201()()2(0)(0)dWiWh =011111()()(0).dWh=By the definition of(,)H z zin(15),we have?202002320231111011()(0)(0)(,)()(0)(0)(,)TTHg qgqo h hHg qg qo hh=+=+where 2122*3*22232*(1)2(1)a qmxqmy qhaqmx+=+?12*23*232222232*(1)()2)2(1)amxq qq qmy q qhaq qmx+=+002222*3*2332*(1)(1)ia qmxqmy qehmx+=+?22*23*2322332*(1)()2)(1)amxq qq qmy q qhmx+=+Substituting 20()Wand 11()W into above equations,we have12311111(,)E=,12322221(,)F=,where 002111001021*11022*(2)(2)(2)(2)(1)iibriibbrirea rybriemx=+021120132(2)/ih b rireabh r a=+0221110203 12(2)(2)/)ibrih rireh a r a=+002322*1110301 322*(2)(2)(1)ih a y ebrihibbhmx =+221*112*()/(1)a ry brmx=+，?13212/abrha=?2321112()/a r br ha=+?323221112*112*()()/(1)hbrbba y h brmx=+Based on the analysis above,we can compute the following quantities 2221120111102001(0)(2),232giCg ggg=+212Re(0),C=120Re(0),Re()C=120200Im(0)Im().Ct +=From the expression of 1(0)C,it is easy to get the values of 22,and2.tOn the other hand,we know that 2determines the direction of the Hopf bifurcation:if 20(0)This indicates that it is a supercritical Hopf bifurcation.Numerical simulations are presented in Fig.1 and Fig.2.From Fig.1,it is clear the origin is asymptotically stable with 00.1=(see Fig.2).REFERENCES 1 C.S.Holling,The functional response of predators to prey density and its role in minicry and population regulation,Mem.Entomolog.Soc.Can.45(1965)3-60.2 C.Sun,M.Han and Y.Lin,Analysis of stability and Hopf bifurcation for a delayed logistic equation,Chaos,Solitons&Fractals,31(2007)672-682.3 W.Wang,L.Chen,A predator-prey system with stage-structure for predator,Comput.Math.Appl.33(1997)83-91.4 K.Cooke,Z.Grossman,Discrete delay,distributed delay and stability switches,J.Math.Anal.Appl.86(1982)592-627.5 B.Hassard,N.Kazarinoff,Y.H.Wan,Theory and Applications of Hopf Bifurcation,London Math Soc.Lect.Notes,Series,41.Cambridge:Cambridge Univ.Press,1981.0.40.200.20.40.30.20.100.10.20.150.10.0500.050.10.150.2x_1x2y 21012310.500.511.50.80.60.40.200.20.40.60.8x1x2y Fig.1 phase portrait with 0.1=Fig.2 phase portrait with 1.8=