Nonlinear Dyn (2013) 72:197–206DOI 10.1007/s11071-012-0703-9

O R I G I NA L PA P E R

Center-focus problem and limit cycles bifurcationsfor a class of cubic Kolmogorov model

Chaoxiong Du · Wentao Huang

Received: 21 November 2012 / Accepted: 5 December 2012 / Published online: 29 December 2012© Springer Science+Business Media Dordrecht 2012

Abstract The problem of limit cycles for the Kol-mogorov model is interesting and significant both intheory and applications. In this paper, we investigatethe center-focus problems and limit cycles bifurca-tions for a class of cubic Kolmogorov model with threepositive equilibrium points. The sufficient and neces-sary condition that each positive equilibrium point be-comes a center is given. At the same time, we showthat each one of point (1,2) and point (2,1) can bi-furcate 1 small limit cycles under a certain condition,and 3 limit cycle can occur near (1,1) at the samestep. Among the above limit cycles, 4 limit cycles canbe stable. The limit cycles bifurcations problem forKolmogorov model with several positive equilibriumpoints are hardly seen in published references. Our re-sult is new and interesting.

Keywords Kolmogorov model · Positive equilibriumpoints · Limit cycles · Poincaré succession function ·Stable · Center problem

C. Du (�)Department of Mathematics, Hunan Shaoyang University,Shaoyang, Hunan, 422000, P.R. Chinae-mail: duchaoxiong@hotmail.com

C. Due-mail: ducx123@126.com

W. HuangSchool of Mathematics and Computational Science, GuilinUniversity of Electronic Technology, Guilin, Guangxi,541004, P.R. China

1 Introduction

In mathematical ecology, more and more mathematicworkers pay attention to the three most fundamentalsystems namely the predator–prey, the competition,and the cooperation systems. A lot of natural predator–prey systems can be discussed and investigated; [1]generalizes some studied results. Theoretically, thesesystems can be reduced to some kinds of ecologicalmodels. The following Kolmogorov models are inter-esting models among them:{

dxdt

= xf (x, y),

dydt

= yg(x, y),(1)

in which f (x, y) and g(x, y) are polynomials. SuchKolmogorov models are widely used in ecology to de-scribe the interaction between two populations. In thatcase, attention is restricted to the behavior of orbitsin the “realistic quadrant” {(x, y) : x > 0, y > 0}. Ofparticular significance in applications is the existenceof limit cycles and the number of limit cycles thatcan arise from positive equilibrium points, because alimit cycle corresponds to an equilibrium state of thesystem, while the existence and stability of limit cy-cles is related to the positive equilibrium points. Atthe same time, the problem on the number of limitcycles gets in close touch with famous Hilbert 16thproblem; Hence, many articles studying Kolmogorovmodels pay more attention to the limit cycles bifurca-tion problem. For example, article [2] studied a class

198 C. Du, W. Wentao

of cubic Kolmogorov systems with three limit cycles;[3] showed this class of the Kolmogorov system couldbifurcate five limit cycles including 3 stable cycles; [4]showed a class of a cubic Kolmogorov system couldbifurcate six limit cycles; [5, 6] studied a general Kol-mogorov model and obtained the conditions for theexistence and uniqueness of limit cycles, at the sametime it classified a series of models. As far as limitcycles of Kolmogorov models are concerned, manygood results have been obtained by analyzing the solepositive equilibrium point’s state. But results on limitcycles bifurcation from several different equilibriumpoints are hardly seen; perhaps it is difficult to investi-gate this kind of problem.

In this paper, we study a class of the following cu-bic Kolmogorov models:

{dxdt

= x(y − 1)[a + bx − 12 (a + b)y] ≡ P(x, y),

dydt

= y(x − 1)[c + dy − 12 (c + d)x] ≡ Q(x,y),

(2)

in which a, b, c, d ∈ R.Clearly, model (2) has three positive equilibrium

points namely (1,2), (2,1), and (1,1). We will focuson the center-focus problems and limit cycles bifur-cations of the three positive equilibrium points. Withthe help of the computer and by carefully calculating,we obtain the expression of focal values for each pos-itive equilibrium point. By analyzing the structure offocal values, we give the sufficient and necessary con-dition that three positive equilibrium points becomethree centers. At the same time, we show that each oneof positive equilibrium points (1,2) and (2,1) can bi-furcate 1 limit cycles under a certain condition whichcan be stable, in addition, positive equilibrium point(1,1) can bifurcate 3 limit cycle of which 2 cyclescan be stable. Our results are concise (especially in theexpressions of focal values) and the proof about exis-tence of limit cycles is algebraic and symbolic. It isworth pointing out the fact that our work (bifurcationsof the Kolmogorov model with three positive equilib-rium points) is new and interesting.

