Research article Special Issues

Hopf bifurcation problems near double positive equilibrium points for a class of quartic Kolmogorov model

  • The Kolmogorov model is a class of significant ecological models and is initially introduced to describe the interaction between two species occupying the same ecological habitat. Limit cycle bifurcation problem is close to Hilbertis 16th problem. In this paper, we focus on investigating bifurcation of limit cycle for a class of quartic Kolmogorov model with two positive equilibrium points. Using the singular values method, we obtain the Lyapunov constants for each positive equilibrium point and investigate their limit cycle bifurcations behavior. Furthermore, based on the analysis of their Lyapunov constants' structure and Hopf bifurcation, we give the condition that each one positive equilibrium point of studied model can bifurcate 5 limit cycles, which include 3 stable limit cycles.

    Citation: Chaoxiong Du, Wentao Huang. Hopf bifurcation problems near double positive equilibrium points for a class of quartic Kolmogorov model[J]. AIMS Mathematics, 2023, 8(11): 26715-26730. doi: 10.3934/math.20231367

    Related Papers:

    [1] Aeshah A. Raezah, Jahangir Chowdhury, Fahad Al Basir . Global stability of the interior equilibrium and the stability of Hopf bifurcating limit cycle in a model for crop pest control. AIMS Mathematics, 2024, 9(9): 24229-24246. doi: 10.3934/math.20241179
    [2] Yudan Ma, Ming Zhao, Yunfei Du . Impact of the strong Allee effect in a predator-prey model. AIMS Mathematics, 2022, 7(9): 16296-16314. doi: 10.3934/math.2022890
    [3] Junjie Mao, Xiaowei Liu . Dynamical analysis of Kaldor business cycle model with variable depreciation rate of capital stock. AIMS Mathematics, 2020, 5(4): 3321-3330. doi: 10.3934/math.2020213
    [4] Sulasri Suddin, Fajar Adi-Kusumo, Mardiah Suci Hardianti, Gunardi . Bifurcation analysis of a diffuse large b-cell lymphoma growth model in germinal center. AIMS Mathematics, 2025, 10(5): 12631-12660. doi: 10.3934/math.2025570
    [5] Binfeng Xie, Na Zhang . Influence of fear effect on a Holling type III prey-predator system with the prey refuge. AIMS Mathematics, 2022, 7(2): 1811-1830. doi: 10.3934/math.2022104
    [6] Wenwen Hou, Maoan Han . Melnikov functions and limit cycle bifurcations for a class of piecewise Hamiltonian systems. AIMS Mathematics, 2024, 9(2): 3957-4013. doi: 10.3934/math.2024194
    [7] Haowen Gong, Huijun Xiang, Yifei Wang, Huaijin Gao, Xinzhu Meng . Strategy evolution of a novel cooperative game model induced by reward feedback and a time delay. AIMS Mathematics, 2024, 9(11): 33161-33184. doi: 10.3934/math.20241583
    [8] Mirela Garić-Demirović, Dragana Kovačević, Mehmed Nurkanović . Stability analysis of solutions of certain May's host-parasitoid model by using KAM theory. AIMS Mathematics, 2024, 9(6): 15584-15609. doi: 10.3934/math.2024753
    [9] Jing Zhang, Shengmao Fu . Hopf bifurcation and Turing pattern of a diffusive Rosenzweig-MacArthur model with fear factor. AIMS Mathematics, 2024, 9(11): 32514-32551. doi: 10.3934/math.20241558
    [10] Jawdat Alebraheem . Asymptotic stability of deterministic and stochastic prey-predator models with prey herd immigration. AIMS Mathematics, 2025, 10(3): 4620-4640. doi: 10.3934/math.2025214
  • The Kolmogorov model is a class of significant ecological models and is initially introduced to describe the interaction between two species occupying the same ecological habitat. Limit cycle bifurcation problem is close to Hilbertis 16th problem. In this paper, we focus on investigating bifurcation of limit cycle for a class of quartic Kolmogorov model with two positive equilibrium points. Using the singular values method, we obtain the Lyapunov constants for each positive equilibrium point and investigate their limit cycle bifurcations behavior. Furthermore, based on the analysis of their Lyapunov constants' structure and Hopf bifurcation, we give the condition that each one positive equilibrium point of studied model can bifurcate 5 limit cycles, which include 3 stable limit cycles.



    The following differential autonomous systems in a planar vector field

    ˙x=F(x,y),˙y=G(x,y) (1)

    have been widely studied and a great deal of attentions have been paid to this problem in many literatures. This activity reflects the breadth of interest in Hilbert's 16th problem and the fact that the above systems are often used as mathematical models to describe real-life problems. Hilbert's 16th problem is to find the maximum number of limit cycles of system (1).

    The predator-prey system, the competition system and the cooperation model are the three most basic types of systems in mathematical ecology. Theoretically, many natural predator-prey systems can be discussed and investigated by some kinds of ecological models. The qualitative properties of differential systems are often used to describe the characteristics of ecosystems, as they have been investigated in some literatures, for example, Liénard systems ([2,3,19,21]), Kolmogorov systems ([2,3,4,5,6,7,8,9,10,11,12,16,17,18,19,20,22,23,24,25,26]) and some other differential systems ([13,14,15]) and so on. The Kolmogorov systems (introduced by A. Kolmogorov in 1936 [12]), as a class of significant ecological models, were initially introduced to describe the interaction between two species occupying the same ecological habitat. The form Kolmogorov models are as follows:

    {dxdt=xf(x,y),dydt=yg(x,y), (2)

    in which f(x,y) and g(x,y) are polynomials in x and y. The variables x and y are often described as the number of species in two ecological populations, dxdt and dydt represent the growth rates of x and y. Hence, attention is often restricted to the behavior of orbits in the 'realistic quadrant' {(x,y):x>0,y>0}. Particular significance in applications is the existence of limit cycles and the number of limit cycles that can occur near positive equilibrium points, because a limit cycle corresponds to an equilibrium state of the system. The existence and stability of limit cycles are closely related to the positive equilibrium points. Hence, many references studying Kolmogorov models pay more attention to the limit cycles problem.

