Agraïments: The second author is partially supported by Dirección de Investigación DIUBB 1204084/R.A ratio-dependent predator-prey model with stage structure for prey was investigated in [8]. There the authors mentioned that they were unable to show if such a model admits limit cycles when the unique equilibrium point E ∗ at the positive octant is unstable. Here we characterize the existence of Hopf bifurcations for the systems. In particular we provide a positive answer to the above question showing for such models the existence of small-amplitude Hopf limit cycles being the equilibrium point E ∗ unstable

Llibre, Jaume

Vidal, Claudio

Jaume Llibre

Claudio Vidal

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Discrete and Continuous Dynamical Systems - Series B

Hopf periodic orbits for a ratio--dependent predator--prey model with stage structure

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http://www.gsd.uab.catDISCRETE AND CONTINUOUS doi:10.3934/dcdsb.2016017DYNAMICAL SYSTEMS SERIES BVolume 21, Number 6, August 2016 pp. X–XXHOPF PERIODIC ORBITS FOR A RATIO–DEPENDENTPREDATOR–PREY MODEL WITH STAGE STRUCTUREJaume LlibreDepartament de Matema`tiques, Universitat Auto`noma de Barcelona08193 Bellaterra, Barcelona, Catalonia, SpainClaudio VidalGrupo de Investigacio´n en Sistemas Dina´micos y Aplicaciones-GISDADepartamento de Matema´tica, Universidad del Bio BioConcepcio´n, Avda. Collao 1202, Chile(Communicated by Shigui Ruan)Abstract. A ratio–dependent predator-prey model with stage structure forprey was investigated in [8]. There the authors mentioned that they wereunable to show if such a model admits limit cycles when the unique equilibriumpoint E∗ at the positive octant is unstable.Here we characterize the existence of Hopf bifurcations for the systems. Inparticular we provide a positive answer to the above question showing for suchmodels the existence of small–amplitude Hopf limit cycles being the equilibriumpoint E∗ unstable.1. Introduction and statement of the main results. Frequently it is assumedin the ratio–dependent predator-prey models that each individual prey admits thesame risk to be attacked by a predator. This hypothesis is not always realistic formany species. Thus there are many species whose individuals along his life passthrough two stages, immature and mature. Here the prey individuals are classifiedin immature or mature, and we assume that the immature ones cannot be attackedby the predators. This assumption is reasonable for many mammals, because theimmature preys concealed in a mountain cave, are raised by their parents and theydo not necessarily go out for seeking food, so the possibility of being attacked bythe predators is negligible.Stage structured models have been studied with attention in these last years.Thus a stage–structured model of single species growth with of immature and ma-ture individuals was stated and analyzed in [1]. Later on in [2] it was also assumedthat the time from immaturity to maturity is itself state dependent. More recently,in the articles [4, 5, 6, 7, 9] the authors considered predator–prey models with stagestructure for prey or predator in order to analyze the influence of a stage structurefor the prey or the predator. Xu, Chaplain and Davidson [8] studied the effect