This paper concludes 3 sections. In Sect. 2, we dis-cuss the each positive equilibrium point’s type and ob-tain the conditions of the center-focus problems. InSect. 3, we respectively compute the focal values ofthe three positive equilibrium points of model (2) andobtain the sufficient and necessary condition that they

become center. Moreover, we discuss the bifurcationsof limit cycles of model (2) and obtain that each oneof the two positive equilibrium points (1,2) and (2,1)

of model (2) can arise 1 small limit cycle and the pos-itive equilibrium points (1,1) can arise 3 small limitcycles under a certain condition. In summation, model(2) can bifurcate 5 small limit cycles from three posi-tive equilibrium points of which 4 limit cycles can bestable. In terms of the number of stable limit cyclesand the results about limit cycles bifurcations for sev-eral positive equilibrium points, our results are new.

2 Positive equilibrium point’s property

For model (2), the linearized system’s coefficient ma-trix in point (x0, y0) is as follows:

A(x0,y0) =⎡⎣ ∂P (x,y)

∂x∂P (x,y)

∂y

∂Q(x,y)∂x

∂Q(x,y)∂y

⎤⎦

(x0,y0)

, (3)

in which

∂P (x, y)

∂x= 1

2(y − 1)(2a + 4bx − ay − by),

∂P (x, y)

∂y= 1

2x(3a + b + 2bx − 2ay − 2by),

∂Q(x, y)

∂x= 1

2y(3c + d − 2cx − 2dx + 2dy),

∂Q(x, y)

∂y= 1

2(x − 1)(2c − cx − dx + 4dy).

Next, we will investigate the nature of the three pos-itive equilibrium points one by one.

2.1 The type of positive equilibrium point (1,1)

For model (2), the linearized system’s coefficient ma-trix in point (1,1) is as follows:

A(1,1) =[

0 a+b2

c+d2 0

]. (4)

The characteristic equation of matrix A(1,1) is as fol-lows:

λ2 − 1

4(a + b)(c + d) = 0. (5)

Center-focus problem and limit cycles bifurcations for a class of cubic Kolmogorov model 199

(1) Suppose that (a + b)(c + d) > 0, then Eq. (4)has two real eigenvalues with the opposite sign,i.e.,

λ1 = 1

2

√(a + b)(c + d),

λ2 = −1

2

√(a + b)(c + d).

It can be seen that positive equilibrium point (1,1)

is a saddle point. Hence, the positive equilibriumpoint (1,1) is unstable.

(2) Suppose that (a + b)(c + d) = 0, then Eq. (4)has iterative eigenvalues 0, and the positive equi-librium point (1,1) is a degenerate node point.Hence, the positive equilibrium point (1,1) is un-stable.

(3) Suppose that (a + b)(c + d) < 0, then Eq. (4)has two conjugate pure imaginary eigenvalues,i.e.,

λ1 = 1

2i√−(a + b)(c + d),

λ2 = −1

2i√−(a + b)(c + d).

At this time, the positive equilibrium point (1,1)

of linearized system of model (2) is a center.Hence, the positive equilibrium point (1,1) ofmodel (2) will perhaps become a center or a fo-cus. Then the study on (1,1) belongs to center-focus problem, we will study this problem in nextsection.

2.2 The type of positive equilibrium point (1,2)

For model (2), the linearized system’s coefficient ma-trix in point (1,2) is as follows:

A(1,2) =[

b − a+b2

c + 3d 0

]. (6)

The characteristic equation of matrix A(1,2) is as fol-lows:

λ(λ − b) + 1

2(a + b)(c + 3d) = 0. (7)

(1) Suppose that b2 − 2(a + b)(c + 3d) > 0, thenEq. (7) has two different real eigenvalues, i.e.,

λ1 = 1

2b − 1

2

√b2 − 2(a + b)(c + 3d),

λ2 = 1

2b + 1

2

√b2 − 2(a + b)(c + 3d).