    On the qualitative analysis and the bifurcation of limit cycles for cubic planar Kolmogorov systems, some papers are as follows: Ref. [1] characterized the center conditions for a cubic Kolmogorov differential system and obtained the condition that the positive equilibrium point can become a fine focus of order five. Ref. [23,24] studied a class of cubic Kolmogorov systems that can bifurcate three limit cycles from the positive equilibrium point (1, 1). Ref. [4] investigated the limit cycles bifurcation problem for a class of cubic Kolmogorov system and showed that this class of the Kolmogorov system could bifurcate five limit cycles including 3 stable cycles. Ref. [22] showed that a class of the cubic Kolmogorov system could bifurcate 6 limit cycles. Ref. [25] obtained the condition of integrability and non-linearizability of weak saddles for a cubic Kolmogorov model. Ref. [6] investigated limit cycles in a class of the quartic Kolmogorov model with three positive equilibrium points. Ref. [7] studied the three-Dimensional Hopf Bifurcation for a class of the cubic Kolmogorov model. Ref. [19] investigated the integrability of a class of 3-dimensional Kolmogorov system and provided the phase portraits.

    In addition to cubic Kolmogorov systems, there is also a lot of literature on the bifurcation of limit cycles for some generalized Kolmogorov systems and higher order Kolmogorov systems. For example: Ref. [10,26] studied a general Kolmogorov model and obtained the condition for the existence and uniqueness of limit cycles and it classified a series of models. Ref. [8] investigated the bifurcation of limit cycles for a class of the quartic Kolmogorov model with two symmetrical positive singular points. Ref. [9] investigated the Hopf bifurcation problem about small amplitude limit cycles and the local bifurcation of critical periods for a quartic Kolmogrov system at the single positive equilibrium point (1, 1) and proved that the maximum number of small amplitude limit cycles bifurcating from the equilibrium point (1, 1) was 7. Ref. [20] considered the Kolmogorov system of degree 3 in R2 and R3, and showed it had an equilibrium point in the positive quadrant and octant, and provided sufficient conditions in order that the equilibrium point will be a Hopf point for the planar case and a zero-Hopf point for the spatial one, and studied the limit cycles bifurcating from these equilibria using averaging theory of second and first order.

    As far as limit cycles of Kolmogorov models are concerned, many good results have been obtained, especially in lower degree system by analyzing a sole positive equilibrium point's state. However, results for Simultaneous limit cycles bifurcating from several different equilibrium points' is less seen, and perhaps it is difficult to investigate this kind of problem. From an ecological perspective, investigation about multiple positive equilibrium points is meaningful, the equilibrium point (a,b) represents that the ratio of the density about predator and prey is a:b, and so it is possible for several equilibrium points to occur in ecosystem.

    In this paper, we study a class of the following quartic Kolmogorov models

    {dxdt=16x(y1)(x224ay6A10x+6A10xy7y2+30ay26a)=P(x,y),dydt=13y(3b+3B10xx2+6x3+9by6B10xy26x2y2y227by2+39xy2+3B10xy216y3+15by3)=Q(x,y), (3)

    in which a,b,A10andB10R.

    Clearly, model (3) has two positive equilibrium points namely, (1, 1) and (2, 1). We will focus on the limit cycles bifurcations of the two positive equilibrium points. By analyzing and proving carefully, we obtain that each one of the two positive equilibrium points can be a 5th-order fine focus. Furthermore, we find the condition that each positive equilibrium point can bifurcate five limit cycles, of which three limit cycles are stable. Our results are concise (especial that in the expressions of focal values) and the proof about existence of limit cycles is algebraic and symbolic.

    This paper includes 4 sections. In Section 2, we introduce the method to compute focal value offered by [18]. In Section 3, we respectively compute the focal values of the two positive equilibrium points of model (3) and obtain the condition that they can be two 5th-order fine focuses. In Section 4, we discuss the bifurcation of limit cycles of model (3) and obtain that each one of the two positive equilibrium points of model (3) can have five small limit cycles; we give an example that three stable limit cycles can occur near each positive equilibrium point under a certain condition.

    In order to use the algorithm of the singular point value to compute focal values and construct the Poincaré succession function, we need to give some properties of focal values and singular point values.

    Consider the following system

    {dxdt=δxy+k=1Xk(x,y),dydt=x+δy+k=1Yk(x,y), (4)

    where Xk(x,y),Yk(x,y) are homogeneous polynomials of degree k on x,y. Under the polar coordinates x=rcosθ,y=rsinθ, system (4) takes the form

    drdθ=rδ+k=2rk1φk+1(θ)1+k=2rk1ψk+1(θ). (5)

    For sufficiently small h, let

    d(h)=r(2π,h)h,r=r(θ,h)=m=1v(θ)hm, (6)

    be the Poincaré succession function and the solution of Eq (6) satisfies the initial value condition r|θ=0=h. It is evident that

    v1(θ)=eδθ>0,vm(θ)=0,m=2,3, (7)

    Lemma 2.1. For system (4) and any positive integer m, among v2m(2π),vk(2π) andvk(π), there exists expression of the relation

    v2m(2π)=11+v1(π)[ξ(0)m(v1(2π)1)+m1k=1ξ(k)mv2k+1(2π)], (8)

    where ξ(k)m are all polynomials of v1(π),v2(π),, vm(π) and v1(2π),v2(2π), ,vm(2π) with rational coefficients.

    In addition to indicating that v2m=0 under the conditions v1(2π)=1,v2k+1(2π) =0,k=1,2,,m1. Lemma 2.1 plays an important role in construction of Poincaré succession function.

    Definition 2.1. For system (4), in the expression (6), if v1(2π)=1, then the origin is called the rough focus (strong focus); if v1(2π)=1, and v2(2π)=v3(2π)==v2k(2π)=0,v2k+1(2π)0, then the origin is called the fine focus (weak focus) of order k, and the quantity of v2k+1(2π) is called the kth focal values at the origin (k=1,2,); if v1(2π)=1, and for any positive integerk,v2k+1(2π)=0, then the origin is called a center.