of2010 Mathematics Subject Classification. Primary: 34D23, 92D25.Key words and phrases. Ratio–dependence, predator-prey model, Hopf bifurcation, averagingtheory.1This is a preprint of: “Hopf periodic orbits for a ratio-dependent predator-prey model with stagestructure”, Jaume Llibre, Claudio Vidal, Discrete Contin. Dyn. Syst. Ser. B, vol. 21(6), 1859–1867, 2016.DOI: [10.3934/dcdsb.2016026]2 JAUME LLIBRE AND CLAUDIO VIDALstage structure for prey on the dynamics of ratio–dependent predator-prey systemby considering the differential systemx˙ = −(b+ r1)x+ ay,y˙ = bx− b1y2 − a1yzy +mz,z˙ = z(−r + a2yy +mz),(1)where x = x(t) represents the density of immature individual preys at time t, andy = y(t) denotes the density of mature individual preys at time t, z = z(t) representsthe density of the predator at time t. As usual the eight parameters a, a1, a2, b, b1,m, r and r1 of the system are positive, see [8] for their meaning.When y∗ > 0 and a2 > r the differential system (1) has a unique equilibriumpointE∗ =(ay∗b+ r1, y∗,(a2 − r)y∗mr)with y∗ =aa2bm− a1(a2 − r)(b+ r1)a2b1m(b+ r1),in the positive octant, i.e., in {(x, y, z) ∈ R3 : x > 0, y > 0, z > 0}.In the conclusions of the paper [8] the authors said: We would like to mentionhere that we are unable to show system (1) admits limit cycles when E∗ exists andis unstable. We leave this for future work. As far as we know this question remainedopen. The main difficulty for solving it was that the differential system (1) dependson 8 parameters.Here we shall characterize the existence of Hopf bifurcations for systems (1).Moreover using the Hopf bifurcation for these systems we provide a positive answerto the above question restricted to the small–amplitude Hopf limit cycles.The following result characterizes when the equilibrium point E∗ has eigenvaluesof the form ρ, ε± ωi.Proposition 1. The equilibrium point E∗ of the differential system (1) has eigen-values of the form ρ, ε± ωi if and only ifa =β(b+ r1)(a22r2 − 2a2r3 + r4 + 2a22rε− 2a2r2ε+ a22ε2 + a2rε2 + a22ω2 + a2rω2)αb,a1 =βa2m(a22r2 − 2a2r3 + r4 + 2a22rε− 2a2r2ε+ a22ε2 + a22ω2)α(a2 − r) ,whereα = −a22br2 + a2b2r2 + 2a2br3 − b2r3 − br4 − a22r2r1 + 2a2br2r1 + 2a2r3r1−2br3r1 − r4r1 + a2r2r21 − r3r21 − 2a22brε+ 4a2br2ε− 2br3ε− 2a22rr1ε+4a2r2r1ε− 2r3r1ε− a22bε2 − a2brε2 + 2a2r2ε2 − 2r3ε2 − a22r1ε2−a2rr1ε2 − a22bω2 − a2brω2 + 2a2r2ω2 − 2r3ω2 − a22r1ω2 − a2rr1ω2,β = b2 + 2br1 + r21 + 2bε+ 2r1ε+ 2ε2 + 2ω2.Proposition 1 is proved in section 2.Hypothesis. In the rest of the paper we assume that the parameters a and a1 arethe ones given in the statement of Proposition 1.Since a = a(b,m, r, r1, ε, ω) and a1 = a1(a2, b,m, r, r1, ε, ω), the parameters a2,b, m, r, r1, ε and ω must be chosen in such a way that a > 0 and a1 > 0, later onwe shall show that such election of the parameters exist. Note that in fact we havechanged the parameters a and a1 by the parameters ε and ω.HOPF PERIODIC ORBITS FOR A PREDATOR–PREY MODEL 3The linearization of the differential system (1) at E∗ when ε = 0 has the pairof conjugate purely imaginary eigenvalues ±ωi, and if the other one eigenvaluesρ is non–zero this is the setting for a Hopf bifurcation. We can expect to see asmall–amplitude limit cycle bifurcating from the equilibrium point E∗. It remainsto compute the first Liapunov