(8)

By analyzing the expression of (8), we have thefollowing results:

(1a) In case b > 0, then (1,2) is a unstable nodepoint.

(1b) In case b < 0, then (1,2) is a stable nodepoint.

(1c) In case b = 0, then (1,2) is a saddle point.

(2) Suppose that b2 − 2(a + b)(c + 3d) = 0, thenEq. (7) has two iterative real eigenvalues λ1,2 =12b, and the positive equilibrium point (1,2) is adegenerate node point. It is stable as follows:

(2a) In case b ≥ 0, then (1,2) is a unstable degen-erate node point.

(2b) In case b < 0, then (1,2) is a stable degener-ate node point.

(3) Suppose that b2 − 2(a + b)(c + 3d) < 0, thenEq. (7) has two conjugate complex eigenvalues,i.e.,

λ1 = 1

2b − 1

2i√

−b2 + 2(a + b)(c + 3d),

λ2 = 1

2b + 1

2i√

−b2 + 2(a + b)(c + 3d).

(9)

By analyzing the expression of (9), we have thefollowing results:

(3a) In case b > 0, then (1,2) is a unstable focus.(3b) In case b < 0, then (1,2) is a stable focus.(3c) In case b = 0, then the positive equilibrium

point (1,2) of linearized system of model (2) isa center. Hence, the positive equilibrium point(1,2) of model (2) will perhaps become a cen-ter or a focus. Then the study on (1,2) belongsto center-focus problem under this condition; wewill study this problem in next section.

200 C. Du, W. Wentao

2.3 The type of positive equilibrium point (2,1)

For model (2), the linearized system’s coefficient ma-trix in point (2,1) is as follows:

A(2,1) =[

0 a + 3b

− 12 (c + d) d

]. (10)

The characteristic equation of matrix A(2,1) is as fol-lows:

λ(λ − d) + 1

2(c + d)(a + 3b) = 0. (11)

(1) Suppose that d2 − 2(c + d)(a + 3b) > 0, thenEq. (11) has two different real eigenvalues, i.e.,

λ1 = 1

2d − 1

2

√d2 − 2(c + d)(a + 3b),

λ2 = 1

2d + 1

2

√d2 − 2(c + d)(a + 3b).

(12)

By analyzing the expression of (12), we have thefollowing results:

(1a) In case d > 0, then (2,1) is a unstable nodepoint.

(1b) In case d < 0, then (2,1) is a stable nodepoint.

(1c) In case d = 0, then (2,1) is a saddle point.

(2) Suppose that d2 − 2(c + d)(a + 3b) = 0, thenEq. (11) has two iterative real eigenvalues λ1,2 =12d , and the positive equilibrium point (2,1) is adegenerate node point. Its stable is as follows:

(2a) In case d ≥ 0, then (2,1) is a unstable degen-erate node point.

(2b) In case d < 0, then (2,1) is a stable degener-ate node point.

(3) Suppose that d2 −2(c+d)(a +3b) < 0, then (11)has two conjugate complex eigenvalues as fol-lows:

λ1 = 1

2d − 1

2i√

−d2 + 2(c + d)(a + 3b),

λ2 = 1

2d + 1

2i√

−d2 + 2(c + d)(a + 3b).

(13)

By analyzing the expression of (13), we have thefollowing results:

(3a) In case d > 0, then (2,1) is a unstable focus.(3b) In case d < 0, then (2,1) is a stable focus.(3c) In case d = 0, then the positive equilibrium

point (2,1) of linearized system of model (2) isa center. Hence, the positive equilibrium point(2,1) of model (2) will perhaps become a cen-ter or a focus. Then the study on (2,1) belongsto center-focus problem under this condition; wewill study this problem in next section.