    By means of transformation

    z=x+yi,w=xyi,T=it,i=1, (9)

    system (4)|δ=0 can be transformed into the following systems

    {dzdT=z+k=2Zk(z,w)=Z(z,w),dwdT=wk=2Wk(z,w)=W(z,w), (10)

    where z,w,T are complex variables and

    Zk(z,w)=α+β=kaαβzαwβ,Wk(z,w)=α+β=kbαβwαzβ.

    It is obvious that the coefficients of system (10) satisfy conjugate condition aαβ=¯bαβ, we call that system (4)|δ=0 and (10) are concomitant.

    Lemma 2.2. (See [16]) For system (10), we can derive successively the terms of the following formal series

    M(z,w)=α+β=0cαβzαwβ, (11)

    such that

    (MZ)z(MW)w=m=1(m+1)μm(zw)k, (12)

    where ckkR,k=1,2,, and to any integer m, μm is determined by following recursion formulas

    c0,0=1,
    when(α=β>0)orα<0,orβ<0,cα,β=0,

    else

    cα,β=1βαα+β+2k+j=3[(α+1)ak,j1(β+1)bj,k1]cαk+1,βj+1, (13)
    μm=2m+2k+j=3(ak,j1bj,k1)cmk+1,mj+1. (14)

    Lemma 2.3. (See[18]) For systems (5)|δ=0, (10) and any positive integer m, the following assertion holds

    v2m+1(2π)=iπ(μm+m1k=1ξ(k)mμk), (15)

    where ξ(k)m(k=1,2,,m1) be polynomial functions of coefficients of system (10).

    For the linearized system of model (3), its coefficient matrix in point (x0,y0) is as follows:

    A(x0,y0)=[P(x,y)xP(x,y)yQ(x,y)xQ(x,y)y](x0,y0),

    in which

    P(x,y)x=16(y1)(6a12A10x+3x224ay+12A10xy7y2+30ay2),P(x,y)y=16x(18a12A10x+x2+14y108ay+12A10xy21y2+90ay2),Q(x,y)x=13y(3B102x+18x26B10y52xy+39y2+3B10y2),Q(x,y)y=13(3b+3B10xx2+6x3+18by12B10xy52x2y6y281by2+117xy2+9B10xy264y3+60by3).

    For the two positive equilibrium points (1, 1) and (2, 1) of model (3), their coefficient matrixes of the linearized system of model (3) become

    A(1,1)=A(2,1)=[0110].

    Clearly, A(1,1) and A(2,1) have the same two eigenvalues ±i. Hence, model (3) can be changed into the system (4) by making some appropriate transformations.

    For convenience, we will respectively compute the focal values of each positive equilibrium point of model (3).

    By means of transformation

    x=u+1,y=v+1, (16)

    model (3) takes the following form

    {dudt=vuv+16(u+1)v(2u+u214v+36av+6A10v+6A10uv7v2+30av2),dvdt=u+uv+13(v+1)(6u39u2+26uv26u2v11v2+18bv2+3B10v2+39uv2+3B10uv216v3+15bv3), (17)

    and the equilibrium point (1,1) of model (3) becomes the origin of (17) correspondingly.

    Under the transformation

    z=u+iv,w=uiv,T=it,i=1, (18)

    system (17) becomes the following complex system

    {dzdT=z+Z2(z,w)+Z3(z,w)+Z4(z,w),dwdT=wW2(z,w)W3(z,w)W4(z,w), (19)

    in which

    Z2(z,w)=112i(22+18a+3A10+18ib+3iB10)w2+16(20+7i18ia3iA10+18b+3B10)wz+112i(364i+18a+3A10+18ib+3iB10)z2,Z3(z,w)=i8[5+18i+(6+5i)a+2A1011b(1i)B10]w3+124[74109i+(4518i)a6iA10+99ib+(3+9i)B10]w2z124[92123i+(45+18i)a+6iA10+99ib(39i)B10]wz2+124[64+i+(15+18i)a+6iA10+33ib(33i)B10]z3,Z4(z,w)=148(14+33i+15a3iA1015b+3iB10)w4+124(29+45i+15a30b+3iB10)w3z+124(743iA10+45b)w2z2124(35+45i+15a+30b+3iB10)wz3+116(2+11i+5a+iA10+5b+iB10)z4,W2(z,w)=112(4+36i18ia3iA1018b3B10)w2+16(207i+18ia+3iA10+18b+3B10)wz112i(22+18a+3A1018ib3iB10)z2,W3(z,w)=124i[164i+(18+15i)a+6A10+33b+(33i)B10]w3+124[92123i(4518i)a+6iA10+99ib+(3+9i)B10]w2z+124[74+109i+(45+18i)a+6iA1099ib+(39i)B10]wz218[185i+(5+6i)a+2iA1011ib+(1i)B10]z3,W4(z,w)=116(211i+5aiA10+5biB10)w4124(3545i+15a+30b3iB10)w3z+124(74+3iA10+45b)w2z2+124(2945i+15a30b3iB10)wz3148(1433i+15a+3iA1015b3iB10)z4.

    Obviously, system (19) belongs to the type of system (10). Then, we can compute the focal values of the origin of (19) (namely the focal values of the equilibrium point (1,1) of model (3)) by using the method of Section 2. According to the recursion formulas (13) and (14) offered by Lemma 2.2, the following theorem holds. For convenience, here we note that B10=λ6b.

    Theorem 3.1. The first five singular point values at the origin of system (19) are as follows:

    μ1=136i(379378a54A10+135b102λ+ 108aλ+18A10λ); μ2=19720iA10f1+129160if2; μ3=141990400ig2+ 14665600iA10(9A10g3+g4); μ4=1226748160000i(486A410g5+ 3A10g6+g7+9A210g8+81A310g9+7290A510g10+3280500A610g11); μ5=12203992115200000i(f3+3A10f4+27A10f5λ+27A10f6λ2+81A10f7λ3+ 243A10f8λ4+2187A10f9λ5+98415A10f10λ6 +98415A10f11λ6);

    in which fi,gi,(i{1,2,11}) are the expressions about A10,a,λ, their expressions can be obtained via computing by reader.