coefficient `1(E∗|ε=0) of (1) at E∗ when ε = 0 andsee that it is non–zero. When `1(E∗|ε=0) < 0 the point E∗ is a weak focus onthe corresponding two–dimensional central manifold of the system (1) and the limitcycle that borns from E∗ is stable restricted to this central manifold, consequentlythe weak focus becomes unstable restricted to the central surface. Due to the factthat the eigenvalues of E∗ are ρ and ε± ωi where the small–amplitude limit cyclewhich appears in the Hopf bifurcation is stable, it follows that ε > 0. In this case wesay that the Hopf bifurcation is supercritical. When `1(E∗|ε=0) > 0 the point E∗ isalso a weak focus of system (1) but the limit cycle that borns from E∗ is unstable,and the weak focus becomes stable, and consequently ε < 0. In this second case wesay that the Hopf bifurcation is subcritical. Here we use the results on the Hopfbifurcation stated in Chapters 3 and 5 of the book of Kuznetsov [3] for computing`1(E∗|ε=0).We have computed in function of all parameters of the system the Liapunovcoefficient `1(E∗|ε=0), but his expression is so huge that we shall need tens of pagesfor writing it. All the steps of the computation of `1(E∗|ε=0) are described in section2. These steps as we shall see can be used for computing effectively `1(E∗|ε=0) fora given set of parameters. Using them we have proved the following result.Theorem 2. The following system ratio–dependent predator–prey model with stagestructurex˙ = −7x+ 7y(ε2 + 14ε+ 50) (6ε2 + 4ε+ 7)−40ε2 − 14ε+ 2 ,y˙ = x− y2 + 2z(2ε2 + 14ε+ 51) (4ε2 + 4ε+ 5)y(y + z) (40ε2 + 14ε− 2) ,z˙ = z(−1 + 2yy + z),(2)has a Hopf bifurcation at the equilibrium point E∗ = (x∗, y∗, z∗) wherex∗ =(−2ε2 + 24ε+ 95) (ε2 + 1) (ε2 + 14ε+ 50) (6ε2 + 4ε+ 7)4 (20ε2 + 7ε− 1)2 =166252+O(ε),y∗ = z∗ =(−2ε2 + 24ε+ 95) (ε2 + 1)2 (−20ε2 − 7ε+ 1) =952+O(ε),when ε = 0. The eigenvalues at E∗ are−7(2ε2 − 24ε− 95)4 (20ε2 + 7ε− 1) = −6654+O(ε) and ε± i.Since `1(E∗|ε=0) = −5796656446586/592156593152863031275 < 0 the Hopf bifur-cation is supercritical and an stable Hopf limit cycle on the central manifold throughE∗ there exist for ε > 0 sufficiently small. So locally the equilibrium point E∗ isunstable on the central manifold when it exhibits the small–amplitude Hopf limitcycle.Theorem 2 is proved in Section 4.We also have the following result.4 JAUME LLIBRE AND CLAUDIO VIDALProposition 3. Assume that a Hopf limit cycle borns at the equilibrium point E∗of the differential system (1), then the real eigenvalue at this equilibrium is negative.Proposition 3 is also proved in Section 4.2. Proof of Proposition 1. The characteristic polynomial of the linear part ofthe differential system (1) at the equilibrium point E∗ isp(λ) = λ3 + d2λ2 + d1λ+ d0,whered2 =1a22m(b+ r1)(− a1a22b+ 2aa22bm+ a22b2m+ a22bmr + a1br2 − a2bmr2−a1a22r1 + 2a22bmr1 + a22mrr1 + a1r2r1 − a2mr2r1 + a22mr21),d1 =1a22m(b+ r1)(− a1a22b2 + aa22b2m− a1a22br + 2aa22bmr + a22b2mr+2a1a2br2 + a1b2r2 − 2aa2bmr2 − a2b2mr2 − a1br3 − 2a1a22br1+aa22bmr1 − a1a22rr1 + 2a22bmrr1 + 2a1a2r2r1 + 2a1br2r1 − 2a2bmr2r1−a1r3r1 − a1a22r21 + a22mrr21 + a1r2r21 − a2mr2r21),d0 = −r(−a2 + r)(−a1a2b+ aa2bm+ a1br − a1a2r1 + a1rr1)a22m.Forcing that the characteristic polynomial p(λ) be equal to (λ− ρ)(λ− ε−ωi)(λ−ε + ωi), we get the expressions for a and a1 given in the statement of