3 Center-focus problem and limit cyclesbifurcations of model (2)

For model (2), if the positive equilibrium point (x0, y0)

is a center of linearized system of model (2), then thispoint (x0, y0) may be a center or a focus of model (2).This problem is called the center-focus problem. Andif (x0, y0) is a fine focus, it will occur several limitcycles in the disturbed model (2). Next, we study thecenter-focus problem and limit cycles bifurcations ofmodel (2) by computing and analyzing Lyapunov con-stants (or focal values).

3.1 Center-focus problem and limit cycles bifurcationof the positive equilibrium point (1,1)

Suppose that the positive equilibrium point (1,1) isa center of linearized system of model (2), it has beenobtained that (a+b)(c+d) < 0; the eigenvalues of thelinearized system’s coefficient matrix in point (1,1)

for model (2) is

λ1,2 = ±1

2i√−(a + b)(c + d).

By means of transformation

u = x − 1, v = y − 1, (14)

model (2) is changed into the following form:⎧⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎩

dudt

= 12 (a + b)v + 1

2 (a + 3b)uv + bu2v

− 12 (a + b)v2 − 1

2 (a + b)uv2,

dvdt

= 12 (c + d)u + 1

2 (c + 3d)uv + duv2

− 12 (c + d)u2 − 1

2 (c + d)u2v.

(15)

Center-focus problem and limit cycles bifurcations for a class of cubic Kolmogorov model 201

Clearly, the equilibrium (1,1) of model (2) becomes

the origin of (15) correspondingly.

Under the transformation,

ξ = u

√− c + d

a + b− v

√−a + b

c + d, η = u + v,

dτ = 1

2

√−(a + b)(c + d)dt,

(16)

system (15) becomes

⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩

dξdτ

= −η + a20ξ2 + a11ξη + a02η

2 + a30ξ3

+ a21ξ2η + a12ξη2 + a03η

3,

dηdτ

= ξ + b20ξ2 + b11ξη + b02η

2 + b30ξ3

+ b21ξ2η + b12ξη2 + b03η

3,

(17)

in which

a30 = 2(a + b)(ad − bc)

(a + b − c − d)3

√− c + d

a + b,

a21 = a2c − 2abc − 3b2c − ac2 − 3bc2 + 3a2d + 2abd − b2d + 2acd − 2bcd + 3ad2 + bd2

(a + b − c − d)3,

a12 = −(a + b)(a2 − b2 − 4bc − c2 + 4ad + d2)

(a + b − c − d)3

√− c + d

a + b,

a03 = −(a + b)(a − b − c + d)(c + d)

(a + b − c − d)3,

a20 = −2(2ac + 3bc + 3ad + 4bd)

(a + b − c − d)2,

a11 = −2(a + b + c + d)(2ac + 3bc + 3ad + 4bd)

(a + b)(a + b − c − d)2

√−a + b

c + d,

a02 = (a + b)2 + 2ac + 4bc + c2 + 4ad + 6bd + 2cd + d2

(a + b − c − d)2,

b30 = − (a + b)(a + 3b − c − 3d)(c + d)

(a + b − c − d)3,

b21 = (a + b)(a2 + 6ab + 5b2 + 2bc − c2 − 2ad − 6cd − 5d2)

(a + b − c − d)3

√− c + d

a + b,

b12 = 2b(a + b)2 + ac(a − c) + 6ad(c + d) + 5b2(c + d) − c2(b + 2d) + (a2 − 2d2)d − (a + b)(6cd + 5d2)

(a + b − c − d)3,

b03 = −2(a + b)(ab + b2 − cd − d2)

(a + b − c − d)3

√− c + d

a + b,

b20 = 2(a + b)(a + 2b + c + 2d)

(a + b − c − d)2

√− c + d

a + b,

b11 = a2 + 4ab + 3b2 + 6ac + 8bc + c2 + 8ad + 10bd + 4cd + 3d2

(a + b − c − d)2,

b02 = −2(a + b)(a + 2b + c + 2d)

(a + b − c − d)2

√− c + d

a + b.

202 C. Du, W. Wentao

According to the method in article [7, 8] to computethe Lyapunov constants (or called focal values) of theorigin of system (17), we obtain the Lyapunov con-stants (or focal values) of the origin of (17) (namelythe focal values of the equilibrium (1,1) of model (2))as follows.