    According to the relation between model (3) and system (17) and system (19), from Theorem 3.1 and Lemma 2.3, the following theorem is visible.

    Theorem 3.2. The simplified expressions of the first five focal values in the equilibrium (1,1) of model (3) (or the first five focal values at the origin of system (19)) are as follows:

    v3=136π(379378a54A10+135b102λ+108aλ+ 18A10λ); v5=19720πA10f1 129160πf2; v7=141990400πg214665600 πA10(9A10g3+g4); v9=1226748160000π(486A410g5+ 3A10g6+g7+9A210g8+81A310g9+7290A510g10+ 3280500A610g11); v11=12203992115200000π(f3+3A10f4+ 27A10f5λ+27A10f6λ2+ 81A10f7λ3+243A10f8λ4+2187A10f9λ5+ 98415A10f10λ6+98415A10f11λ6). From Theorem 3.2, we have the following theorems.

    Theorem 3.3. The equilibrium point (1,1) of model (3) can be a 5th-order fine focus at most.

    Proof. According to Definition 2.1, we need to prove that there exits a group of real values about λ,A10,a,b such that v3=v5=v7=v9=0,v110.

    At first, we prove v3=v5=v7=v9=0 have real solutions. Let v3=0, then we have

    b=1135(379+378a+54A10+102λ108aλ18A10λ). (20)

    From the quality of resultant, if f(x,y)=0,g(x,y)=0 have solutions, then the resultant of f(x,y),g(x,y) with respect to x or y will vanish. When computer soft Mathematica 6.0 is used, f(x,y),g(x,y) with respect to x is shown as Resultant[f,g,x]. Hence, v3=v5=v7=v9=0 hold if and only if Eq (20) holds and

    {r57=Resultant[v5,v7,A10]=0,r59=Resultant[v5,v9,A10]=0. (21)

    While Eq (21) hold if and only if

    r579=Resultant[r57,r59,a]=0. (22)

    By computing, we obtain

    r579=Resultant[r57,r59,a]=2401(5365λ+10λ2)2(42611966λ+200λ2)2g(λ),

    in which g(λ) is a 242 degrees function on λ. It can be seen that Eq (22) has some real solutions such as λ=120(65±2105) et al. Hence, v3=v5=v7=v9=0 have real solutions.

    In fact, we can find 13 groups of real number solutions such that v3=v5=v7=v9=0, namely

    1)A100.459518722796388637074,a0.20757605414524699732908423106,

    λ187.1523806676270387309,b100.90105602889687249731131061;

    2)A1051.84584918202522022025,a7.9357437035495313851150889631,

    b7.006788358368472491711,λ6.31370574128761387341294162;

    3)A1025.494503512525795111362,a3.07139640695089280406061269,

    b1.532179840913880774192,λ1.72922556020556141044796397;

    4)A1025.070512253424586307122,a3.33783060983953401572921236,

    b3.263720382764871832166,λ4.72919037845355221552810321;

    5)A1014.449047061417717199600,a3.08815966599950902620342510,

    b0.629369782476961469600,λ3.25754589079607174405111000;

    6)A108.111924106836898022434,a1.53372638845377554695210010,

    b0.126446067636650042023,λ0.53142294384083763082100010;

    7)A105.621924167862105789322,a0.00165824258524943110012000,

    b0.514536038089562571070,λ8.50571390132676346023508968;

    8)A103.0566036738431139791551,a0.98949281181308481001231800,

    b1.3119509650856638974769,λ4.97304633639086336473699900;

    9)A106.6897266686712602090401,a0.25129280165975847841010671,

    b1.7878517956323023614398,λ14.74192519032294176080864107;

    10)A101.6274879010489748450201,a0.18419310366272218518672210,

    b0.4716932080651445815482,λ5.39966835957640020711011671;

    11)A100.1394463380540504608910,a0.08831387583182001769245266,

    b1.6869344282004140469154,λ5.80208273810907688645907382;

    12)A108.0555918046326485797027,a0.68203278918102018306728290,

    b0.8147669629127921204300,λ3.84321467839847163723015678;

    13)A107.2153236483212033683720,a1.12357380119512738273882722,

    b2.6297408958156117722512,λ6.32346437261518950112432918.

    Next, we prove v110 if v3=v5=v7=v9=0.

    Let r1=Resultant[v5,v11,A10],r2=Resultant[v7,v11,A10],r3= Resultant[v9,v11,A10], and r12=Resultant[r1,r2,a],r13=Resultant[r1,r3,a].

    If v3=v5=v7=v9=v11=0, then r123=Resultant[r12,r13,λ]=0. In fact, by computing we obtain r123=Resultant[r12,r13,λ]=30765900988602388330650400.

    Hence, v110 if v3=v5=v7=v9=0, then the equilibrium point (1,1) of model (3) can be a 5th-order fine focus at most.

    Next, we compute the focal values of the positive equilibrium point (2, 1) of model (3). By means of transformation

    x=˜u+2,y=˜v+1, (23)

    model (3) takes the following form:

    {d˜udt=˜v˜u˜v2+˜v6(˜u+2)(4˜u+˜u214˜v+36a˜v+12A10˜v+6A10˜u˜v7˜v2+30a˜v2),d˜vdt=˜u+˜u˜v+13(˜v+1)(6˜u3+9˜u226˜u˜v26˜u2˜v+28˜v2+18b˜v2+6B10˜v2+39˜u˜v2+3B10˜u˜v216˜v3+15b˜v3), (24)

    and the equilibrium point (2,1) of model (3) becomes the origin of system (24) correspondingly. Under the transformation