Proposition1. Moreoverρ =(a2 − r)r(b+ r1)αa2(− a2b2r + a22r2 − 2a2r3 + r4 − 2a2brr1 − a2rr21+2a22rε− 2a2brε− 2a2r2ε− 2a2rr1ε+ a22ε2 − a2rε2 + a22ω2 − a2rω2).Hence, the proposition follows.3. The computation of the Liapunov coefficient. The following result comesfrom [3].Theorem 4. Let x˙ = F (x) be a differential system in Rn having E∗ as an equilib-rium point. Consider the third order Taylor approximation of F around E∗ givenby F (x) = Ax +12!B(x, x) +13!C(x, x, x) + O(|x|4). Assume that A has a pair ofpurely imaginary eigenvalues ±ωi. Let q be the eigenvector of A corresponding tothe eigenvalue ωi, normalized so that q.q = 1, where q is the conjugate vector of q.Let p be the adjoint eigenvector such that AT p = −ωip and p.q = 1. If I denotesthe n× n identity matrix, then `1(E∗|ε=0) is equal to12ωRe(p · C(q, q, q)− 2p ·B(q,A−1B(q, q)) + p ·B(q, (2ωiI −A)−1B(q, q))). (3)The linear part of system (1) at the equilibrium E∗ isA =−b− r1 a 0ba1(a22 − r2)(b+ r1)− 2aa22bma22m(b+ r1)−a1r2a220(a2 − r)2a2mr(r − a2)a2 ,HOPF PERIODIC ORBITS FOR A PREDATOR–PREY MODEL 5and its eigenvalues are ε± ωi and ρ = −δ(a2 − r)r(b+ r1)/(σa2), whereδ = r4 − 2a2r3 + a22r2 − 2a2εr2 − a2b2r − a2r21r − a2ε2r − a2ω2r − 2a2br1r+2a22εr − 2a2bεr − 2a2r1εr + a22ε2 + a22ω2,σ = br4 + r1r4 + b2r3 + r21r3 + 2ε2r3 + 2ω2r3 − 2a2br3 − 2a2r1r3 + 2br1r3+2bεr3 + 2r1εr3 − a2b2r2 − a2r21r2 − 2a2ε2r2 − 2a2ω2r2 + a22br2 + a22r1r2−2a2br1r2 − 4a2bεr2 − 4a2r1εr2 + a2bε2r + a2r1ε2r + a2bω2r + a2r1ω2r+2a22bεr + 2a22r1εr + a22bε2 + a22r1ε2 + a22bω2 + a22r1ω2.Now we compute the bi– and tri–linear functions B and C of Theorem 4, and weobtainB(x1, y1, z1, x2, y2, z2) =(B1, B2, B3),where Bi = Bi(x1, y1, z1, x2, y2, z2) areB1 = 0,B2 = − b1ηa22(a1a32by1y2 − aa32bmy1y2 − 2a1a2br2y1y2 + a1br3y1y2 + a1a32r1y1y2−2a1a2r2r1y1y2 + a1r3r1y1y2 − 2a1a2bmr2y2z1 + 2a1bmr3y2z1−2a1a2mr2r1y2z1 + 2a1mr3r1y2z1 − 2a1a2bmr2y1z2 + 2a1bmr3y1z2−2a1a2mr2r1y1z2 + 2a1mr3r1y1z2 + a1bm2r3z1z2 + a1m2r3r1z1z2),B3 =b1r(b+ r1)ηa2(a22y1y2 − 2a2ry1y2 + r2y1y2 − 2a2mry2z1 + 2mr2y2z1−2a2mry1z2 + 2mr2y1z2 +m2r2z1z2),η = a1a2b− aa2bm− a1br + a1a2r1 − a1rr1;andC(x1, y1, z1, x2, y2, z2, x3, y3, z3) =(C1, C2, C3),where Ci = Ci(x1, y1, z1, x2, y2, z2, x3, y3, z3) areC1 = 0,C2 = −a1b21m(a2 − r)r2(b+ r1)2η2a22(a2y1y2y3 − ry1y2y3 + a2my2y3z1−3mry2y3z1 + a2my1y3z2 − 3mry1y3z2 + a2my1y2z3 − 3mry1y2z3),C3 =a2a1C2.Computing the normalized eigenvector q of A when ε = 0, associated to theeigenvalue ωi, we obtain q = (q1, q2, q3) whereq1 =im(b+ r1)(ib+ ir1 + ω)(a2r − r2 + ia2ω)(a22r2 − 2a2r3 + r4 + a22ω2 + a2rω2)−α0µb(a2 − r)2 ,q2 =m(a2r − r2 + ia2ω)µ(a2 − r)2 ,q3 =aµ,where α0 = α|ε=0 andµ =[1 +m2(a22r2 − 2a2r3 + r4 + a22ω2)(a2 − r)4 +m2(b+ r1)2α20b2(a2 − r)4 (b2 + 2br1 + r21 + ω2)(a22r2 − 2a2r3 + r4 + a22ω2)(a22r2 − 2a2r3 + r4 + a22ω2 + a2rω2)2]1/2.6 JAUME LLIBRE AND CLAUDIO VIDALThe adjoint eigenvector p = (p1, p2, p3) isp1 =bνb+ r1 − iω ,p2 = ν,p3 =(mr2(a2r − r2 + ia2ω)(b2 + 2br1 + r21 + 2ω2)να0(a2 − r) ,whereν =α0µ(a2 − r)2(b+ r1 + iω)2mων1,withν1 = −ia22b3r2 + ia32br3 + ia2b3r3 − 3ia22br4 + 3ia2br5 − ibr6 − 3ia22b2r2r1+ia32r3r1 + 3ia2b2r3r1 − 3ia22r4r1 + 3ia2r5r1 − ir6r1 − 3ia22br2r21+3ia2br3r21 − ia22r2r31 + ia2r3r31 − a32br2ω + a22b2r2ω + 2a22br3ω−a2b2r3ω − a2br4ω − a32r2r1ω + 2a22br2r1ω + 2a22r3r1ω − 2a2br3r1ω−a2r4r1ω + a22r2r21ω − a2r3r21ω + ia32brω2 − 2ia22br2ω2 + ia2br3ω2+ia32rr1ω2 − 2ia22r2r1ω2 + ia2r3r1ω2 − a32bω3 − a22brω3 + 2a22r2ω3−2a2r3ω3 − a32r1ω3 − a22rr1ω3.In short, all the elements appearing in Theorem 4 are given explicitly for thedifferential system (1), only remains to compute the Liapunov coefficient `1(E∗|ε=0)using formula (3). As we mentioned in the introduction we have computed explicitly`1(E∗|ε=0) in function of the coefficients of the system (1), using the algebraicmanipulator mathematica, but this expression occupies tens of pages.4. Proofs of Theorem 2 and Proposition 3.Proof of Theorem 2. We shall use the notations and definitions introduced in sec-tion 3. Then, for the differential system (2) we have the matrixA =−7 −7(ε2 + 14ε+ 50) (6ε2 + 4ε+ 7)2 (20ε2 + 7ε− 1) 01160ε3 + 642ε2 + 370ε+ 6354 (20ε2 + 7ε− 1)(2ε2 + 14ε+ 51) (4ε2 + 4ε+ 5)4 (20ε2 + 7ε− 1)012−12,having the eigenvalues given in the statement of Theorem 2.The normalized eigenvector q of A when ε = 0 associated to the eigenvalue ωi isq =(−441 + 637i√600274,−√2300137(1 + 2i),−√2300137),The adjoint eigenvector p = (p1, p2, p3) isp1 =4559− 15988i11056025√3001372,p2 =637− 4659i442241√3001372,p3 =−507705 + 172635i884482√3001372.HOPF PERIODIC ORBITS FOR A PREDATOR–PREY MODEL 7The bi–linear function B(x1, y1, z1, x2, y2, z2) = (B1, B2, B3) with Bi = Bi(x1,y1, z1, x2, y2, z2) is given byB1 = 0,B2 =12660(−875y1y2 − 3570y2z1 − 3570y1z2 + 1785z1z2),B3 =1190(−y1y2 + 2y2z1 + 2y1z2 − z1z2);and the tri–linear function C(x1, y1, z1, x2, y2, z2, x3, y3, z3) = (C1, C2, C3) withCi = Ci(x1, y1, z1, x2, y2, z2, x3, y3, z3) is given byC1 = 0,C2 = − 517220(y1y2y3 − y2y3z1 − y1y3z2 − y1y2z3),C3 =118050(y1y2y3 − y2y3z1 − y1y3z2 − y1y2z3).According to Theorem 4 for computing `1(E∗|ε=0), we need to compute first A−1and a (2iI −A)−1. Note that now ω = 1. We have thatA−1 =− 89133−7019892519− 2665− 2955119− 2665− 2951319 ,and(2iI −A)−1 =7 + 2i −1225 0−1 6354+ 2i25540 −1212+ 2i .The first, second and third terms of `1(E∗|ε=0) given in Theorem 4 arep · C(q, q, q) = 950436 + 129948i239582861065685,−2p ·B(q, A−1B(q, q)) = −152367568 + 104825176i4552074360248015,p ·B(q, (2iI −A)−1B(q, q)) = 928154260416 + 1952607677788i93498409445188899675.Consequently we get`1(E∗|ε=0) = − 5796656446586592156593152863031275.This completes the proof of the theorem.Proof of Proposition 3. From the expression of ρ, given in section 2, and for |ε|sufficiently small it follows that ρ > 0 if and only ifα0(−a2b2r + a22r2 − 2a2r3 + r4 − 2a2brr1 − a2rr21 + a22ω2 − a2rω2) > 0. (4)Here we have used that a2 > r, recall that this is due to the fact that the equilibriumpoint E∗ must be in the positive octant.8 JAUME LLIBRE AND CLAUDIO VIDALBy Proposition 1 and for |ε| sufficiently small we get that a > 0 if and only ifα0(a22r2 − 2a2r3 + r4 + a22ω2 + a2rω2) > 0. (5)Note that β|ε=0 > 0, see its definition in the statement of Proposition 1.On the other hand, the coordinate x of the equilibrium point E∗ is positive ifand only−(−a2b2r + a22r2 − 2a2r3 + r4 − 2a2brr1 − a2rr21 + a22ω2 − a2rω2)·(a22r2 − 2a2r3 + r4 + a22ω2 + a2rω2) > 0. (6)Now, since the x–coordinate of E∗ and the parameter a are positive, the inequal-ities (5) and (6) hold. Then, from (4) we obtain that ρ < 0. That is, the realeigenvalue of the equilibrium point E∗ is negative for |ε| sufficiently