Theorem 3.1 The first three Lyapunov constants (orfocal values) of the origin of (17) (or the positive equi-librium point (1,1) of (2)) are as follows:

V3 = πbc − ad

2(a + b)(a + b − c − d)

√−a + b

c + d,

V5 = πabc(a + 3b)(7a + 5b)(a + c)

12(a + b)3(a − c)2

√− 1

ac,

(1) In case 7a + 5b = 0, then

V7 = −π77c(a + c)

2(a − c)2

√−a

c,

(2) In case a + 3b = 0, then

V7 = 0.

Among the above expressions of V2k+1, k = 5,7, wehave already let V2k−1 = 0.

Considering (a + b)(c + d) < 0 and Theorem 3.1,we have the following two theorems.

Theorem 3.2 The first three Lyapunov constants (orfocal values) of the origin of (17) vanish if and only ifone of the following two conditions holds:

C1: d = − c

3, b = −a

3;

C2: d = 7a

5, b = −7a

5, c = −a.

Theorem 3.3 The origin of (17) (or the positive equi-librium point (1,1) of (2)) becomes a fine focus of thirdorder if and only if

d = −7c

5, b = −7a

5, c �= −a.

From Theorem 3.2, we can obtain the followingtheorems.

Theorem 3.4 The origin of (17) (or the positive equi-librium point (1,1) of (2)) is a center if and only if oneof the conditions of Theorem 3.2 holds.

Proof At first, we prove the necessary condition.In case the origin of (17) (or the positive equilib-

rium point (1,1) of (2)) is a center, then its all Lya-punov constants vanish, so one of the conditions ofTheorem 3.2 holds.

Next, we prove the sufficient condition.Suppose that C1 holds, then system (15) becomes{

dudt

= − 13a(u + 1)v(u + v − 1),

dvdt

= − 13c(v + 1)u(u + v − 1).

(18)

System (18) has a first integral

f1 = cu − av − c ln |u + 1| + a ln |v + 1|.Hence, if C1 holds, then the origin of (17) (or the pos-itive equilibrium point (1,1) of (2)) is a center.

Suppose that C2 holds, then system (15) becomes{dudt

= − 15a(u + 1)v(7u − v + 1),

dvdt

= 15a(v + 1)u(7v − u + 1).

(19)

Under the transformation,

u1 = v, v1 = u, t1 = −t,

the form of system (19) is invariant, so the origin of(17) is a center; moreover, the positive equilibriumpoint (1,1) of (2) is a center. Proof end. �

According to the relation between Lyapunov con-stants (or focal values) and limit cycles bifurcations inarticle [8], we have the following theorem.

Theorem 3.5 Suppose that the origin of (17) (or thepositive equilibrium point (1,1) of (2)) is a fine focusof 3 order, the positive equilibrium point (1,1) of (2)can bifurcate three limit cycles. Two limit cycles arestable if V7 < 0, one limit cycle is stable if V7 > 0.

3.2 Center-focus problem and limit cycles bifurcationof the positive equilibrium point (1,2)

Suppose that the positive equilibrium point (1,2) isa center of the linearized system of model (2); it hasbeen obtained that b = 0, and a(c + 3d) > 0. Theeigenvalues of the linearized system’s coefficient ma-trix in point (1,2) for model (2) is

λ1,2 = ±√

2

2i√

a(c + 3d).

Center-focus problem and limit cycles bifurcations for a class of cubic Kolmogorov model 203

By means of transformation

u = x − 1, v = y − 2, (20)

model (2) is changed into the following form:

⎧⎪⎨⎪⎩

dudt

= − 12av − 1

2av(u + v + uv),

dvdt

= (c + 3d)u − 12u[2(c + d)u − (c + 7d)v

+ (c + d)uv − 2dv2].(21)

Clearly, the equilibrium (1,2) of model (2) becomesthe origin of (21) correspondingly.