    ˜z=˜u+i˜v,˜w=˜ui˜v,T=it,i=1, (25)

    system (24) becomes its concomitant complex system, i.e.,

    {d˜zdT=˜z+Z2(˜z,˜w)+Z3(˜z,˜w)+Z4(˜z,˜w),d˜wdT=˜wW2(˜z,˜w)W3(˜z,˜w)W4(˜z,˜w), (26)

    in which

    Z2(˜z,˜w)=124i(74+33i+72a+24A10+36ib+12iB10)˜w2+16(37+14i36ia12iA10+18b+6B10)˜w˜z+124i(18+43i+72a+24A10+36ib+12iB10)˜z2,Z3(˜z,˜w)=18i[12i+(6+10i)a+4A1011b(2i)B10)˜w3+124[13+26i+(9018i)a12iA10+99ib+(3+18i)B10]˜w2˜z124[49+12i+(90+18i)a+12iA10+99ib(318i)B10]˜w˜z2+124[17+22i+(30+18i)a+12iA10+33ib(36i)B10]˜z3,Z4(˜z,˜w)=148(1433i15a+3iA10+15b3iB10)˜w4+124(29+45i+15a30b+3iB10)˜w3˜z+124(743iA10+45b)˜w2˜z2124(35+45i+15a+30b+3iB10)˜w˜z3+116(2+11i+5a+iA10+5b+iB10)˜z4,W2(˜z,˜w)=124i(1843i+72a+24A1036ib12iB10)˜w2+16(3714i+36ia+12iA10+18b+6B10)˜w˜z124i(7433i+72a+24A1036ib12iB10)˜z2,W3(˜z,˜w)=124i[2217i+(18+30i)a+12A10+33b+(63i)B10]˜w3+124[49+12i(9018i)a+12iA10+99ib+(3+18i)B10]˜w2˜z+124[1326i+(90+18i)a+12iA1099ib+(318i)B10]˜w˜z218[112i+(10+6i)a+4iA1011ib+(12i)B10]˜z3,W4(˜z,˜w)=116(211i+5aiA10+5biB10)˜w4124(3545i+15a+30b3iB10)˜w3˜z+124(74+3iA10+45b)˜w2˜z2+124(2945i+15a30b3iB10)˜w˜z3148(1433i+15a+3iA1015b3iB10)˜z4.

    Obviously, system (26) belongs to the class of system (10). Next, we begin to compute the focal values of the origin of system (24) (namely the focal values of the equilibrium point (2,1) of model (3)). According to the recursion formulas of Lemma 2.2, we have the following result. For convenience, we note that B10=λ13b176.

    Theorem 3.4. The first five singular point values at the origin of system (24) are as follows:

    ˜μ1=112i(312a+110A10+45b+8λ1+144aλ1+48A10λ1);

    ˜μ2=14860i(h1+λ1h2+108λ21h3);

    ˜μ3=127993600i(15240960λ41h41440λ31h5+12λ21h6+λ1h7+h8);

    ˜μ4=1604661760000i(10484051097600λ61h9207360λ51h1041472λ41h11 +192λ31h12+4λ21h13λ1h14+h15);

    ˜μ5=17836416409600000i(6216291333365760000λ81h16+149299200λ71h17+1244160 ×λ61h186912λ51h191728λ41h20+48λ31h21+4λ21h22+λ1h23+h24);

    in which hi,(i{1,2,24}) are the expressions about A10,a.

    Considering the relation between model (3) and system (24) and system (26), from Theorem 3.4 and Lemma 2.3, the following theorem is visible.

    Theorem 3.5. The first five focal values of the equilibrium point (2,1) of model (3) (or the first five focal values at the origin of system (24)) are as follows:

    ˜v3=112π(312a+110A10+45b+8λ1+144aλ1+48A10λ1);

    ˜v5=14860π(h1+λ1h2+108λ21h3);

    ˜v7=127993600π(15240960λ41h41440λ31h5+12λ21h6+λ1h7+h8);

    ˜v9=1604661760000π(10484051097600λ61h9207360λ51h1041472λ41h11 +192λ31h12+4λ21h13λ1h14+h15);

    ˜v11=17836416409600000π(6216291333365760000λ81h16+149299200λ71h17+1244160 ×λ61h186912λ51h191728λ41h20+48λ31h21+4λ21h22+λ1h23+h24).

    From Theorem 3.5, we have

    Theorem 3.6. The equilibrium point (2,1) of model (3) can be a 5th-order fine focus at most.

    Proof. According to Definition 2.1, we need to prove that there exits a group of real values about λ1,A10,a,b such that ˜v3=˜v5=˜v7=˜v9=0,˜v110.

    At first, we prove that ˜v3=˜v5=˜v7=˜v9=0 have real number solutions.

    Let ˜v3=0, we have

    b=245(156a+55A10+4λ1+72aλ1+24A10λ1). (27)

    By using computer soft Mathematica 6.0 to compute, ˜v3=˜v5=˜v7=˜v9=0 hold if and only if Eq (27) holds and

    {˜r57=Resultant[˜v5,˜v7,A10]=0,˜r59=Resultant[˜v5,˜v9,A10]=0. (28)

    While Eq (28) hold if and only if

    ˜r579=Resultant[˜r57,˜r59,a]=0. (29)

    By computing, we obtain

    ˜r579=Resultant[r57,r59,a]=(3107+3210λ1+720λ21)2(13964+10293λ1+1800λ21)2˜g(λ1),

    in which ˜g(λ) is a 242 degrees function on λ1. It can be seen that Eq (29) has some real solutions such as λ1=1240(535±9465) et al. Hence, ˜v3=˜v5=˜v7=˜v9=0 have real number solutions.

    In fact, we can find 15 groups of real number solutions such that ˜v3=˜v5=˜v7=˜v9=0, namely

    1)A104643.3018425501515263,a1550.8973397671423233,

    b1036.3362485482110284,λ143.0579290316570603;

    2)A100.9959258074047267,a0.0496867911910842,

    b215.6922502839121132,λ1197.5784671350668527;

    3)A1017.6221151647949041,a5.8674840306887030,

    b5.1961268584644113,λ114.0933015765297504;

    4)A1022.4133844512194665,a7.6232290779662400,

    b1.4651057037980100,λ15.1151041647768187;

    5)A106.0154299290528854,a3.1637820182235810,

    b7.3080323729222197,λ14.1189107630353050;

    6)A105.3540843597677715,a2.210452369910683,

    b1.9959489233153501,λ12.748983659910321;

    7)A103.2879140223160507,a1.636028779039390,

    b1.4395307705732038,λ12.489840572873740;

    8)A104.1024413004886361,a2.121698633718532,

    b1.3456036840775022,λ11.287647593274435;

    9)A102.7095465588712933,a1.607999835283884,

    b0.1502206861914896,λ11.921623601914863;

    10)A100.0859310460513803,a0.219273746634790,

    b1.1721730119425250,λ12.988862274797442;

    11)A100.3976086548746663,a0.119165192985334,

    b1.0985939724503711,λ12.946168239121486;

    12)A100.3522522540516692,a0.044767143636023,

    b0.9813497899428434,λ13.089686791404887;

    13)A100.5942698666771841,a0.294891555364813,

    b1.1887051809847815,λ12.669570213107159;

    14)A101.0727163914454935,a0.107785966220006,

    b1.1631832585231690,λ12.719197614164823;

    15)A101.5505080546244919,a1.101693440557772,

    b2.0337942792540713,λ13.472776931203230. .