small. Hencethe proposition is proved.5. Conclusions. Xu, Chaplain and Davidson [8] studied the effect of stage struc-ture for prey on the dynamics of ratio–dependent predator-prey system by consid-ering the differential systemx˙ = −(b+ r1)x+ ay,y˙ = bx− b1y2 − a1yzy +mz,z˙ = z(−r + a2yy +mz),(7)where x = x(t) represents the density of immature individual preys at time t, andy = y(t) denotes the density of mature individual preys at time t, z = z(t) representsthe density of the predator at time t. As usual the eight parameters a, a1, a2, b, b1,m, r and r1 of the system are positive. It is verified that when y∗ > 0 and a2 > rthe differential system (7) has a unique equilibrium pointE∗ =(ay∗b+ r1, y∗,(a2 − r)y∗mr)with y∗ =aa2bm− a1(a2 − r)(b+ r1)a2b1m(b+ r1),in the positive octant, i.e., in {(x, y, z) ∈ R3 : x > 0, y > 0, z > 0}. In theconclusions of the paper [8] the authors said: We would like to mention here thatwe are unable to show system (1) admits limit cycles when E∗ exists and is unstable.We leave this for future work. As far as we know this question remained open. Inthis work we answer affirmatively this question, see Theorem 2. The main difficultyfor solving it was that the differential system (7) depends on 8 parameters, ourmain result show the existence of a Hopf bifurcation at the equilibrium point E∗for appropriate values of the parameters.Acknowledgments. The first author is partially supported by a MINECO grantMTM2013-40998-P, an AGAUR grant number 2013SGR-568, the grants FP7-PEO-PLE-2012-IRSES 318999 and 316338. The second author is partially supported byDireccio´n de Investigacio´n DIUBB 1204084/R.REFERENCES[1] W. G. Aiello and H. I. Freedman, A time delay model of single–species growth with stagestructure, Math. Biosci., 101 (1990), 139–153.[2] W. G. Aiello, H. I. Freedman and J. Wu, Analysis of a model representing stage-structuredpopulation growth with state–dependent time delay, SIAM J. Appl. Math., 52 (1992), 855–869.HOPF PERIODIC ORBITS FOR A PREDATOR–PREY MODEL 9[3] Y. Kuznetsov, Elements of Applied Bifurcation Theory, Applied Mathematical Sciences, Vol.112, Springer–Verlag, New York, 2004.[4] Z. Li, M. Han and F. Chen, Global stability of a predator–prey system with stage structureand mutual interference, Discrete Contin. Dyn. Syst. Ser. B, 19 (2014), 173–187.[5] K. G. Magnusson, Destabilizing effect of cannibalism on a structured predator–prey system,Math. Biosci., 155 (1999), 61–75.[6] W. Wang and L. Chen, A predator–prey system with stage structure for predator, Comput.Math. Appl., 33 (1997), 83–91.[7] R. Xu, Global convergence of a predator–prey model with stage structure and spatio–temporaldelay, Discrete Contin. Dyn. Syst. Ser. B, 15 (2011), 273–291.[8] R. Xu, M. A. J. Chaplain and F. A. Davidson, Persistence and global stability of a ratio–dependent predator–prey model with stage structure, Appl. Math. Comput., 158 (2004),729–744.[9] X. Zhang and L. Chen, The stage–structured predator–prey model and optimal harvestingpolicy, Math. Biosci., 168 (2000), 201–210.Received October 2014; revised March 2016.E-mail address: jllibre@mat.uab.catE-mail address: clvidal@ubiobio.cl

Hopf periodic orbits for a ratio-dependent predator-prey model with stage structure

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