Under the transformation

u = ξ + η

2, v = − 1√

2(ξ − η)

√c + 3d

a,

dτ = 1√2

√a(c + 3d)dt,

(22)

system (21) becomes

⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩

dξdτ

= −η + a20ξ2 + a11ξη + a02η

2 + a30ξ3

+ a21ξ2η + a12ξη2 + a03η

3,

dηdτ

= ξ + b20ξ2 + b11ξη + b02η

2 + b30ξ3

+ b21ξ2η + b12ξη2 + b03η

3,

(23)

in which

a30 = −4cd + 12d2 + (3c + 7d)√

2a(c + 3d)

16a(c + 3d),

a21 = 4cd + 12d2 + (c + 5d)√

2a(c + 3d)

16a(c + 3d),

a12 = −a30, a03 = −a21,

a20 =4c2 + 20cd + 24d2 − (c + 3d)(c − d)

√2(c + 3d)/a

8(c + 3d)2,

a11 = c + d + (c + 3d)√

2(c + 3d)/a

2(c + 3d),

a02 =

− 12c2 + 4cd + 24d2 + (c + 3d)(3c + 13d)√

2(c + 3d)/a

8(c + 3d)2,

b30 = 4cd + 12d2 − (c + 5d)√

2a(c + 3d)

16a(c + 3d),

b21 = −4cd − 12d2 + (3c + 7d)√

2a(c + 3d)

16a(c + 3d),

b12 = −b30, b03 = −b21,

b20 = 12d2 + 4cd + (3c + 13d)(c − d)√

2(c + 3d)/a

8(c + 3d)2,

b11 = − c + d − (c + 3d)√

2(c + 3d)/a

2(c + 3d),

b02 =

− 4c2 + 24d2 + 20cd + (c + 3d)(c − d)√

2(c + 3d)/a

8(c + 3d)2.

According to the method in article [7, 8] to computethe Lyapunov constants (or called focal values) of theorigin of system (17), we obtain the Lyapunov con-stants (or focal values) of the origin of (23) (namelythe focal values of the equilibrium (1,2) of model (2))as follows.

Theorem 3.6 The first Lyapunov constant (or focalvalue) of the origin of (23) (or the positive equilibriumpoint (1,2) of (2)) is as follows:

V 3 = d(c + d)π

2√

2(c + 3d)√

a(c + 3d),

in which a(c + 3d) > 0.

From Theorem 3.6, we have the following two the-orems.

Theorem 3.7 The first Lyapunov constant (or focalvalue) of the origin of (23) vanishes if and only if oneof the following two conditions holds:

C3: d = 0;C4: d = −c.

Theorem 3.8 The origin of (23) (or the positive equi-librium point (1,2) of (2)) becomes a fine focus of oneorder if and only if d(c + d) �= 0.

From Theorem 3.7, we can obtain the followingtheorems.

Theorem 3.9 The origin of (21) (or the positive equi-librium point (1,2) of (2)) is a center if and only if oneof the conditions of Theorem 3.7 holds.

Proof At first, we prove the necessary condition.

204 C. Du, W. Wentao

In case the origin of (23) (or the positive equilib-rium point (1,2) of (2)) is a center, then its all Lya-punov constants vanish, so one of the conditions ofTheorem 3.7 holds.

Next, we prove the sufficient condition.Suppose that C3 holds, then system (21) becomes

{dudt

= − 12a(u + 1)v(v + 1),

dvdt

= − 12c(v + 2)u(u − 1).

(24)

System (24) has a first integral

f2 = 3c(u + 1) − 1

2c(u + 1)2 − 2c ln |u + 1|

− 3a(v + 2) + 1

2a(v + 2)2 + 2a ln |v + 2|.

Hence, if C3 holds, then the origin of (21) (or the pos-itive equilibrium point (1,2) of (2)) is a center.

Suppose that C4 holds, then system (21) becomes

{dudt

= − 12a(u + 1)v(v + 1),

dvdt

= −c(v + 2)u(v + 1).(25)

System (25) has a first integral

f3 = a(v+2)−2a ln |v+2|−2c(u+1)+2c ln |u+1|.

Hence, if C4 holds, then the origin of (21) (or the pos-itive equilibrium point (1,2) of (2)) is a center. Proofend. �

According to the relation between Lyapunov con-stants (or focal values) and limit cycles bifurcations inarticle [8], we have the following theorem.

Theorem 3.10 Suppose that the origin of (23) (or thepositive equilibrium point (1,2) of (2)) is a fine focusof 1 order, the positive equilibrium point (1,2) of (2)can bifurcate 1 limit cycles, and it is stable if V 3 < 0,and it is unstable if V 3 > 0.