    Next, we prove ˜v110 if ˜v3=˜v5=˜v7=˜v9=0

    Let

    ˜r1=Resultant[˜v5,˜v11,A10],˜r2=Resultant[˜v7,˜v11,A10],˜r3=Resultant[˜v9,˜v11,A10], and ˜r12=Resultant[˜r1,˜r2,a],˜r13=Resultant[˜r1,˜r3,a].

    If ˜v3=˜v5=˜v7=˜v9=0, then ˜r123=Resultant[˜r12,˜r13,λ1]=0. By computing, we obtain: ˜r123=Resultant[˜r12,˜r13,λ1]=546558871293257498128812854709220.

    Hence, ˜v110 if ˜v3=˜v5=˜v7=˜v9=0, then the equilibrium point (2,1) of model (3) can be a 5th-order fine focus at most. Proof end.

    After finding focal values of two positive equilibrium points of model (3), we will consider the limit cycle bifurcation near (1,1) and (2,1) of perturbed model (3).

    Lemma 4.1. Let J1 be the Jacobin of the function group (v3,v5,v7,v9) with respect to the variables (a,λ,A10,b), if the equilibrium (1,1) of model (3) is a 5th-order fine focus, then J10.

    Proof. Suppose that J1=0, next we deduce a contradictory result. The Jacobin of the function group (v3,v5,v7,v9) with respect to the variables (a,λ,A10,b) has the following form

    J1=|v3av3A10v3λv3bv5av5A10v5λv5bv7av7A10v7λv7bv9av9A10v9λv9b|.

    Obviously, J1 is a function with respect to the variables (a,λ,A10,b). Because the equilibrium point (1,1) of model (3) is a 5th-order fine focus, then v3=v5=v7=v9=0. Suppose that J1=0, then the resultant of J1,vi,(i{3,5,7,9}) with respect to the variable b will become 0.

    Let R1=Resultant[J1,v3,b],R2=Resultant[J1,v5,b],R3=Resultant[J1,v7,b], R4=Resultant[J1,v9,b], then Ri=0,i{1,2,3,4}.

    While Ri=0,i{1,2,3,4} will deduce that R12=Resultant[R1,R2,A10]=0, R13=Resultant[R1,R3,A10]=0, R14=Resultant[R1,R4,A10]=0. Similarly, R12=R13=R14=0 deduce that R23=Resultant[R12,R13,a]=0 and R24=Resultant[R12,R14,a]=0. In the same way, R23=R24=0 deduce that R34=Resultant[R23,R24,λ]=0. In fact, with help of computer, we obtain that

    R34=24231418616323066317496248857998624904907753623104170.

    R340 pushes J10. Proof end.

    Similarly, we can obtain the following lemma.

    Lemma 4.2. Let J2 be the Jacobin of the function group (˜v3,˜v5,˜v7,˜v9) with respect to the variables (a,λ1,A10,b), if the equilibrium point (2,1) of model (3) is a 5th-order fine focus, then J20.

    Theorem 4.1. Suppose that (1,1) is a 5-th order fine focus of system (3), then by small perturbations of the parameter group (a,λ,A10,b), the point (1,1) of perturbed model (3) can bifurcate at least 5 small amplitude limit cycles.

    Proof. From lemma 4.1, J10, while (1,1) is a 5-th order fine focus of system (3), according to the theory of reference [14], the conclusion of Theorem 4.1 holds. Proof end.

    Similarly, we have the following theorem.

    Theorem 4.2. Suppose that (2,1) is a 5-th fine focus of system (3), then by small perturbations of the parameter group (a,λ1,A10,b), the point (2,1) of perturbed model (3) can bifurcate at least 5 small amplitude limit cycles.

    Next we will give a case that (1,1) of the model (3) can bifurcate 5 limit cycles of which 3 limit cycles are stable.

    Obviously, vi,(i=3,5,7,9,11) are functions about (a,λ,A10,b). The course of Theorem 3.3 shows there exists a group solutions (a,λ,A10,b) = (˜a,˜λ,~A10,˜b) such that vi=0,i{3,5,7,9},v110. Hence, we may as well let

    v3=ϵ1,v5=ϵ2,v7=ϵ3,v9=ϵ4, (30)

    in which ϵi,i{1,2,3,4} are a group of arbitrary given real small parameters.

    According to existence theorem of implicit function and the result of Lemma 4.1, Eq (30) has a group of solutions as follows:

    a=a(ϵ1,ϵ2,ϵ3,ϵ4),λ=λ(ϵ1,ϵ2,ϵ3,ϵ4),A10=A10(ϵ1,ϵ2,ϵ3,ϵ4),b=b(ϵ1,ϵ2,ϵ3,ϵ4). (31)

    From (30) and (31), clearly the following theorem holds.

    Theorem 4.3. Suppose that (a,λ,A10,b) disturb by way of (31), then v3=ϵ1,v5=ϵ2,v7=ϵ3,v9=ϵ4, in which ϵi,i{1,2,3,4} are a group of arbitrary given real small parameters.

    Next, we let ϵi,i{1,2,3,4} be some special values, we have the following theorem.