3.3 Center-focus problem and limit cycles bifurcationof the positive equilibrium point (2,1)

Suppose that the positive equilibrium point (2,1) is acenter of linearized system of model (2); it has beenobtained that d = 0 and c(a + 3b) > 0. The eigen-values of the linearized system’s coefficient matrix in

point (2,1) for model (2) is

λ1,2 = ±1

2i√

c(a + 3b).

By means of transformation,

u = x − 2, v = y − 1, (26)

model (2) is changed into the following form:{dudt

= 12 (u + 2)v(a + 3b + 2bu − av − bv),

dvdt

= − 12cu(u + 1)(v + 1).

(27)

Clearly, the equilibrium (2,1) of model (2) becomesthe origin of (27) correspondingly.

Under the transformation,

u = ξ + η

2, v = −

√2

4

√c

a + 3b(ξ − η),

dτ =√

c(a + 3b)√2

dt,

(28)

system (27) becomes

⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩

dξdτ

= −η + a20ξ2 + a11ξη + a02η

2 + a30ξ3

+ a21ξ2η + a12ξη2 + a03η

3,

dηdτ

= ξ + b20ξ2 + b11ξη + b02η

2 + b30ξ3

+ b21ξ2η + b12ξη2 + b03η

3,

(29)

in which

a30 = 2√

2b(a + 3b) − (3a + 7b)√

c(a + 3b)

16√

2(a + 3b)2,

a21 = 2√

2b(a + 3b) − (3a + 5b)√

c(a + 3b)

16√

2(a + 3b)2,

a12 = −a30, a03 = −a21,

a20 =√

2(a2 + 2ab − 3b2) + (4a + 8b)√

c(a + 3b)

8√

2(a + 3b)2,

a11 = −√

2(a2 + 6ab + 9b2) − (a + b)√

c(a + 3b)

2√

2(a + 3b)2,

a02 = −√

2(3a2 + 22ab + 39b2) − 4b√

c(a + 3b)

8√

2(a + 3b)2,

b30 =√

2(2ab + 6b2) + (a + 5b)√

c(a + 3b)

16√

2(a + 3b)2,

Center-focus problem and limit cycles bifurcations for a class of cubic Kolmogorov model 205

b21 =√

2(2ab + 6b2) + (3a + 7b)√

c(a + 3b)

8√

2(a + 3b)2,

b12 = −b30, b03 = −b21,

b20 =√

2(3a2 + 22ab + 39b2) + 4b√

c(a + 3b)

8√

2(a + 3b)2,

b11 =√

2(a2 + 6ab + 9b2) + (a + b)√

c(a + 3b)

2√

2(a + 3b)2,

b02 =√

2(a2 + 2ab − 3b2) − (4a + 8b)√

c(a + 3b)

8√

2(a + 3b)2.

According to the method in article [7, 8] to computethe Lyapunov constants (or called focal values) of theorigin of system (17), we obtain the Lyapunov con-stants (or focal values) of the origin of (29) (namelythe focal values of the equilibrium (2,1) of model (2))as follows.

Theorem 3.11 The first Lyapunov constant (or focalvalue) of the origin of (29) (or the positive equilibriumpoint (2,1) of (2)) is as follows:

V3 = b(a + b)π

4√

2(a + 3b)2√

ca+3b

,

in which c(a + 3b) > 0.

From Theorem 3.11, we have the following twotheorems.

Theorem 3.12 The first Lyapunov constant (or focalvalue) of the origin of (29) vanishes if and only if oneof the following two conditions holds:

C5: b = 0;C6: b = −a.

Theorem 3.13 The origin of (29) (or the positive equi-librium point (2,1) of (2)) becomes a fine focus of oneorder if and only if b(a + b) �= 0.

From Theorem 3.12, we can obtain the followingtheorems.

Theorem 3.14 The origin of (27) (or the positive equi-librium point (2,1) of (2)) is a center if and only if oneof the conditions of Theorem 3.12 holds.

Proof At first we prove the necessary condition.In case the origin of (27) (or the positive equilib-

rium point (2,1) of (2)) is a center, then its all Lya-punov constants vanish, so one of the conditions ofTheorem 3.12 holds.

Next we prove the sufficient condition.Suppose that C5 holds, then system (27) becomes{

dudt

= − 12a(u + 2)v(v − 1),

dvdt

= − 12c(v + 1)u(u + 1).

(30)

System (30) has a first integral

f4 = 3c(u + 2) − 1

2c(u + 2)2 − 2c ln |u + 2|

− 3a(v + 1) + 1

2a(v + 1)2 + 2a ln |v + 1|.

Hence, if C5 holds, then the origin of (27) (or the pos-itive equilibrium point (2,1) of (2)) is a center.

Suppose that C6 holds, then system (27) becomes{dudt

= −a(u + 1)(u + 2)v,

dvdt

= − 12c(v + 1)u(u + 1).

(31)

System (31) has a first integral

f5 = 4a2(v + 1 + ln |v|) − c(u + 2 − 2 ln |u|).

Hence, if C6 holds, then the origin of (27) (or the pos-itive equilibrium point (2,1) of (2)) is a center. Proofend. �

According to the relation between Lyapunov con-stants (or focal values) and limit cycles bifurcations inarticle [8], we have the following theorem.

Theorem 3.15 Suppose that the origin of (29) (or thepositive equilibrium point (2,1) of (2)) is a fine focusof 1 order, the positive equilibrium point (1,2), of (2)can bifurcate one limit cycles, and it is stable if V3 < 0,it is unstable if V3 > 0.

3.4 The number of limit cycles and the number ofstable limit cycles for disturbed model (2)

By the above analysis about the Lyapunov constantsof three positive equilibrium points and discussion ontheir limit cycles bifurcations behavior, we have thefollowing theorem.

206 C. Du, W. Wentao

Theorem 3.16 Disturbed model (2) can bifurcate5 small limit cycles from three positive equilibriumpoints at the same step, of which 4 limit cycles can bestable.

Proof According to Theorem 3.1, Theorem 3.6, andTheorem 3.11, respectively, the three positive equilib-rium points of model (2) can be fine focus of threeorder, fine focus of one order and fine focus of one or-der at the same time, hence disturbed model (2) canbifurcate 5 small limit cycles from three positive equi-librium points at the same step. By analyzing the factthat three positive equilibrium points of model (2) be-come fine focus of three order, fine focus of one orderand fine focus of one order at the same time, we have

ac < 0, a(c + 3d) > 0, c(a + 3b) > 0. (32)

Next, we discuss the number of stable cycles by ana-lyzing (32). According to (32), it must be investigatedas the following two cases.

(1) Suppose that a > 0, from (32), we have

c < 0, c + 3d > 0, a + 3b < 0,

which deduce

c < 0, d > 0, b < 0.

By analyzing the Lyapunov constants of the three pos-itive equilibrium points of model (2), and studying thesign of the following expressions:

V7 = −π77c(a + c)

2(a − c)2

√−a

c,

V 3 = d(c + d)π

2√

2(c + 3d)√

a(c + 3d),

V3 = b(a + b)π

4√

2(a + 3b)2√

ca+3b

,

we have V7 < 0,V 3 < 0, V3 < 0 if the following con-ditions hold:

a + c < 0, c + d < 0, a + b > 0. (33)

Obviously, when the conditions (32) and (33) hold,model (2) can bifurcate 4 stable limit cycles of which2 stable cycles are near (1,1) and other two stable cy-cles are near (1,2) and (2,1), respectively.

(2) Suppose that a < 0, similarly, we have V7 < 0,

V 3 < 0, V3 < 0 if the following conditions hold:

a + c > 0, c + d < 0, a + b < 0. (34)

Moreover, when the conditions (32) and (34) hold,model (2) can also bifurcate 4 stable limit cycles ofwhich 2 stable cycles are near (1,1) and other two sta-ble cycles are near (1,2) and (2,1), respectively. Proofend. �

Acknowledgements This research is partially supported bythe National Natural Science Foundation of China (11261013,11071222) and the Hunan provincial Natural Science Founda-tion of China (12JJ3008), and the Research Fund of Hunanprovincial education department (11B113).

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