    Theorem 4.4. Suppose that the coefficients of model (3) disturb via v3=21076c5πϵ8+o(ϵ9),v5=7645c5πϵ6+o(ϵ7),v7=1023c5πϵ4+o(ϵ5), v9=55c5πϵ2+o(ϵ3),c5=v11, then the point (1, 1) of model (3) can bifurcate 5 small limit cycles which are near to circles (x1)2+(y1)2=k2ϵ2,k=1,2,3,4,5 in which 3 limit cycles can be stable.

    Proof. Suppose that v3=21076c5πϵ8+o(ϵ9),v5=7645c5πϵ6+o(ϵ7),v7=1023c5πϵ4+o(ϵ5),v9=55c5πϵ2+o(ϵ3),c5=v11, then we have

    v1(2π,ϵ,δ)=e2πδ=1+c0πϵ10+o(ϵ11),v3(2π,ϵ,δ)=c1πϵ8+o(ϵ9),v5(2π,ϵ,δ)=c2πϵ6+o(ϵ7),v7(2π,ϵ,δ)=c3πϵ4+o(ϵ5),v9(2π,ϵ,δ)=c4πϵ2+o(ϵ3),v11(2π,ϵ,δ)=c5+o(ϵ),

    in which

    c5=v11|ϵ=0, and c0=14400c5,c1=21076c5,c2=7645c5,c3=1023c5,c4=55c5.

    At this time, according to (6), Poincarˊe succession function for the point (1, 1) of model (3) is as follows:

    d(ϵh)=r(2π,ϵh)ϵh=(v1(2π,ϵ,δ)1)ϵh+v2(2π,ϵ,δ)(ϵh)2+v3(2π,ϵ,δ)(ϵh)3++v11(2π,ϵ,δ)(ϵh)11+=πϵ11h[g(h)+ϵhG(h,ϵ)],

    in which

    g(h)=c0+c1h2+c2h4+c3h6+c4h8+j0h10=c6(h21)(h24)(h29)(h216)(h225)=14400c5+21076c5h27645c5h4+1023c5h655c5h8+c5h10,

    and G(h,ϵ) is analytic at (0,0).

    Obviously, g(h)=0 has 5 simple positive zero points 1,2,3,4,5. From implicit function theorem, the number of positive zero points of equation d(ϵh)=0 is equal to one of g(h)=0, and these positive zero points are close to 1,2,3,4,5 when 0<|ϵ|1. The above analysis shows there are 5 small limit cycles in a small enough neighborhood of (1,1) of model (3), which are near to circles (x1)2+(y1)2=k2ϵ2,k=1,2,3,4,5.

    Obviously, if v11<0, then the point (1, 1) of model (3) can bifurcate 3 stable cycles which are near circles (x1)2+(y1)2=k2ϵ2,k=1,3,5. While by analyzing the 13 groups of solutions showed in the proof course of Theorem 3.3, this kind of solutions exist such as the second group of solution. Hence, the result of Theorem 4.4 holds. Proof end.

    Remark: We can also find the solution such that v3=v5=v7=v9=0,v11>0 among the 13 groups of solutions showed in the proof course of Theorem 3.3. Hence, the point (1, 1) of model (3) can also bifurcate 2 stable cycles which are near circles (x1)2+(y1)2=k2ϵ2,k=2,4.

    Similarly, we can obtain the following theorem.

    Theorem 4.5. The point (2, 1) of model (3) can bifurcate 3 stable limit cycles under certain condition.

    The work of this paper focuses on investigating the limit cycle bifurcation of a class of the quartic Kolmogorov model, which is an interesting and significant ecological model both in theory and applications. We used the singular values method to compute focal value. First, we give the relation between focal value and singular point value at the origin, which is necessary for us to investigate bifurcations of limit cycles. Sceond, we give the singular point values' recursive formulas and respectively compute the focal values of the two positive equilibrium points of model (3) and obtain the condition that they can be two 5th-order fine focuses. Next, we discuss the bifurcation of limit cycle of model (3) and obtain that each one of the two positive equilibrium points of model (3) can have five small limit cycles. Then, we show a case that each positive equilibrium point can bifurcate 3 stable limit cycles at most by making use of algebraic and symbolic proof.

    The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

    The authors would like to thank the editor and the referees for their valuable comments and suggestions, which improved the quality of our paper. This research is partially supported by the Research Fund of Hunan provincial education department (22A719) and the National Natural Science Foundation of China (12061016).

    The authors declare that they have no competing interest.



    [1] A. Algaba, C. García, J. Giné, Nondegenerate centers and limit cycles of cubic Kolmogorov systems, Nonlinear Dyn., 91 (2018), 487–496. https://doi.org/10.1007/s11071-017-3883-5 doi: 10.1007/s11071-017-3883-5
    [2] X. Chen, J. Llibre, Z. Zhang, Suffificient conditions for the existence of at least n or exactly n limit cycles for the Liénard differential systems, J. Differ. Equations, 242 (2007), 11–23. https://doi.org/10.1016/j.jde.2007.07.004 doi: 10.1016/j.jde.2007.07.004
    [3] H. Chen, M. Han, Y. Xia, Limit cycles of a Liénard system with symmetry allowing for discontinuity, J. Math. Anal. Appl., 468 (2018), 799–816. https://doi.org/10.1016/j.jmaa.2018.08.050 doi: 10.1016/j.jmaa.2018.08.050
    [4] C. Du, W. Huang, Center-focus problem and limit cycles bifurcations for a class of cubic Kolmogorov model, Nonlinear Dyn., 72 (2013), 197–206. https://doi.org/10.1007/s11071-012-0703-9 doi: 10.1007/s11071-012-0703-9
    [5] C. Du, Y. Liu, W. Huang, Limit cycles bifurcations for a class of Kolmogorov model in symmetrical vector field, Int. J. Bifurcat. Chaos, 24 (2014), 1450040. https://doi.org/10.1142/S0218127414500400 doi: 10.1142/S0218127414500400
    [6] C. Du, Y. Liu, Q. Zhang, Limit cycles in a class of quartic Kolmogorov model with three positive equilibrium points, Int. J. Bifurcat. Chaos, 25 (2015), 1550080. https://doi.org/10.1142/S0218127415500807 doi: 10.1142/S0218127415500807
    [7] C. Du, Q. Wang, W. Huang, Three-Dimensional Hopf bifurcation for a class of cubic Kolmogorov model, Int. J. Bifurcat. Chaos, 24 (2014), 1450036. https://doi.org/10.1142/S0218127414500369 doi: 10.1142/S0218127414500369
    [8] J. Gu, A. Zegeling, W. Huang, Bifurcation of limit cycles and isochronous centers on center manifolds for a class of cubic Kolmogorov systems in R3, Qual. Theory Dyn. Syst., 22 (2023), 42. https://doi.org/10.1007/s12346-023-00745-8 doi: 10.1007/s12346-023-00745-8
    [9] D. He, W. Huang, Q. Wang, Small amplitude limit cycles and local bifurcation of critical periods for a quartic Kolmogorov system, Qual. Theory Dyn. Syst., 19 (2020), 68. https://doi.org/10.1007/s12346-020-00401-5 doi: 10.1007/s12346-020-00401-5
    [10] X. Huang, L. Zhu, Limit cycles in a general kolmogorov model, Nonlinear Anal. Theor., 60 (2005), 1394–1414. https://doi.org/10.1016/j.na.2004.11.003 doi: 10.1016/j.na.2004.11.003
    [11] M. Han, Y. Lin, P. Yu, A study on the existence of limit cycles of a planar system with 3rd-degree polynomials, Int. J. Bifurcat. Chaos, 14 (2004), 41–60. https://doi.org/10.1142/S0218127404009247 doi: 10.1142/S0218127404009247
    [12] A. Kolmogorov, Sulla teoria di Volterra della lotta per lésistenza, Giornale dell'Istituto Italiano degli Attuari, 7 (1936), 74–80.
    [13] A. Q. Khan, S. A. H. Bukhari, M. B. Almatrafi, Global dynamics, Neimark-Sacker bifurcation and hybrid control in a Leslie's prey-predator model, Alex. Eng. J., 61 (2022), 11391–11404. https://doi.org/10.1016/j.aej.2022.04.042 doi: 10.1016/j.aej.2022.04.042
    [14] A. Q. Khan, F. Nazir, M. B. Almatrafi, Bifurcation analysis of a discrete Phytoplankton CZooplankton model with linear predational response function and toxic substance distribution, Int. J. Biomath., 16 (2023), 2250095. https://doi.org/10.1142/S1793524522500954 doi: 10.1142/S1793524522500954
    [15] A. Q. Khan, M. Tasneem, M. B. Almatrafi, Discrete-time COVID-19 epidemic model with bifurcation and control, Math. Biosci. Eng., 19 (2022), 1944–1969. https://doi.org/10.3934/mbe.2022092 doi: 10.3934/mbe.2022092
    [16] Y. Liu, Theory of center-focus for a class of higher-degree critical points and infinite points, Sci. China Ser. A-Math., 44 (2001), 365–377. https://doi.org/10.1007/BF02878718 doi: 10.1007/BF02878718
    [17] Y. Liu, H. Chen, Formulas of singular point quantities and the first 10 saddle quantities for a class of cubic system, Acta Math. Appl. Sin., 25 (2002), 295–302.
    [18] Y. Liu, J. Li, Theory of values of singular point in complex autonomous differential system, Sci. China Ser. A-Math., 3 (1990), 10–24.
    [19] J. Llibre, Y. Martínez, Dynamics of a family of Lotka-Volterra systems in R3, Nonlinear Anal., 199 (2020), 111915. https://doi.org/10.1016/j.na.2020.111915 doi: 10.1016/j.na.2020.111915
    [20] J. Llibre, Y. Martínez, C. Valls, Limit cycles bifurcating of Kolmogorov systems in R2 and in R3, Commun. Nonlinear Sci., 91 (2020), 105401. https://doi.org/10.1016/j.cnsns.2020.105401 doi: 10.1016/j.cnsns.2020.105401
    [21] J. Llibre, X. Zhang, Limit cycles of the classical Liénard differential systems: A survey on the Lins Neto, de Melo and Pughs conjecture, Expo. Math., 35 (2017), 286–299. https://doi.org/10.1016/j.exmath.2016.12.001 doi: 10.1016/j.exmath.2016.12.001
    [22] N. G. Lloyd, J. M. Pearson, E. Saéz, I. Szántó, A cubic Kolmogorov system with six limit cycles, Comput. Math. Appl., 44 (2002), 445–455. https://doi.org/10.1016/S0898-1221(02)00161-X doi: 10.1016/S0898-1221(02)00161-X
    [23] Z. Lu, B. He, Multiple stable limit cycles for a cubic kolmogorov system, Chinese Journal of Engineering Mathematics, 4 (2001), 115–117.
    [24] N. G. Lloyd, J. M. Pearson, E. Saez, I. Szanto, Limit cycles of a cubic kolmogorov system, Appl. Math. Lett., 9 (1996), 15–18. https://doi.org/10.1016/0893-9659(95)00095-X doi: 10.1016/0893-9659(95)00095-X
    [25] Y. Wu, C. Zhang, Integrability and non-linearizability of weak saddles in a cubic Kolmogorov model, Chaos Soliton. Fract., 153 (2021), 111514. https://doi.org/10.1016/j.chaos.2021.111514 doi: 10.1016/j.chaos.2021.111514
    [26] Y. Yuan, H. Chen, C. Du, Y. Yuan, The limit cycles of a general Kolmogorov system, J. Math. Anal. Appl., 392 (2012), 225–237. https://doi.org/10.1016/j.jmaa.2012.02.065 doi: 10.1016/j.jmaa.2012.02.065
  • Reader Comments
  • © 2023 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)
通讯作者: 陈斌, bchen63@163.com
  • 1. 

    沈阳化工大学材料科学与工程学院 沈阳 110142

  1. 本站搜索
  2. 百度学术搜索
  3. 万方数据库搜索
  4. CNKI搜索

Metrics

Article views(1339) PDF downloads(70) Cited by(0)

Other Articles By Authors

/

DownLoad:  Full-Size Img  PowerPoint
Return
Return

Catalog